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G1BIQM/G13QTI Introduction to Quantum Mechanics. 1 The road to quantum mechanics Modern quantum mechanics is the culmination of a series of experimental discoveries and partially successful theories which arose in the period 1900-1926. When the catharsis eventually came in 1926, an elegant self-consistent picture emerged, but in the period leading up to it progress was haphazard and the participants were often confused and frustrated. A complete and historically faithful account of this period, with all of its confusion and half-understood theories, is not the most helpful way of understanding the foundations of quantum mechanics, at least at a ﬁrst pass.1 We will therefore restrict our attention to a somewhat selective account of the most important discoveries, capturing the essential elements necessary for an understanding of the basis of quantum theory but passing over any technical details that are not strictly necessary. Instead of following these developments in strict chronological order, it is helpful to separate the main elements into two logical components, • particle-wave duality and • Bohr’s theory of Hydrogen (or “Old Quantum Mechanics”). o Particle-wave duality is the essential element we will need to “discover” Schr¨dinger’s equation, which can be seen as the root of modern quantum mechanics. This discovery is not a derivation however (and cannot be, since quantum mechanics is a fundamentally new theory). Its veracity is checked by seeing how well it describes real-world observations, and this is where Bohr’s theory will play its role in our logical development. Bohr had produced in 1913 a set of ad o hoc rules which could explain some atomic spectra (and not others). Schr¨dinger’s equation is convincing in large part because Bohr’s theory emerges as a natural result of it. Even better, problems which are impervious to Bohr’s incomplete theory become treatable and modern quantum mechanics emerges as a complete and self-consistent theory of matter. 1.1 A new fundamental constant The seed for quantum mechanics is conventionally held to be the theoretical explanation by Planck in 1900 of blackbody radiation. This concerns measurements of the frequency- distribution of light radiated from a hot object or from within a hot cavity. In qualitative terms, the theory endeavours to explain why a hot object radiates heat in the form of infrared 1 This is not to say that the topic is to be avoided. This is a fascinating episode in the history of science and is well worth investigating, if only at a recreational level. Two books in particular can be recommended which give very readable accounts of this period. One is the biography of Niels Bohr by Abraham Pais, entitled Niels Bohr’s Times: in Physics, Philosophy and Polity. The other is The Making of the Atomic Bomb by Richard Rhodes. A copy each of these can be found in the George Green library and they are highly recommended as bed-time reading. 1 radiation, something really hot glows red so that visible light is radiated and something really really hot radiates light of all colours and appears “white hot”. Theoretical predictions based on the existing theory of electromagnetism had produced nonsensical results. They had, for example, predicted an “ultraviolet catastrophe,” whereby the light intensity decayed so slowly with increasing frequency that the total predicted energy of the radiated light was inﬁnite. Un- surprisingly the theory failed to agree with experiment at high frequencies, though agreement at lower frequencies was good. To understand what Planck did to get around this, we need to recall a couple of elements of electromagnetic theory. The classical theory of light holds that it is a wave-ﬁeld describing oscillations in the electric and magnetic ﬁelds. The simplest plane-wave solutions are of the form E(x, t) = Re E0 ei(k·x−ωt) B(x, t) = Re B0 ei(k·x−ωt) (1) in which the wavevector is in the direction of propagation of the wave and is related to the wavelength through |k| = 2π/λ. For the record, recall also that the circular frequency ω is related to the ordinary frequency ν through ω = 2πν. In classical theory, the energy per unit volume of such a ﬁeld is of the form energy / = const. × amplitude2 (2) unit volume where the constants depend on the units being used and need not concern us. The important point is that we can set up wave ﬁelds with arbitrarily small energies simply by having arbitrarily small wave amplitudes. Planck’s contribution is to suggest a radical departure from this classical picture whose sole justiﬁcation is initially that it leads to a prediction for blackbody radiation consistent with experiment. His assertion is that, in blackbody radiation, there is a fundamental “quantum” of energy for any given frequency given by E = hν (3) and that, eﬀectively, the total energy in any given volume must be an integer multiple of this. The quantity h here is a new fundamental constant, now referred to as Planck’s constant. At very high frequencies the electromagnetic ﬁeld is then faced with the option of having a relatively large energy or having none at all, and is forced to opt for the latter. Eﬀectively then, the high frequencies are cut oﬀ and the ultraviolet catastrophe is avoided. A detailed description of Planck’s analysis would take us too far oﬀ course and we will not pursue it here. Instead let us note it as the ﬁrst hint of a new theory and jump to 1905, when Einstein used Planck’s formula to explain the so-called photoelectric eﬀect and in so doing oﬀered what is eﬀectively the modern interpretation of E = hν. 1.2 Photons In the photoelectric eﬀect, electrons escape from the surface of a metal after absorbing energy from light shone on it. In order to escape the attraction of the surface, an electron has to acquire 2 an energy W which is called the work function and varies from metal to metal. Once again, the classical formalism fails to explain experimental observations. A particular example is the fact that if the frequency is too low, no electrons escape no matter how intense the light is — in classical theory we could make light of any frequency suﬃciently energetic to eject electrons simply by making it more intense. Einstein’s assumptions were that light consisted of discrete particle-like packets of energy called photons and that each photon carried an energy E = hν. Photoelectric emission occurred when a photon collided with an electron and transferred all of its energy to it. The electron can then escape with a kinetic energy Eesc = hν − W. Note in particular that the electron cannot escape if hν < W . A complete explanation of the photoelectric eﬀect then becomes possible on the basis of this formula. Figure 1: The intensity of classical wave ﬁeld is interpreted in quantum mechanics as a measure of the density of photons. We have therefore arrived at the ﬁrst element in particle-wave duality. Light, previously, thought of as an entirely wave-like phenomenon, can display particle-like behaviour. These seem like entirely diﬀerent ideas, so how can they be consistent with one another, and in particular how can (2) be compatible with (3)? The following is the modern interpretation. Imagine a pulse of light, with a reasonably well-deﬁned frequency and wavelength, as illustrated in Fig. 1. Suppose now that we take a small region and blow it up as in the ﬁgure. What the classical ﬁeld represents is a course-grained or smoothened picture of what is actually a swarm of photons, all moving in more or less the same direction. The energy density is then energy / # of photons = hν × . unit volume / unit volume Notice that this can be made consistent with (2) if we suppose that the number of photons per unit volume takes a similar form, # of photons = const. × amplitude2 (4) / unit volume 3 (with diﬀerent constants of course). On macroscopic length scales or for high intensities the number of photons in a typical region is quite large and the discreteness of the photons is unim- portant in much the same way that molecules are unimportant in traditional ﬂuid mechanics — a purely classical course-grained theory then suﬃces. On microscopic scales however, such as correspond to the absorption of a photon by an electron, the particle nature of light becomes important — this is the domain of quantum mechanics. Let us anticipate somewhat and note what the ﬁnal interpretation of (4) will be. We can think of the quantum-mechanical limit as corresponding to the case where the photons are sparse. We can still retain (4) in that case, but we must interpret the left-hand side as the probability of ﬁnding a photon in a given region (which might be small). In particular, the mathematics of the classical theory of light (Maxwell’s equations) will carry over to the quantum theory but the interpretation of what the ﬁelds represent will be diﬀerent. 1.3 Photons have momentum The next step in the story is almost a direct consequence of E = hν but did not come about until the 1920’s. Compton used the fact that photons should have momentum to explain observations of γ-rays which had been scattered from free electrons. If we think of the photon as a particle of zero mass, then in relativistic theory it should have a momentum related to its energy by E = cp. Combining this with Planck’s formula we then note that the momentum should be hν h p= = . (5) c λ By asserting this logical extension of Planck’s formula Compton was able to explain observations made of photons scattering oﬀ electrons. In “Compton scattering” a very high energy photon (or γ-ray) collides with a stationary electron and is deﬂected. In so doing it donates some of its energy to the electron and its wavelength increases. Using (5) and conservation of momentum and energy, Compton was able to come up with an explicit formula for the wavelength change as a function of the angle of deﬂection. The fact that this prediction agreed with observations ﬁrmly established the idea that photons could be treated as particles with momentum as well as energy and that (5) gave the momentum as a function of wavelength. Finally, we note that it has become customary to work with the following rescaling of Planck’s constant, h ¯ h= 2π (pronounced h-bar) instead of with h itself. In terms of this constant, Planck’s formula and the momentum equation can be written in the forms ¯ E = hω ¯ p = hk (6) 4 respectively, where the momentum equation is now written in vector form associating a vector momentum with a plane wave propagating in the direction of k. 1.4 If light can be particles, matter can be waves De Broglie made the contribution (in his doctoral thesis) in 1924 of asking whether, if the rela- tionships in (6) are natural for light, they might not work in the opposite direction for matter. This would associate a plane wave with any particle propagating with a deﬁnite momentum and energy. In other words, if light has a dual particle-wave nature, we might ﬁnd the same to hold for matter. Even though we used the speciﬁc properties of light to deduce p = hk as ¯ ¯ a consequence of E = hω, de Broglie argued that from the point of view of relativistic theory it seemed natural that the same formulas should hold without modiﬁcation for matter (see Appendix 1.8). Asserting wave-particle duality for matter as well as light is a fairly wild conjecture (so wild and speculative that de Broglie was only grudgingly awarded a PhD for it), but experimental conﬁrmation came in 1926 with experiments by Davisson and Germer showing interference eﬀects could be made by passing electrons through crystals. The observation of interference is the classic signature of waves and provides indisputable evidence of the wave-particle duality of matter. It has been repeated many times since with diﬀerent sorts of particles. Assuming that (6) can be used for matter as well as light is a crucial step in the development of quantum mechanics. Once we make this leap we are pointed ﬁrmly in the direction of the o full theory and all the elements are in place to see where Schr¨dinger’s equation comes from. First however we will step back and follow another thread in which equally stark deviations from the classical picture were appearing. 1.5 Atomic spectra and Old Quantum Mechanics It had been known since the nineteenth century that light emitted from excited atoms had frequencies restricted to a set of well-deﬁned values. These atomic spectra were characteristic of each element and, once again, could not be explained with classical physics. The simplest spectrum belongs to the simplest element, Hydrogen. In 1913 Bohr produced a complete and accurate account of the spectrum of Hydrogen using a set of ad hoc rules that completely violated classical notions. His model for Hydrogen starts with an electron following a circular orbit around a much heavier proton, attracted by the electrostatic coulomb force. He then assumes that • The angular momentum of the electron about the centre is an integer multiple of h, ¯ h L = n¯ . This forces the orbit radius and energy to be restricted to discrete sets of values, rn = n 2 a0 5 and E0 En = − 2n2 respectively. Explicit calculation gives [4π 0 ]¯ 2 h a0 = ≈ 0.053nm me e 2 (called the Bohr radius) and e2 E0 = ≈ 2 × 13.6eV [4π 0 ]a0 (exercise!). The term in square brackets is omitted if cgs units are used. We may say that the orbits are quantised. • Transitions between these quantised orbits occur in discrete jumps and when they do, a single photon is emitted which carries away all of the energy diﬀerence. The frequencies of the emitted light are therefore restricted to the values 1 1 hνn→m = En − Em = E0 2 − 2 . 2m 2n The set of frequencies νn→m explains the hydrogen spectrum completely. It therefore seems immediately clear that Bohr’s model carries a signiﬁcant element of truth. However, many aspects of it are disturbing. There is no reason to suppose that angular momentum should be quantised other than that it leads to a explanation of Hydrogen. If an electron were orbiting a proton in classical physics, it would be expected to lose energy to electromagnetic radiation because it is an accelerating charge and would eventually fall into the centre. Bohr’s explanation is simply that it doesn’t happen because it isn’t observed. Furthermore, while the theory can explain Hydrogen (and other single-electron ions) beau- tifully, it is not at all clear how more complicated atoms could be treated, even in principle. Classical orbits are not circular or even closed in such systems and a self-consistent calculation of energies cannot be made. A theory which can only be applied to very simple systems is clearly not completely satisfactory. Some generalisation was possible. Somerfeld extended Bohr’s calculation to allow for ellip- tical orbits (and got the same set of energies). More generally, it was found that one could specify a set of rules for any system that was classically integrable or separable, by applying the action quantisation condition pdq = nh around any closed coordinate curve in phase space. This process is called old quantum mechan- ics and generalises Bohr’s quantisation of angular momentum somewhat but it is still of no use in understanding the majority of physical systems, which are not integrable or separable. Note that in light of de Broglie’s relationship, we can in retrospect give some intuitive expla- nation of the action quantisation. pdq/h = λ−1 dq simply counts the number of wavelengths that ﬁt into an orbit, which should be an integer if standing waves were set up. However Bohr knew nothing of this since de Broglie’s assertion came much later. 6 1.6 Appendix: notation for plane waves The canonical example of a travelling wave is the function u(x, t) = u0 cos(kx − ωt) describing a plane wave in one dimension. Here u0 is the amplitude telling us how large the oscillations are. We refer to k as the wavenumber, which is related to the spatial period or wavelength λ by 2π k= λ and ω is the frequency which is related to the temporal period T by 2π ω= . T We have, of course, u(x + λ, t) = u(x, t) = u(x, t + T ). In the case of frequency the following version is also used 1 ω ν= = . T 2π If we need to distinguish between ν and ω, we refer to ω as the circular frequency and to ν simply as the frequency although in practice the word frequency is often used for both. For some reason there is no symbol in common use for the spatial analogy 1/λ of ν. We can also write the travelling wave in the form u(x, t) = u0 cos k(x − ct) where ω c= k is the phase velocity of the wave. Notice that, for example, crests of the wave corresponding to x − ct = 2πn move to the right with velocity c. It is often a huge advantage in manipulating plane waves to write them in the complex form u(x, t) = Re u0 ei(kx−ωt) . This is so common that one often thinks of a plane wave as the complex function u(x, t) = u0 ei(kx−ωt) with the understanding that at the end of a calculation physical answers are obtained by taking the real part. Using the complex version means that diﬀerential relations are often quite simple. One has ∂u = iku ∂x 7 for example. In classical problems, this use of complex notation is merely a device to ease mathematical manipulation and the things functions like u(x, t) represent are ultimately real. In quantum mechanics we will ﬁnd the novel aspect that the function that we are trying to calculate is genuinely complex. The whole theory would be extraordinarily unwieldy if we tried to formulate it without complex numbers, which is unlike classical problems where using complex solutions is helpful, but not absolutely necessary. In view of its relevance to quantum theory, we will adopt the complex convention from now on. k k.x= const. Figure 2: Plane waves. The world is three-dimensional of course so we need in general to calculate with functions u(x, t) depending on x = (x, y, z). The three-dimensional version of the (complex) plane wave is u(x, t) = u0 ei(k·x−ωt) = u0 ei(kx x+ky y+kz z−ωt) where k = (kx , ky , kz ) is called the wave vector. This is called a plane wave because at any given instant it is constant on the planes deﬁned by the condition k · x = const. It is not hard to show that u(x, t) has the same value on any two planes separated by a (perpendicular) distance 2πn nλ = k where n is an integer and we call k = |k| 8 the wavenumber. The interpretation is that planes deﬁned by the condition k · x = const. represent wave fronts and that these propagate in the direction of the vector k with a phase velocity ω c= k as in the one-dimensional case. The direction of the wavevector k therefore tells us the direction in which wave fronts travel and it magnitude k tells us the the speed at which they do so. To see this note that if x is on a given wavefront at time t = 0 and y is on the plane this wavefront evolves into a time t later, then k · x = k · y − ωt, so k · (y − x) = ωt. This means that ωt d⊥ == ct k where d⊥ is the projection of y − x on the direction of k. So c gives the rate of separation of the planes containing x and y as claimed. k.y = ω t k.x = 0 d y x Figure 3: How to get d⊥ . 1.7 The wave equation The wave equation in one-dimension is ∂2u ∂2u − c2 2 = 0. ∂t2 ∂x 9 It is easily veriﬁed that the plane wave u(x, t) = ei(kx−ωt) is a solution provided ω c= . k In fact one can show that any function of the form u(x, t) = f (x − ct) + g(x + ct) is a solution but this more general form will not concern us here because it does not work for other equations we are interested in (whereas plane waves do). In three dimensions the wave equation is ∂2u ∂2u ∂2u ∂2u ∂2u − c2 − − 2 = 2 −c 2 2 u = 0. ∂t2 ∂x2 ∂y 2 ∂z ∂t Here it is easily veriﬁed that the three-dimensional plane-wave u(x, t) = ei(k·x−ωt) is a solution provided ω c= . |k| These one and three-dimensional wave equations are the ﬁrst port of call whenever we try to understand wave problems. They are ubiquitous in applied maths and mathematical physics and describe many physical wave problems, including waves on a string, sound waves and electromagnetic waves (although as in the case of electromagnetic waves solutions might rep- resent components of a vector rather than giving a complete scalar answer). They do not apply directly to quantum mechanics but they do provide an important indication of how a quantum-mechanical theory might work. In the next Chapter we will look for analogous partial diﬀerential equations that are consistent with the constraints we can place on matter waves. 1.8 Appendix: de Broglie’s relation is natural in relativity Given the association of a frequency ω with an energy E through Planck’s formula, the asso- ciation between momentum and wavevector is very natural in light of the theory of relativity, independently of any of the properties of light. Vectors in relativity often come as part of a four-component package, called four-vectors, the most basic of which is the four-vector X = (x, ct) representing the position of an event in spacetime or the relative displacement between two events. Similar combinations are the four-momentum P = (p, E/c) 10 or the four-wavevector K = (k, ω/c). Physical laws are naturally stated as relationships between four-vectors. If we are told that energy is proportional to frequency, it then appears inevitable in relativity that the relationship can be extended to the corresponding fourvectors. We should therefore expect ¯ P = hK to hold as a matter of principle. Written in separate timelike and spacelike components, this is simply (6) above. 2 Wave Mechanics 2.1 o Discovering the Schr¨dinger equation Once all the elements of particle-wave duality are in place, the next step is to try to come up with an explicit partial diﬀerential equation describing the wave properties of massive particles such as electrons. In particular, this equation would play the same role for electrons that Maxwell’s equations (and the wave equation derived from them) play for photons. In fact, it is very useful to keep this analogy with the wave equation for light in mind as we seek the wave equation for electrons. We already know the answer in that case. Each of the components of E and B satisﬁes the wave equation ∂2ψ ∂2ψ ∂2ψ ∂2ψ = c2 2 ψ = c2 + 2 + 2 . ∂t2 ∂x2 ∂y ∂z How might we “deduce” this equation from the particle properties of photons, in such a way that the case of electrons might be treated similarly? All we have to go on for the moment is that every plane-wave solution, ψ(x, t) = ei(k·x−ωt) ¯ ¯ is associated with a particle travelling in free space with momentum p = hk and energy E = hω. In fact, we could rewrite the solution in terms of these variables as, h ψ(x, t) = ei(p·x−Et)/¯ . (7) If we are given a solution with well-deﬁned momentum and energy we can read oﬀ the compo- nents of momentum by applying the vector of operators h∂ h∂ h∂ ¯ ¯ ¯ ¯ h ˆ p ˆ ˆ p = (ˆx , py , pz ) = , , = i ∂x i ∂y i ∂z i ˆ (giving pψ = pψ) and similarly associate energy with the derivative ∂ h E ∼ i¯ ∂t 11 h (so that i¯ ∂ψ/∂t = Eψ). We see now that the wave equation for light is simply a statement that the energy-momentum equation for photons E 2 = c2 p2 = c2 (p2 + p2 + p2 ) x y z should apply to the wave-ﬁeld as an operator equation. Let us try to do the same thing for electrons. In the nonrelativistic limit, the energy- momentum equation for a particle of mass m is p2 E= . 2m A plane wave solution of the form (7) should then be a solution of ∂ψ 1 2 2 2 h2 2 ¯ h i¯ = ˆ ˆ ˆ px + p y + p z ψ = − ψ. ∂t 2m 2m Now we simply say that any wave-ﬁeld (which we could express as a linear superposition of plane-waves using Fourier transform methods) associated with an electron or any other nonrelativistic particle with mass should be a solution of the same linear equation. o We have eﬀectively written down Schr¨dinger’s equation. Before discussing its properties, however, a generalisation is needed. All of our discussion so far has involved particles propa- gating freely in space, without external forces acting on them. We will more generally expect the energy-momentum relation to involve a potential function, p2 E= + V (x). 2m It seems natural to generalise the wave equation for particles in such cases to ∂ψ h2 2 ¯ h i¯ =− ψ + V (x)ψ. ∂t 2m o This is the Schr¨dinger equation. Note that we might write it in the form ∂ψ ˆ h i¯ = Hψ ∂t ˆ where H is the diﬀerential operator 2 ˆ ¯ h 2 H=− + V (x) 2m which will be called the Hamiltonian operator, or Hamiltonian for short. Before discussing the solutions of this equation in more detail, some remarks are in order. o • We’ve written the Schr¨dinger equation in scalar form, whereas we know that the equation governing the wave properties of photons had a vector character. Why do we assume that such a simple form holds when we already know the case of light to be more complicated? The simple answer is that we’re just guessing. We’ve written down the simplest equation we can and now hope that it corresponds to physical reality. Only after such a comparison is made does it really gain credibility (and it will). 12 • The plane-wave solutions we’ve written down were complex. Normally such complex so- lutions are a mathematical device to simplify the calculation and to get physical solutions o we must take the real or imaginary part (see (1), for example). The Schr¨dinger equation gives us, for the ﬁrst time in nature, a theory whose solutions are intrinsically complex. This might be taken as the ﬁrst indication that the physical interpretation of ψ(x, t), when it comes, will be novel. ¯ • Notice that, in the case of photons, h cancels when we turn the energy-momentum relation into a diﬀerential equation. This is one of the accidents that allows the wave theory of light to be formulated in such a way that quantum mechanics does not appear. Had the ¯ photon had mass, it seems like Maxwell’s equations might well have had h-dependent terms in them and quantum mechanics might have appeared at an earlier stage. There is a second accident, however, which is more subtle but probably more important. It is the nature of photons that we can pile lots of them into the same solution and eﬀectively have the waveﬁeld describing the density of large numbers of photons simultaneously. In this case the relevant ﬁelds, E and B, become physically measurable quantities. There is an exclusion principle in the case of electrons, however, stating that a given wave-ﬁeld can only describe one electron at a time. It will never then be measurable as a true density of electrons and does not have a classical limit as a physical ﬁeld. There are certain particles, called bosons, for which we can allow the same wavefunction to describe many individuals simultaneously. It turns out to be very hard experimentally, requiring very low temperatures in particular, but has recently become possible and is currently a hot topic (referred to as Bose-Einstein condensation). 2.2 Looking for solutions It is conventional to call ψ(x, t) the wavefunction. We still have no idea (oﬃcially) what it rep- o resents, but let us ﬁrst satisfy ourselves that the Schr¨dinger equation is promising as a physical theory. The way we do this is to note that when we look for solutions we automatically arrive o at a generalisation of old quantum mechanics. In particular, Schr¨dinger was able to apply it to Hydrogen and, in very short order, was able to rederive Bohr’s quantisation conditions. Not only that, but it becomes obvious how we might in principle try to solve any other problem, even if in practice the general solution procedure might be very hard or intractable. o Later in the module we will see the detailed solution of the Schr¨dinger equation for Hy- drogen but at this early stage it is not helpful to delve so much into technical detail. Instead we will try to see what the solution strategy is and look in general terms at how the energy- quantisation of old quantum mechanics emerges. The ﬁrst step is to take advantage of the fact that the system is time-independent (we assume, for example, that the potential does not depend on t). If there is a nontrivial potential V (x) the plane waves ψ(x, t) = ei(k·x−ωt) o are no longer solutions of the Schr¨dinger equation but we can still ﬁnd solutions with a har- monic time dependence h e−iωt = e−iEt/¯ . 13 That is, we look for solutions which separate into functions of space and time, of the form, h ψ(x, t) = e−iEt/¯ ϕ(x). o Substituting in the Schr¨dinger equation gives 2 ˆ ¯ h 2 Hϕ = − ϕ + V (x)ϕ = Eϕ. 2m This is called the time-independent Schr¨dinger equation. Notice that it has the form of an o eigenvalue problem. When we substitute forms for V (x) corresponding to various physical problems and impose reasonable boundary conditions (usually ϕ(x) → 0 as |x| → ∞), we often ﬁnd solutions corresponding to a sequence of eigenvalues ˆ Hϕn (x) = En ϕn (x) n = 1, 2, . . . . (8) o ˆ In particular Schr¨dinger found for the hydrogen atom that H had the eigenvalues E0 En = − 2n2 which correspond to Bohr’s set of allowed energies. This is very strong evidence that the o Schr¨dinger equation is on the right track. Furthermore, we know now how to tackle any other problem in principle — write down the Hamiltonian operator and look for its eigenvalues. These eigenvalues are then interpreted physically as the values of energy that quantum mechanics allows the system to have. Much of modern physics reduces in practice to solving (8). In this module we will soon see how to solve it for some simple one-dimensional problems and the physically important cases of the simple harmonic oscillator (which underlies much of condensed matter physics, the theory of lasers and much more) and Hydrogen (which is at the basis of our understanding of atoms in general and therefore all of chemistry in particular). Before embarking on that programme, o though, let us reconsider the Schr¨dinger equation in general terms and try to come up with some form of interpretation for the wavefunction ψ(x, t). 2.3 The beginnings of an interpretation o Once we start ﬁnding solutions of the Schr¨dinger equation, we soon become utterly convinced that it is “right” because it so eﬀortlessly reproduces and generalises the quantisation rules of old quantum mechanics. It should make us uneasy, however, that we still have not really said what ψ(x, t) is supposed to represent. In looking for an interpretation, it is again useful to make the analogy with light. Consider (4) once again. The square modulus of the wavefunction of light is like a density (of photons). Let’s see if the same might not be true of ρ(x, t) = |ψ(x, t)|2 = ψ ∗ (x, t) ψ(x, t), 14 which is the nearest analogy we have for electrons. An intriguing hint that this might work as a density is that we can deﬁne a vector j(x, t) such that the following continuity equation is satisﬁed, ∂ρ + · j = 0. ∂t The continuity equation arises whenever we have ﬂow of some quantity in space. Think of a ﬂuid where ρ(x, t) represents the mass density. Then the current j(x, t) is a vector ﬁeld directed everywhere along the ﬂuid ﬂow whose magnitude is the rate at which ﬂuid passes through unit area normal to the ﬂow. Physically, the continuity equation says that the rate of change of density in some small region (∂ρ/∂t) is balanced by the inﬂux of ﬂuid from outside ( · j). The existence fo a continuity equation involving ρ(x, t) = |ψ(x, t)|2 in this way therefore suggests that |ψ(x, t)|2 is a density of some sort but again it should be stressed that this is not a “proof” in any sense, but simply an encouragement to press ahead and see what comes of it. To get a continuity equation, let us deﬁne ¯ h ¯ h j= (ψ ∗ ψ − ψ ψ ∗ ) = Im ψ ∗ ψ. 2im m o Then, assuming ψ(x, t) satisﬁes the (time-dependent) Schr¨dinger equation, we have ∗ ∂ ∗ ∂ ∂ (ψ ψ) = ψ ∗ ψ + ψ ψ ∂t ∂t ∂t 1 ∗ h2 ¯ 2 h2 ¯ 2 = ψ − ψ+Vψ −ψ − ψ∗ + V ψ∗ h i¯ 2m 2m ¯ h 2 2 = − ψ∗ ψ−ψ ψ∗ 2im Using the identity 2 2 · (ϕ ψ − ψ ϕ) = ϕ ψ−ψ ϕ we then ﬁnd that ∂ ∗ h¯ (ψ ψ) = − · (ψ ∗ ψ − ψ ψ ∗ ) , ∂t 2im which is precisely the identity we were aiming for. o Schr¨dinger found the ﬂuid analogy suggested by the continuity equation so tempting that he initially thought that the particle was genuinely smeared over space and that ρ(x, t) represented a literal mass density. Such an interpretation is rapidly seen to be inconsistent with physical reality, however. In practice, wavefunction solutions are often found to spread very rapidly in space whereas electrons, when observed, always seem to be point-like. Such an interpretation would not be consistent with the observation of particle tracks in cloud chambers, for example. 2.4 Born’s interpretation of the wavefunction The ﬁnal interpretation is usually credited to Born. It is now accepted that ρ(x, t) is merely the probability that the particle will be found at a given position at a given time. Before looking for 15 it, we can have no idea where it is, other than that we are more likely to ﬁnd it where ρ(x, t) is large. If we make the same measurement on many identicle systems, all described by the same wavefunction — extremely diﬃcult in practice but always possible as a thought experiment — we will obtain a diﬀerent result every time. Quantum mechanics will never tell us what happens in any individual experiment. It only says what happens on average when the results of many identical experiments are collated. Many people feel cheated by this interpretation and think that we should be able to give a more complete description of reality. The interpretation above seems to work nonetheless — no one has ever constructed an experiment for which it is inadequate — and however strange it seems it does end up giving a logically consistent theory. We will develop this interpretation and the formalism that goes with it in more detail later — the fully developed version is often referred to as the “Copenhagen interpretation” ( the people who developed it were often associated with Bohr’s school in Copenhagen) or the “orthodox interpretation.” In quantum mechanics we deal with one electron at a time. In that case we demand that the probability that the electron can be found somewhere is unity, ρ(x, t) dV = |ψ(x, t)|2 dV = 1. o Since the Schr¨dinger equation is linear, we are free to multiply any solution by a constant in order to ensure that this condition is satisﬁed. This process is called normalisation, and the wavefunction thus obtained is said to be normalised. Deﬁning the inner product between two arbitrary functions to be ψ|ϕ = ψ ∗ (x) ϕ(x) dV the normalisation condition can be written as ψ|ψ = 1. Note that the quantity we think of as physical, ρ(x, t), does not change if we multiply the wavefunction by a complex phase ψ(x, t) → eiθ ψ(x, t). This is an example of gauge-invariance. There is never any reason to choose one phase con- vention over any other and this indicates ψ(x, t) is not directly a directly measurable quantity. We can only probe it indirectly by making measurements and making deductions about what the density ρ(x, t) must have been like. (Note also that, after the measurement the state of the system will have changed because the measurement process will have disturbed the system). Note that, in the case of a solution h ψ(x, t) = e−iEt/¯ ϕ(x) o obtained from the time-independent Schr¨dinger equation, the density ρ(x, t) = |ψ(x, t)|2 = |ϕ(x)|2 16 is independent of time. For this reason such solutions are often referred to as stationary states. They allow a hand-waving argument for why an electron in an atom does not radiate energy because of acceleration and fall into the centre. The charge distribution −eρ(x) associated with a stationary state is independent of time, even though it is associated with a classically moving electron. In classical electromagnetism such a stationary charge distribution does not radiate energy. This is yet another hole in Bohr’s formalism (sort of) ﬁlled in, though it should be emphasised that a completely rational treatment of radiation needs a more sophisticated treatment of light which we will not be going into. 2.5 Appendix: the continuity equation A very common situation in applied maths is where we follow the evolution in time of a density ρ(x, t) of “stuﬀ” where, depending on the application, “stuﬀ” can represent mass, heat, electric charge and so on. Whenever stuﬀ is moving around we also have an associated current which is represented mathematically by a vector ﬁeld j(x, t). Its direction tells us the direction in which stuﬀ is moving at a given point and time and its magnitude |j| tells us how much stuﬀ ﬂows in the direction of j per unit time per unit area perpendicular to j. A universal feature of such a current-density combination is that, if the total amount of stuﬀ is conserved, they satisfy the continuity equation ∂ρ + · j = 0. (9) ∂t The fact that we can get this equation in the quantum mechanical case is an important step in interpreting the wavefunction and here we will see where it comes from in the general context. For simplicity, let assume a one-dimensional model where ρ(x, t) is the amount of stuﬀ per unit length. Current can only ﬂow in one direction in one dimension so j has only one component, which we denote by j(x, t). This tells us the net amount of stuﬀ ﬂowing per unit time past the point x. The current is positive if stuﬀ is ﬂowing to the right and negative if it is ﬂowing to the left. The amount of stuﬀ in the interval (a, b) at time t is b N (t) = ρ(x, t) dx. a If the interval (a, b) is kept ﬁxed, the rate of change of this quantity is dN b ∂ρ(x, t) = dx. dt a ∂t This tells us how much stuﬀ enters the interval (a, b) per unit time. If stuﬀ is neither created nor destroyed — that is, if there is a law of conservation of stuﬀ — then this must be accounted for by the net current entering the interval at a and leaving at b. That is dN = j(a) − j(b). dt But we have, b ∂j(x, t) j(a) − j(b) = − dx. a ∂x 17 So putting all this together means that b ∂ρ(x, t) b ∂j(x, t) dx = − dx. a ∂t a ∂x Since this holds for any interval (a, b) we must have ∂ρ ∂j + = 0. ∂t ∂x This is the continuity equation in one dimension. This generalises in three dimensions to ∂ρ ∂jx ∂jy ∂jz + + + = 0, ∂t ∂x ∂y ∂z which is nothing but (9) in component form. The derivation in three dimensions follows the same principle but we look at the rate of change of stuﬀ in a volume rather than in an interval and where we used simple integration above, we use vector calculus and the divergence theorem to manipulate this rate of change. 18 3 Some one-dimensional problems o The big problem to be solved once the Schr¨dinger equation is written down is the hydrogen atom. This allows us reproduce Bohr’s results, but with greater understanding, and points the way to solving more complicated problems. A full solution of Hydrogen requires more sophisti- cated techniques than we have at our disposal at present, however, so we put this problem oﬀ until later in the module. One-dimensional problems already exhibit the essential features we need and we begin by solving some examples which can be solved by fairly elementary tech- niques (ordinary diﬀerential equations with constant coeﬃcients). Note that these problems are often of interest in their own right and not merely as an academic exercise — problems such as the ones we solve here are often used in modelling semiconductor devices and other systems of practical interest. We start with the general one-dimensional problem. A particle moving under the inﬂuence o of a potential V (x) has the time-independent Schr¨dinger equation h2 d2 ψ(x) ¯ − + V (x)ψ(x) = Eψ(x). 2m dx2 Notice that we use the symbol ψ for the time-independent wavefunction in this Chapter. It is convenient to rewrite this in the form 2m(E − V (x)) ψ (x) + ψ(x) = 0. h2 ¯ Deﬁne the local wavenumber k(x) and momentum p(x) implicitly by p(x)2 2m(E − V (x)) k(x)2 = 2 = . h ¯ h2 ¯ o Then the Schr¨dinger equation becomes ψ (x) + k(x)2 ψ(x) = 0. Quantum mechanics in one dimension therefore reduces to solving second-order linear diﬀer- ential equations. We can of course only ﬁnd explicit solutions in special cases and we will restrict ourselves in this Chapter to problems where k(x) is locally constant. Before tackling these explicit problems, however, it is worth noting that the general qualitative features of the solution can be neatly related to the behaviour of classical trajectories in various regions of the line. We state here what the situation is without proof but note that these features will be evident in the explicit problems we do later. The essential features of any solution ψ(x) near a given x can be read from the value of k(x)2 there and in particular from its sign. We identify two kinds of behaviour. • The classically allowed region is deﬁned by the condition E > V (x). Here there is suﬃcient energy there for the particle to climb the potential and have enough left over for a positive kinetic energy. In particular we can ﬁnd meaningful solutions of the classical equations of o motion in which the particle moves here (hence the name). In the Schr¨dinger equation, 2 we ﬁnd that k(x) > 0 in the classically allowed region and k(x) is real. The solutions are oscillatory as we will see shortly for the case where k is constant. 19 • The classically forbidden region is deﬁned by the condition E < V (x). There is not enough energy for the particle to enter such regions, which would necessitate a negative kinetic energy. In the Schr¨dinger equation, we have k(x)2 < 0 so k(x) is imaginary. o While this region is forbidden to classical solutions, there is nothing to prevent us from o ﬁnding solutions to the Schr¨dinger equation there. We ﬁnd merely that the solutions are of a diﬀerent nature to those in the allowed region. Instead of oscillating they depend evanescently on x. In the case of constant potential, for example, we ﬁnd that solutions in the forbidden region are real exponentials of type e±κx , where k = iκ. One of the surprising features of quantum mechanics is that, because ψ(x) need not vanish in classically forbidden regions, we ﬁnd a nonzero probability of ﬁnding the particle there. Physical eﬀects of this feature are referred to as tunnelling. The idea is that we imagine the particle penetrating into the hillside of a potential as if through a tunnel. In particular, we often ﬁnd that there is a possibility that the particle can pass completely through a potential barrier even though it does not have enough energy to go over the top. This is the essential mechanism for atomic nuclei to decay, for example, and the unpredictability inherent in quantum mechanics is at the root of our inability to know when any given nucleus will decay. Let us now look at some concrete examples. 3.1 A particle in a box o The most important feature of the Schr¨dinger equation is that it often leads to solutions only for a discrete set of quantised energies. This will generally happen in problems where a particle would classically be conﬁned to a region of ﬁnite volume in space. The simplest model exhibiting this behaviour corresponds to particle being conﬁned to a box in one dimension. We suppose the box occupies the interval −a < x < a and that the potential vanishes in this interval so that classically the particle moves as a free particle there. The particle is absolutely forbidden from leaving the box so we insist that the wavefunction vanishes outside this interval. We also demand that the wavefunction be continuous. In other words, ψ(x) should be a solution of the equation ψ (x) + k 2 ψ(x) = 0 for −a < x < a (10) √ where k = h 2mE/¯ , subject to the boundary conditions ψ(x) → 0 as x → ±a. (11) o This problem is sometimes alternatively stated as solving the Schr¨dinger equation subject to the inﬁnite square-well potential 0 for |x| < a, V (x) = ∞ for |x| > a. The inﬁnite potential outside the box forces the wavefunction to vanish there and gives us the boundary conditions (11) above. 20 21 can be compared with the potential function V (x). and they are commonly represented graphically by horizontal lines on a vertical scale which where we now use the length L = 2a of the box. Each of these is referred to as an energy level 2m 2mL2 En = = , ¯ 2 h 2 kn ¯ h 2 π 2 n2 In particular we get solutions only of the energy is one of the quantised values respectively. for n = 2, 4, 6, · · · B sin kn x ψ(x) = A cos kn x for n = 1, 3, 5, · · · for positive integers n, and these are of the forms 2a , k = kn = nπ We ﬁnd solutions only if A = 0 = sin ka or B = 0 = cos ka. so A cos ka = 0 = B sin ka and for nontrivial solutions we have either 0 = A cos ka − B sin ka, 0 = A cos ka + B sin ka The boundary conditions give ψ(x) = A cos kx + B sin kx. Equation (10) has the general solution potential is often referred to as an inﬁnite square well potential. as one with a potential which goes to ∞ outside the box. Since a graph is rectangular, the Figure 4: A schematic representation of a particle in a box. We may interpret the problem x=−a x=a ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ £¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ ¡£¡£¡£¡£¡£¡£¡£¡¤ £ £ £ £ £ £ £ V=0 ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ £¤¡¡¡¡¡¡¡¡¤£ ¢¡¡¡¡¡¡¡¡¢¢ ¡ ¡¡¡¡¡¡¡¡ ¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ £ £ £ £ £ £ £ £ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡£ ¡£¡£¡£¡£¡£¡£¡£¡¤ £¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤£ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¢¡¡¡¡¡¡¡¡¢¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡¡¡¡ ¤¡£¡£¡£¡£¡£¡£¡£¡£ £¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡£¤ ¢¡¡¡¡¡¡¡¡¢ ¡ ¡¢¢¡¢¢¡¢¢¡¢¢¡¢¢¡¢¢¡¢¢¡ £¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤£ £ £ £ £ £ £ £ £¤¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡¤ ¢¡¡¡¡¡¡¡¡ ¡ ¢¡¡¡¡¡¡¡¡¢¢¡ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢ ¢¢ ¡ ¡£¡£¡£¡£¡£¡£¡£¡£ £¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ ¡£¡£¡£¡£¡£¡£¡£¡¤ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¡ ¢¡¡¡¡¡¡¡¡¢¢¡ £¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤£ ¡£¡£¡£¡£¡£¡£¡£¡£ ¡¡¡¡¡¡¡¡ £¤¡¡¡¡¡¡¡¡¤ ¤¡¡¡¡¡¡¡¡£ ¢¡¡¡¡¡¡¡¡¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ £ £ £ £ £ £ £ £ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡ £¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤£ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡¡¡¡¡¡¡¡¢¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡¡¡¡ ¤¡£¡£¡£¡£¡£¡£¡£¡£ ¤¡¡¡¡¡¡¡¡¤£¤ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ££¡¤£¡¤£¡¤£¡¤£¡¤£¡¤£¡¤£¡ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡£ ¢¡¡¡¡¡¡¡¡¢¢ ¡ ¡¡¡¡¡¡¡¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡¡¡¡£¤ £¡£¡£¡£¡£¡£¡£¡£¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡¡¡¡¡¡¡¡¢¡ 22 kinetic energy) but it is reﬂected if the energy is too low. penetrable walls — with enough energy it punches through and escapes (albeit with reduced a particle bouncing between brick walls, this one might be thought of as a particle conﬁned by where V0 is a constant, assumed positive in this section. If the previous problem is visualised as for |x| > a, 0 V (x) = for |x| < a, −V0 A second example of a conﬁning potential is the ﬁnite square well deﬁned by the conditions Particle in a ﬁnite square well 3.2 so the ﬁrst state with n = 1 is even, the next with n = 2 is odd, and so on. (12) ψn (−x) = (−1)n+1 ψn (x), A noteworthy feature of these solution is that they alternate in symmetry. That is L L for n = 2, 4, 6, · · · sin 2 nπx ψn (x) = L L for n = 1, 3, 5, · · · cos 2 nπx normalised wavefunctions are therefore 2/L. The and it is easily seen that we can achieve this in each case by choosing A = B = −a −∞ |ψ(x)|2 dx = |ψ(x)|2 dx = 1 that ∞ a It remains to determine the constants A and B in the wavefunction. These are chosen so Figure 5: The energy levels for the particle in a box. ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ £¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ ¤¡£¡£¡£¡£¡£¡£¡£¡£¤ £ £ £ £ £ £ £ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ££¡£¡£¡£¡£¡£¡£¡£¡ ¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡£¤ ¡£¡£¡£¡£¡£¡£¡£¡¤ £¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤£ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡£¡£¡£¡£¡£¡£¡£¡ £¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡£ ¡£¡£¡£¡£¡£¡£¡£¡£¤ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ £¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡ ¡£¡£¡£¡£¡£¡£¡£¡£¤ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¤£¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£¡ ¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡£ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¤¡£¡£¡£¡£¡£¡£¡£¡¤ £¡¡¡¡¡¡¡¡¤£ ¤¡¡¡¡¡¡¡¡£ En ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¡¡¡¡¡¡¡¡¢ ¢¡¡¡¡¡¡¡¡ £ £ £ £ £ £ £ £ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡ £¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡£ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¤¡£¡£¡£¡£¡£¡£¡£¡£¤ ¤¡¡¡¡¡¡¡¡£¤ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢¡¡¡¡¡¡¡¡ ¢ ££¡£¡£¡£¡£¡£¡£¡£¡ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡£¤¡ £¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤£ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡£¡£¡£¡£¡£¡£¡£¡£ ¤£¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ £¡£¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¡¡¡¡¡¡¡¡£¤£ £¡£¡£¡£¡£¡£¡£¡£¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡¡¡¡ ¢ V=0 II V=0 I III V=−V0 x=−a x=a Figure 6: The ﬁnite square well. Here we consider energies in the range −V0 < E < 0 o corresponding to conﬁnement. The Schr¨dinger equation can be written ψ (x) + k 2 ψ(x) = 0 in |x| < a ψ (x) − κ2 ψ(x) = 0 in |x| > a with √ 2m(E + V0 ) −2mE k= and κ = . ¯ h ¯ h As before we insist that the solution should be continuous. We also impose continuity on the derivative, in particular at x = ±a where the potential is discontinuous — otherwise the current would be discontinuous there, and we would have to account for the creation of particles at x = ±a. o We can ﬁnd continuous solutions to the Schr¨dinger equation for any value of E. However, only at a discrete set of values can we ﬁnd solutions which are square-integrable. In the present context, square-integrability means we insist that the wavefunction decays exponentially in the forbidden regions I and III of ﬁgure 6. The solution is therefore of the form κx Ce for x < −a, ψ(x) = A cos kx + B sin kx for −a < x < a, De−κx for x > a. Now, one of the exercises shows that since the potential is symmetric with respect to reﬂection about the origin, V (−x) = V (x), the solution must be either an even or an odd function of x (as in (12)). Let us consider the even case ﬁrst. An even solution is of the form κx De for x < −a, ψ(x) = A cos kx for −a < x < a, De−κx for x > a. 23 Imposing continuity of the wavefunction and its derivative then gives κa = ka tan ka. This transcendental equation has a solution only for a ﬁnite number of energies E n which are the quantised energy levels of the system. We obtain a similar condition κa = −ka cot ka. for the energy levels corresponding to odd solutions. The details of this calculation are left as an exercise as outlined in the problem set. The important point is that by demanding that the solution be square-integrable, eﬀectively impos- ing the condition that it decay at inﬁnity, we ﬁnd solutions only for a discrete set of energies. This is a general feature of problems where the classical motion is conﬁned to a region in space of ﬁnite volume (or more properly of length in one dimension). 3.3 Scattering from a step In the examples so far we have found that the energy is quantised when the classical motion is conﬁned to a ﬁnite region. The situation changes if we consider problems where the particle is not conﬁned classically. We illustrate such scattering problems with the case of a step potential of the form 0 for x < 0, V (x) = V0 for x > 0. illustrated in ﬁgure 7. b d a c V=V0 V=0 x=0 Figure 7: The step potential, and a schematic representation of the various coeﬃcients in the general solution. In this case classical motion is always unbound. If 0 < E < V0 , a particle coming from the left is reﬂected at the step at x = 0 because it does not have enough energy to enter the region x < 0 and returns to negative inﬁnity. If E > V0 the particle has enough energy to go everywhere. If it comes from the left, then on reaching x = 0 it loses kinetic energy and 24 slows down, buts keeps going. Notice that classical particles are therefore either reﬂected or transmitted at the step with probability one, depending on the value of E. o In the wave-mechanical picture we solve the Schr¨dinger equation to ﬁnd solutions of the form aeikx + be−ikx for x < 0, ψ(x) = ceik0 x + de−ik0 x for x > 0, with √ 2mE 2m(E − V0 ) k= and k0 = . ¯ h ¯ h We treat the case E > V0 here. As before, we impose continuity on the wavefunction and its derivative. These conditions give, respectively, a+b = c+d k(a − b) = k0 (c − d). It is convenient to represent this information using a transfer matrix, which tells us how to ﬁnd the coeﬃcients for the solution on the right given the coeﬃcients for the solution on the left, c 1 k0 + k k0 − k a = . d 2k0 k0 − k k0 + k b To specify the solutions any further we need to ask how the solutions behave at inﬁnity. In the case of bound states, simply stating that the solution should not blow up as x → ±∞ is enough to determine the remaining coeﬃcients (up to an overall normalisation constant) and give quantisation of energy. In the present case of scattering states, we get equally valid solutions no matter what the values of the remaining constants. Such solutions are not normalisable, so eﬀectively they represent an inﬁnite number of particles, but we can still give physical meaning to the solutions. They represent a steady stream of particles hitting the step, some going to the right (as represented by the solutions eikx and eik0 x ) and some going to the left (as represented by the solutions e−ikx and e−ik0 x ). Let us compute the current on the left-hand side where ψ(x) = aeikx + be−ikx . Then ¯ h j(x) = Im ψ ∗ (x)ψ (x) m ¯ h = Im ik |a|2 − |b|2 + ab∗ e2ikx − a∗ be−2ikx m ¯ hk = |a|2 − |b|2 m = v |a|2 − |b|2 25 where v is the velocity of particles on the left. Then the part aeikx represents right-going particles and is responsible for the positive current v|a|2 while be−ikx represents left-going par- ticles and is responsible for the negative component −v|b|2 of the current. We get a similar decomposition of the current on the right. A particularly interesting case is where we send a steady stream of particles from the left. These hit the step and may be reﬂected back to the left or transmitted forward to the right. The essential feature of this situation is that we do not have particles coming towards the step from the right and d = 0 in the solution above. It is traditional to normalise such a scattering state so that it is written in the form eikx + re−ikx for x < 0, ψ(x) = teik0 x for x > 0, where r and t are referred to as the reﬂection and transmission coeﬃcients respectively. These coeﬃcients are related to each other using the transfer matrix as follows t 1 k0 + k k0 − k 1 = 0 2k0 k0 − k k0 + k r and this gives in turn k − k0 2k r= and t = . k + k0 k + k0 Notice that in contrast to the classical situation, where all particles are transmitted when E > V0 , the quantum-mechanical solution indicates that some particles are reﬂected since the reﬂection coeﬃcient is nonzero. A calculation of the current as above yields (1 − |r|2 )v for x < 0, j(x) = |t|2 v0 for x > 0, ¯ ¯ where v = hk/m is the velocity on the left and v0 = hk0 /m the velocity on the right. It is easy to verify using the expressions for r and t above that the current is the same on both sides. This allows us to write v0 1 = |r|2 + |t|2 . v 2 We then interpret R = |r| as the probability that a particle is reﬂected by the step and T = (v0 /v)|t|2 as the probability that it is transmitted. We conclude by noting brieﬂy what features of the above solution are diﬀerent if 0 < E < V 0 (the details of this are left as an exercise). The solution on the left is the same as before but we must substitute iκ = i 2m(V0 − E) for k0 in the solution on the right. The expressions for the reﬂection and transmission coeﬃcients then become k − iκ 2k r= and t = . k + iκ k + iκ Notice in particular that |r|2 = 1, 26 which indicates that all particles are reﬂected. They do not have enough energy classically enter the region x > 0 and quantum-mechanically, the wavefunction is a decaying exponential ψ(x) = te−κx there. While they can penetrate for a bit, even quantum-mechanically all particles are eventually sent back. Note that the expressions we derived for the current cannot be easily transcribed from the E > V0 case when x > 0 because when we take the imaginary part in the formula for j(x), the replacement of k0 by iκ changes things. We ﬁnd in fact that the current vanishes on both sides in this case, which means that particles moving to the right are always balanced by particles moving to the left. Finally, we ﬁnish by emphasising that in this scattering system we have found that the energy is not quantised. Equally valid solutions can be found for all positive energies. This is a general feature of problems in which particles can escape to inﬁnity. 4 State space, observables and the formalism of quan- tum mechanics We are now in a position to start describing the general principles of quantum mechanics. This is usually done by formulating a series of postulates. These take the interpretations of quantum mechanical results which have seemed “not unreasonable” to us as we have developed the basic equations, and formulate them so that they are more precise and valid for general situations. These there is no proof for these postulates, of course, but they form a self-consistent interpretation of quantum mechanics which, in the three-quarters of a century since its birth, has proved consistent with every observation made of a quantum mechanical system. We ﬁrst state them for a single particle moving in a potential. Once the mathematical tools have been developed further, these can be generalised in a natural way to any quantum system. Note that the order and numbering of the postulates below is not universal and we might well ﬁnd it useful to rearrange the contents when we consider more general systems later. 4.1 State space We begin by describing two postulates which establish how we describe the state of a quantum- mechanical system. Postulate 1: The state of a one-particle system is represented by a wavefunction ψ(x, t), depending on position x and in general evolving in time. Everything we can hope to know about the system is determined from ψ(x, t). Postulate 2: We associate a probability density ρ(x, t) = |ψ(x, t)|2 27 with ψ(x, t) so that the probability of ﬁnding the particle inside a region D of space is p(x ∈ D) = ρ(x, t)dV. D In much of what follows we consider wavefunctions at a ﬁxed instant in time, so it may be convenient to suppress time in the notation and refer simply to ψ(x) or even to ψ. We will consider states which describe a single particle (and will leave aside for now the issue of scattering states of the type encountered in the previous Chapter). In that case we can meaningfully associate a wavefunction ψ to the state of a particle if 2 ψ = |ψ|2 dV < ∞, the integration being taken over all space. Such a wavefunction is said to be square-integrable or normalisable and allows us to deﬁne a normalised wavefunction ˆ ψ ψ= ψ for which the total probability of ﬁnding a particle somewhere in space is one. In addition we might want to consider further conditions such as continuity or smoothness. It turns out that in the ﬁrst instance it is convenient to impose only the condition that the function be suﬃciently unpathological that it be integrable — we will not go into the technical details here — and impose additional constraints such as continuity or diﬀerentiability as we need them later. The set of integrable and normalisable wavefunctions is closed under addition and multi- plication by complex constants. That is, if ψ and ϕ can describe a quantum state then so can χ = αψ + βϕ for any complex constants α and β. So state space has the structure of a vector space. Fur- thermore we can deﬁne on state space an inner product ϕ|ψ = ϕ(x)∗ ψ(x) dV. This inner product plays a key role in quantum mechanics. The key properties of any inner product are • Linearity: χ|αϕ + βψ = α χ|ϕ + β χ|ψ ∗ • Conjugate symmetry: ϕ|ψ = ψ|ϕ • Positivity and nondegeneracy: ψ|ψ ≥ 0, with equality iﬀ ψ = 0. So we have established that state space is naturally thought of as an inner product space, that is, a vector space with an inner product. In fact it has the slightly stronger property of being a Hilbert space. A Hilbert space is an inner product space with certain nice properties to do with 28 the convergence of sequences. We will never use these additional properties in this module, but it is standard terminology in quantum mechanics to refer to state space simply as “Hilbert space” and we will use the same terminology, if only to communicate with text books. In honour of this terminology we often denote state space by the symbol H. A useful property of the inner product which follows from the conditions above is the Schwarz inequality: | ϕ|ψ | ≤ ψ · ϕ with equality iﬀ αϕ + βψ = 0 for some α and β, where in general we denote, for any ψ, its norm by ψ = ψ|ψ . In particular this means that ϕ|ψ is ﬁnite for any two square-integrable states ψ and ϕ. An important tool in the analysis of quantum-mechanical systems is the orthonormal basis. An orthonormal basis is a sequence of functions, ϕn (x), n = 1, 2, 3, · · · , which satisfy ϕn |ϕm = δnm (so they are orthonormal) and which are such that any state ψ can be written as a linear combination of them ψ= c n ϕn n (so they form a basis). In working with an orthonormal basis of functions it is useful keep in mind a simple geometrical analogy with unit vectors in I 3 . R Example: Consider the expansion x = xi + yj + zk 3 R of an arbitrary vector in I in terms of the unit vectors i, j and k. The unit vectors are orthonormal with respect to the dot product because i · j = j · k = k · i = 0 and i · i = j · j = k · k = 1. Given the vector x, we can ﬁnd the components x y and z by forming the projections x = i · x, y = j · x and z = k · x. A similar construction works for orthonormal bases in state space except that the compo- nents are complex and there are more of them (inﬁnitely many in fact). Given the expansion ψ = n cn ϕn , we can compute the coeﬃcients cn from the inner products ϕn |ψ = ϕn | c m ϕm m = cm ϕn |ϕm m = cm δnm m = cn . 29 It is useful to note that we can therefore represent any state in the form ψ= ϕn ϕn |ψ . n Other useful identities follow similarly from the orthonormality of the basis. Let the states ψ and χ have the expansions ψ= c n ϕn and χ = an ϕ n . n n Then their inner product can be written χ|ψ = an ϕ n | c m ϕm n m = a∗ cm ϕn |ϕm n nm = a∗ cm δnm n nm = a∗ c n n n = χ|ϕn ϕn |ψ , n which is similar to the identity x 1 · x 2 = x 1 x2 + y 1 y2 + z 1 z 2 for 3D vectors, except that we have in addition the complex conjugation of a set of components. A special case of this is that the norm of a state is 2 ψ = ψ|ψ = |cn |2 n = | ϕn |ψ |2 n which is analogous to |x|2 = x2 + y 2 + z 2 . 4.2 Observables, operators and the Hermitian conjugate The term observable is used in quantum mechanics for any property of a system which we might hope to measure and give physical meaning to (the practicality of any such observation will not concern us). In practice, the observables we meet will be positions, momenta, energies and things which are functions of these quantities. We have already seen that momentum 30 components and energy are associated in quantum mechanics with operators which act linearly on wavefunctions. The vector of operators ¯ h∂ h∂ h∂ ¯ ¯ ¯ h ˆ p ˆ ˆ p = (ˆx , py , pz ) = , , = i ∂x i ∂y i ∂z i corresponds to the three components of momentum and the Hamiltonian operator ¯2 ˆ = h 2 H + V (x) 2m ˆ corresponds to energy. The eigenvalues of H in particular corresponded to the energies “allowed” by quantum mechanics. The next two postulates assert that this carries over to any observable. ˆ Postulate 3: To every observable O there corresponds a Hermitian operator O acting on wave- functions ψ(x). ˆ Postulate 4: A measurement of the observable O can only give a value in the spectrum of O. If ˆ O has discrete eigenvalues ˆ Oϕn = λn ϕn , the allowed values of O are therefore quantised. Furthermore if the particle is in a state described by an eigenfunction ϕn , then a measurement must yield the value λn with certainty. We need to explain some terminology (particularly “Hermitian”) before these postulates fully make sense, but let us begin by listing operators corresponding to some common observables: h ∂ ¯ ˆ • Momentum: px ψ = ψ i ∂x 2 ˆ ¯ h 2 • Kinetic energy: T ψ = − ψ 2m 2 ˆ ¯ h 2 • Total energy: Hψ = − + V (x) ψ 2m ˆ • Potential energy: V ψ = V (x)ψ ˆ • Position: xψ = xψ These operators have in common the property that they are linear. That is ˆ ˆ ˆ O(αψ + βϕ) = αOψ + β Oϕ for all allowed states ψ and ϕ and complex constants α and β. They generalise the concept of a linear transformation (or multiplication of a vector by a matrix) to inﬁnite-dimensional Hilbert spaces. These operators might involve diﬀerentiation and other operations that mean that that 31 are not deﬁned for arbitrary states. In general we expect therefore to have to restrict their action to a domain D ⊂ H. We generally hope that the domain is at least is large enough that we can approach any state arbitrarily closely while remaining within D but even for common operators D will be a proper subset of H. Example: In one dimension the momentum operator ¯ h ∂ ˆ p= i ∂x acts on diﬀerentiable functions ψ(x) such that ∞ ˆ pψ 2 = h2 ¯ |ψ (x)|2 dx < ∞. −∞ ˆ We exclude from the domain of p functions which are not diﬀerentiable and functions which become unnormalisable when a derivative is taken. The need to specify domains is a severe complication of dealing with inﬁnite-dimensional spaces. A careful treatment is outside the scope of the module and we will pass over the issue for the most part. In many of our calculations the speciﬁcation of the domains of operators such as the Hamiltonian is hidden in our speciﬁcation of the boundary conditions we impose on wavefunctions. For example, when we quantised the inﬁnite square well we restricted ourselves to functions which were continuous and which vanished on the boundary. An important property of the operators that represent observables is that they are Hermitian ˆ and we will now describe what this property represents. An operator O is Hermitian if ˆ ˆ ϕ|Oψ = Oϕ|ψ for all states ϕ and ψ in its domain. An important property of such operators is that they have ˆ ˆ real eigenvalues. Let ϕn be a proper eigenstate of O, by which we mean that Oϕn = λn ϕn and ˆ ˆ Then because O is Hermitian, ϕn is a normalisable state in the domain of O. ˆ ˆ ϕn |Oϕn = Oϕn |ϕn and because ϕn is an eigenvector this becomes ϕn |λn ϕn = λn ϕn |ϕn so λn ϕn |ϕn = λ∗ ϕn |ϕn n and we ﬁnd that λn = λ∗ . It is then reasonable to state that these eigenvalues are the results n ˆ that can be obtained from a measurement of the observable O. Had O not been Hermitian it would in general have had complex eigenvalues and they could not have represented values of an observable. Example: In one dimension the position operator is Hermitian. ϕ|ˆψ x = ϕ(x)∗ (xψ(x)) dx 32 = (xϕ(x))∗ ψ(x)dx = ˆ xϕ|ψ . Example: In one dimension the momentum operator is Hermitian. Integrate by parts and use the fact that of the states are normalisable then they vanish as x → ±∞: ∞ ¯ h ϕ|ˆψ p = ϕ(x)∗ψ (x) dx −∞ i ¯ h ¯ h ∞ = [ϕ(x)∗ ψ(x)]∞ − −∞ ϕ (x)∗ ψ(x)dx i i −∞ ∗ ∞ ¯ h = ϕ (x) ψ(x)dx −∞ i = ˆ pϕ|ψ . The examples above illustrate one more complication that arises with inﬁnite-dimensional operators. Not all eigensolutions are proper. Consider the momentum operator in one dimen- sion. We can easily write an equation h h peip0 x/¯ = p0 eip0 x/¯ ˆ h that has the structure of an eigenvalue equation, but notice that the eigenfunction e ip0 x/¯ is not normalisable and therefore not in the Hilbert space. If p0 is a real number, however, then eip0 x/¯h is “not too far” outside H — we can construct normalisable states which approximate e ip0 x/¯ h ˆ and are approximate eigenfunctions of p. Such (real) numbers are included, along with the proper eigenvalues, in the spectrum of an operator. An operator corresponding to an observable will in general have a spectrum consisting of real numbers which represent the possible values that result from a measurement of that observable. Proper eigenvalues make up the discrete part of the spectrum — generally corresponding to isolated points on the real line. Improper eigenvalues like p0 above make up the continuous spectrum and form a continuous rather than a discrete set as the name suggests. A precise statement and demonstration of these facts is beyond the scope of this module and we will simply assume that an operator representing an observable has a real spectrum, possibly combining discrete and continuous parts, in which discrete parts correspond to proper eigenvalues and continuous parts correspond to improper eigenvalues as in the case of the momentum operator above.2 We have established that Hermitian operators have real proper eigenvalues. We can also show that the eigenstates corresponding to two such eigenvalues are orthogonal. Let ˆ Oϕn = λn ϕn ˆ and Oϕm = λm ϕm 2 ˆ If we make certain claims about the domains of the Hermitian operator O then we can prove that the spectrum is real. Operators for which this can be done are called self-adjoint. In many text books, the terms Hermitian and self-adjoint are used interchangeably, but there is a diﬀerence in the domains they can have and they are not exactly the same thing. The postulates should properly state that observables are represented by self-adjoint operators but we will not make the distinction in this module and will use the terms interchangeably. 33 be two proper eigensolutions. Then ˆ ˆ ϕn |Oϕm = Oϕn |ϕm ⇒ λm ϕn |ϕm = λ∗ ϕn |ϕm n ⇒ (λm − λn ) ϕn |ϕm = 0. If the eigenvalues are distinct then this implies ϕn |ϕm = 0 and the eigenvectors are orthogonal as promised. Eigenvectors which correspond to the same degenerate eigenvalue are not necessarily orthogonal from the outset. However, it is not diﬃcult to show that, given a set of eigenvectors with the same eigenvalue, we can can choose linear combinations of them which are orthogonal. This is useful for the following reason. Given a Hermitian operator, we can choose the eigenvectors so that they form an orthonormal set ϕn |ϕm = δnm . ˆ In quantum mechanics, if O has a purely discrete spectrum then the proper eigenvectors are complete. That is, any state can be written as a linear combination of them and they form an orthonormal basis — which we will often refer to as an eigenbasis. ˆ h Example: Consider the space of functions ψ(θ) on the interval 0 < θ < 2π and let L = −i¯ ∂/∂θ act on functions with a smooth periodic extension ψ(θ + 2π) = ψ(θ). ˆ One can show that L is Hermitian and that the eigenfunctions eimθ ϕm (θ) = √ m = · · · , −1, 0, 1, · · · 2π form an orthonormal set (exercise). Then any function ψ(θ) can be represented as a linear combination ∞ ∞ 1 ψ(θ) = c m ϕm = √ cm eimθ m=−∞ 2π m=−∞ where 1 2π cm = ϕm |ψ = √ e−imθ ψ(θ)dθ. 2π 0 In this case the eigenfunctions form an orthonormal basis as promised and representing arbitrary functions as linear combinations of them recaptures the idea of Fourier series. Finally, related to the idea of a Hermitian operator is the Hermitian conjugate of an operator. ˆ ˆ Let A be an operator, not necessarily Hermitian. We say A† is the Hermitian conjugate of A if ˆ ˆ ˆ ϕ|Aψ = A† ϕ|ψ 34 for all suitable ψ and ϕ. Once again the situation is complicated by the need to specify domains. This is an issue which we will pass over and account for by the use of the word “suitable”. Example: Deﬁne a translation operator which acts on one-dimensional wavefunctions as follows ˆ Ta ψ(x) = ψ(x − a). Then for any ϕ and ψ ˆ ϕ|Ta ψ = ϕ(x)∗ ψ(x − a)dx = ϕ(x + a)∗ ψ(x )dx (x = x − a) = ˆ T−a ϕ|ψ ˆ† ˆ and we ﬁnd that Ta = T−a . Notice that an operator is Hermitian if ˆ ˆ A† = A. We list below some properties of the Hermitian conjugate. These can be shown without diﬃ- culty of we assume that any time an operator acts on a state, the result is well deﬁned (that is, the state is in the domain of the operator). ˆ ˆ ˆ ˆ (i) (A + B)† = A† + B † ˆ ˆ (ii) (αA)† = α∗ A† ˆ ˆ ˆ ˆ so in particular (αA + β B)† = α∗ A† + β ∗ B † . ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ (iii) (AB)† = B † A† (since ϕ|ABψ = A† ϕ|Bψ = B † A† ϕ|ψ ) ˆ ˆ (iv) (A† )† = A ˆ ˆ (v) (An )† = (A† )n N † N (vi) ˆn cn A = ˆ c∗ ( A † ) n n n=0 n=0 ˆ ˆ Finally we note that A and B are Hermitian then so are ˆ (vii) An ˆ ˆ (viii) A + B 35 N (ix) ˆ cn An for real cn . n=0 4.3 Measurement and the evolution of the wavefunction The last two postulates tell us what happens when we try to make a measurement of a quantum system and how it evolves between measurements. ˆ Postulate 5: If the observable O is measured then the result must be in the spectrum of O. If λn is a proper eigenvalue, then the probability that λn is obtained when a system with wavefunction ψ is observed is | ψ|ϕn |2 ˆ 2 pn = = ψ|ϕn , ψ|ψ ˆ where ϕn is the corresponding eigenstate and ψ is the normalised wavefunction corresponding to ψ. If λn is obtained, then immediately after the measurement the particle is in a state corresponding to ϕn . A full statement of the postulate would tell us how to deal with results in the continuous spectrum, so that in particular it subsumes Postulate 2 as a special case. For now however ˆ we restrict ourselves to the discrete spectrum. Let us assume in particular that O has a fully discrete spectrum and that the proper eigenstates ϕn form a complete set. Then if we assume ˆ that ψ = ψ is normalised we can expand it as ψ= c n ϕn n where ψ|ψ = |cn |2 = 1. n It seems natural therefore to interpret pn = |cn |2 as a probability and the postulate asserts that this is indeed what it is. Immediately after the measurement, we know with certainty that the observable has the value λn . We can only conclude that the state is then described by ϕn and not ψ. The interpretation is that in observing the system we have interfered with it and changed the state. The process whereby ψ changes to ϕn is called “reduction” or the “collapse of the wavefunction”. The essential inability to detach the observer from the system being observed is a central feature of quantum mechanics and is intimately connected with its inherent unpredictability. Even in the absence of measurement, an isolated quantum system evolves in time. The ﬁnal postulate tells us how this happens. Postulate 6: The evolution in time of the wavefunction of an isolated quantum system is governed o by the Schr¨dinger equation ∂ψ ˆ h i¯ = Hψ ∂t 36 ˆ where H is a Hermitian Hamiltonian operator. Note that in particular, “isolated” here means that we do not make observations or measure- o ments, which would lead to a collapse of the wavefunction outside the remit of the Schr¨dinger equation. Notice also that we have not speciﬁed the form of the Hamiltonian operator. For a particle moving in a potential we have already seen that it is 2 ˆ ¯ h 2 H=− + V (x), 2m but the postulate leaves open the possibility that we might deal with more general systems, with other Hamiltonians. This is similar to classical mechanics where we state Newton’s laws of motion without specifying what the forces between bodies are. Determining the explicit form of a given force (such as the inverse square law of gravitation) is then a question of coming up with a description of a given system rather than part of the framework of mechanics itself. ˆ Even if we don’t give a general prescription for H, the fact that it is Hermitian is an important feature. It means that probabilities are conserved and the evolution is consistent with the probabilistic interpretation we have given to wavefunction amplitudes. Consider for example o the evolution of ϕ|ψ where both ϕ and ψ evolve according to the Schr¨dinger equation, d ∂ψ ∂ϕ ϕ|ψ = ϕ| + |ψ dt ∂t ∂t 1 ˆ 1 ˆ = ϕ| Hψ + Hϕ|ψ h i¯ h i¯ 1 ˆ ˆ = ϕ|Hψ − Hϕ|ψ h i¯ = 0. In particular in the case ϕ = ψ we ﬁnd that the total probability ψ|ψ of ﬁnding the particle somewhere in space does not change with time, which is clearly necessary for the consistency of our interpretation. 4.4 Expectation values and uncertainties Suppose we are repeatedly able to prepare a quantum system in an identical state ψ and make repeated observations of an observable O. The astonishing fact is that even though the ﬁrst postulate asserts that the system cannot be speciﬁed in any more detail and is fully determined quantum-mechanically, we should expect to get diﬀerent results each time. That is, identical experiments conducted on an identical state will yield diﬀerent results even when our experimental techniques are perfectly accurate and we know the state with inﬁnite precision. What quantum mechanics does allow us to predict are the statistics of such measurements. ˆ Let us assume for simplicity that the spectrum of O is discrete and therefore that we can ˆ construct an orthonormal basis of eigenfunctions ϕn of O. Let the system be in a normalised 37 state ψ= c n ϕn . n Since the probability that an eigenvalue λn is obtained in a measurement is pn = |c2 |, n the average result obtained in the series of measurements of O, which in quantum mechanics is called the expectation value and denoted by O , is O = λn p n = λn |c2 |. n n n This can neatly be expressed in terms of the wavefunction itself. Notice that ˆ Oψ = ˆ cn Oϕn = λn c n ϕn , n n so we can therefore write O = ˆ c∗ (λn cn ) = ψ|Aψ n n and this can be computed without reference to the eigenbasis. In fact, the right-hand side can ˆ be computed in general even if O does not have a discrete spectrum. One can show that that this matrix element expression gives the average result of measurement even in that case (see appendices) and we will let this deﬁne the expectation value in general. In this context we take ˆ comfort from the fact that, because O is Hermitian, the expectation value ˆ ˆ O = ψ|Aψ = Aψ|ψ = O ∗ is self-evidently real. Finally, if we prefer to leave open the possibility that we might work with unnormalised wavefunctions, then we ﬁnd the expectation value using ˆ ψ|Aψ ˆ ψ|Aψ O = = . ψ 2 ψ|ψ Even though we can readily compute the average result of these measurements given ψ, we do not know in advance of any individual measurement what we will get and the outcome is uncertain. We quantify the uncertainty by deﬁning ∆O2 = (O − O )2 . That is, the uncertainty ∆O, which measures how much our measurements are spread around the average value, is the variance of the results we ﬁnd. Notice that we can simplify this as follows ∆O2 = (O − O )2 2 = O2 − 2 O O + O 2 = O2 − O . ˆ Notice also that we can predict O with certainty precisely when ψ is an eigenstate of O (see exercises). 38 4.5 The commutator An important diﬀerence between classical observables and their quantum operator counterparts is that the operators do not commute. For example, for any state ψ(x) in one dimension ¯ h ¯ h ˆˆ pxψ(x) = (xψ(x)) = ˆˆ xpψ(x) = xψ (x), i i or, in other words, ˆ ˆ ˆˆ xp = px. ˆ ˆ We formalise this diﬀerence by deﬁning, for any two operators A and B, the commutator ˆ ˆ ˆˆ ˆˆ [A, B] = AB − B A. Commutators are of primordial importance in quantum mechanics and their algebra plays a role that is sometimes analogous to that played by calculus in classical mechanics. Indeed the ﬁrst formulations of quantum mechanics (such as matrix mechanics) were intimately connected with this idea. A particularly important example is the commutator between position and momentum op- erators. From the calculation above we can see that x ˆ h [ˆ, p] = i¯ where the right-hand side is interpreted as the operator which multiplies a wavefunction by h i¯ . Calculation of more complicated examples is often aided by making use of the following identities (a) ˆ ˆ ˆ ˆ ˆ ˆ ˆ A, αB + β C = α A, B + β A, C , (b) ˆ ˆ ˆ ˆ A, B = − B, A (c) ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ A, B C = A, B C + B A, C , (d) ˆ ˆ ˆ A, B, C ˆ ˆ ˆ ˆ ˆ ˆ + B, C, A + C A, B = 0, whose proof is either obvious or left as an exercise (see problem sheets). We note in particular that ˆ ˆ † ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ A, B = (AB)† − (B A)† = B † A† − A† B † = B † , A† , ˆ ˆ so if A and B are Hermitian then so is ˆ 1 ˆ ˆ C= A, B . h i¯ ˆ ˆ ˆ In the case where A and B are the position and momentum operators, for example, C is the identity operator. 39 4.6 Positive deﬁnite operators With certain observables such as kinetic energy, we expect a measurement always to yield a positive or at least a nonnegative result. It will be useful in later calculations to be able to characterise the operators associated with such observables. ˆ A Hermitian operator A is said to be semipositive deﬁnite if, for all states ψ in its domain, ˆ ψ|Aψ ≥ 0. We sometimes write in this case, ˆ A ≥ 0. ˆ If equality holds if and only if ψ = 0, then we say that A is positive deﬁnite. Notice that by substituting an eigenfunction for ψ we immediately ﬁnd that any proper eigenvalue λ n of a semipositive deﬁnite operator satisﬁes λn ≥ 0 while the eigenvalues of a positive deﬁnite operator satisfy λn > 0. Therefore semipositivity or positivity is reﬂected in the allowed outcomes of a measurement of the corresponding observable. In deducing that certain operators are (semi)positive, we make use of the following obser- vations. ˆ ˆ ˆ ˆ • If A and B are (semi)positive, then so is A + B. ˆ ˆ ˆ • If A is any operator then A† A is semipositive. ˆ ˆ • As a special case, we ﬁnd that the square A2 of a Hermitian operator A is semipositive. The ﬁrst of these is obvious. For the second, note ﬁrst that for any ψ in an appropriate domain, ˆ ˆ ˆ ˆ ˆ ˆ ψ|A† Aψ = Aψ|Aψ = A† Aψ|ψ . ˆ ˆ The outer forms tell us that A† A is Hermitian. From the middle form, and from the property ˆ ˆ Aψ|Aψ ≥ 0 ˆ ˆ of the inner product, we deduce that A† A is semipositive deﬁnite as claimed. From these properties we can quickly come up with a number of physical examples. ˆ ˆ • An operator V acting on three-dimensional wavefunctions by V ψ(x) = V (x)ψ(x) is semi- positive iﬀ V (x) ≥ 0 for all x. • The kinetic energy ˆ p2 ˆ Tx = x 2m in a single degree of freedom is semipositive. 40 • The kinetic energy ˆ p2 ˆ p2 ˆ p2 ˆ T = x + y + z 2m 2m 2m in three degrees of freedom is semipositive. ˆ • If the potential energy V is semipositive then so is the Hamiltonian ˆ ˆ ˆ H =T +V. • For any observable O we have ∆O2 = (O − O )2 ≥ 0 so the deﬁnition of uncertainty makes sense. 4.7 The uncertainty principle For any observable we can ﬁnd some states for which the outcome of a measurement can be predicted with certainty. A particular example is where we prepare the quantum system in a proper eigenstate. Even if there are no proper eigensolutions, we can prepare states for which the measurement can be predicted with arbitrary accuracy. For example, an observation of momentum for a normalised state of the form 2 h ϕp0 (x) = Ceip0 x/¯ − x would with high probability give a result close to p0 if is small. Likewise, for small the wavefunction 2 ψx0 (x) = Ce−(x−x0 ) / is tightly peaked around x = x0 and a measurement of position will give a result close to x0 . The uncertainty principle tells us, however, that we cannot hope to know both position and momentum arbitrarily accurately at the same time. That is, we can construct states for which the uncertainty in either position or momentum can be made arbitrarily small but we cannot make both uncertainties small for the same state. This can be stated compactly by writing an inequality ¯ h ∆x∆p ≥ 2 which must be satisﬁed by the uncertainties ∆x and ∆p in position and momentum for an arbitrary state. If we can predict the momentum very accurately then the wavefunction must be very extended spatially and we must be very uncertain of the particle’s position, and vice versa. It is instructive to think about this for the states ϕp0 (x) and ψx0 (x) above. This incompatibility between position and momentum is the most famous expression of the ˆ ˆ uncertainty principle but it can be stated (and proved) rather more generally. If A and B are 41 any two observables, then for a given state ψ the uncertainties ∆A and ∆B are constrained by the inequality ¯ h ∆A∆B ≥ | C | , (13) 2 ˆ where C is the Hermitian operator ˆ 1 ˆ ˆ C= A, B . h i¯ To show this let us assume that A =0= B . (If this is not initially the case we can deﬁne new observables A =A− A and B = B − B which have the same uncertainties and for which the assumption is true.) We now apply the Schwarz inequality ϕ|ϕ χ|χ ≥ | ϕ|χ |2 to ˆ ϕ = Aψ ˆ and χ = Bψ, giving ˆ ˆ ˆ ˆ ˆ ˆ Aψ|Aψ Bψ|Bψ ≥ | Aψ|Bψ |2 . ˆ ˆ Because A and B are Hermitian we can write ˆ ˆ ˆˆ ψ|A2 ψ ψ|B 2 ψ ≥ | ψ|ABψ |2 and taking a square root gives ˆˆ ∆A∆B ≥ | ψ|ABψ |. (14) ˆˆ Now, in general AB is not Hermitian, but we can write it in the form ˆˆ ˆ ˆ AB = X + i Y where ˆ 1 ˆˆ ˆˆ ˆ i ˆˆ ˆˆ X= AB + B A and Y = − AB − B A 2 2 are. The square modulus of ˆˆ ψ|ABψ = X + i Y is then ˆˆ | ψ|ABψ |2 = X 2 + Y 2 2 ≥ Y 2 i ˆ ˆ = − [A, B] 2 2 ¯ h = C . 2 Combined with (14) this gives the uncertainty principle (13) as required. 42 4.8 Ehrenfest’s Theorem Since classical mechanics works so well in describing our everyday experience of the macroscopic world, we should hope to see it emerge as a limiting case of quantum theory. How this happens is not a simple matter, however, and even today is an active ﬁeld of investigation. While a full understanding of the classical limit is hard, we can relatively easily recover classical-looking ¯ equations if we consider a limit h → 0 (or, more properly, a limit where properties of the ¯ ¯ system with the same dimensions as h become large compared to h). One of the most basic of these correspondences is Ehrenfest’s Theorem, which says that the centre of a wavepacket whose position and momentum are well localised (subject to the constraints imposed by the uncertainty principle) follows a classical trajectory. We will state it in a somewhat more general form. If the wavefunction ψ evolves according o to the time-dependent Schr¨dinger equation, then the expectation value of any observable A evolves according to dA 1 ˆ ˆ = [A, H] . dt h i¯ We prove this before interpreting it. We have dA d ˆ = ψ|Aψ dt dt ∂ψ ˆ ˆ ∂ψ = |Aψ + ψ|A ∂t ∂t 1 ˆ ˆ 1 ˆˆ = Hψ|Aψ + ψ| AHψ i¯h h i¯ 1 ˆ ˆ 1 ˆˆ = − ψ|H Aψ + ψ|AHψ h i¯ h i¯ 1 ˆ ˆ = ψ|[A, H]ψ , h i¯ ˆ as required (we have assumed that A does not depend explicitly on time). Let us apply this to the one-dimensional Hamiltonian ˆ p2 ˆ H= + V (x) 2m and the position and momentum observables. It can be shown (exercise) that the relevant commutators are 1 ˆ p x [ˆ, H] = h i¯ m 1 p [ˆ, H] = −V (x) h i¯ where the operator V (x) acts by multiplication on the wavefunction. Notice that F (x) = −V (x) is the force acting on the particle and we can interpret the corresponding operator as 43 the force operator. Ehrenfest’s theorem then says dx p = dt m dp = F . dt Notice that these look very much like the classical equations of motion for a particle moving under the inﬂuence of an external force F (x) = −V (x). We have made no approximations so far, but let us now imagine a wavepacket whose position and momentum are localised as far as allowed by the uncertainty principle. For example, we might know each with uncertainties ∆x = ∆p = h/2. If these uncertainties are small compared to the characteristic scales of ¯ length and momentum for our system then we might approximate F ≈ F( x ) and Ehrenfest’s theorem says that the average position x and momentum p follow classical trajectories. This is often taken to be a demonstration that the classical world emerges from quantum ¯ theory in a limit h → 0. If we think of wavepackets as being slightly fuzzy particles, then as long as the fuzziness is on a scale that is small compared to macroscopic scales they behave eﬀectively as classical particles. The small numerical value of h = 1.05×10−27 g cm2 s−1 in typical ¯ macroscopic units makes this interpretation very tempting. It is misleading, however, because wavepackets often spread very rapidly and even optimally localised wavepackets can spread to macroscopic dimensions in a short space of time, especially if the dynamics is unstable. For example, rough calculation shows that a pencil standing on its tip, with uncertainties that are minimised subject to the constraints of the uncertainty principle, will fall over on a time scale of a few seconds. A similarly initialised chaotically tumbling moon will become completely delocalised on a time scale of decades, which is a very short time compared to the lifetime of such systems. We therefore need more to explain why the objects we see around us seem sharp and behave classically. 4.9 Appendix: ﬁnite-dimensional spaces Everything we have done with operators on a Hilbert space can be specialised to matrices acting on column vectors, regarded as elements of a ﬁnite-dimensional vector space. This matrix notation is a convenient way of representing a state space with a ﬁnite basis ϕn , n = 1 · · · N. It is conventional in the ﬁnite-dimensional case to replace these with the symbols ei , i = 1 · · · n, however, and we will adopt that notation here. 44 We consider in particular the n-dimensional vector space V whose elements can be written as column vectors, v1 v = . . . . (15) vn Let us deﬁne, for each column vector v an adjoint or Hermitian conjugate ∗ ∗ v † = v ∗T = (v1 · · · vn ), (16) which is the row vector formed by taking the complex conjugate of its transpose. Then we may write u|v = u† v = u∗ v i i (17) i for the inner product of any two vectors in V . It is easily seen that this has the properties of linearity, conjugate symmetry and positivity demanded of an inner product. There is an obvious orthonormal basis for V . The column vectors 1 0 0 0 1 0 e1 = . . , e2 = . . , ··· en = . . . . . 0 0 1 form a basis for V since we can express any vector in the form v1 . v = . = v 1 e1 + · · · + v n en . vn and in fact are easily seen to form an orthonormal basis. (Note that we may in general want to consider other orthonormal bases, however). Note that for any orthonormal basis we can represent any vector u in the form u= ci e i i where ci = ei |u , just as in the case of a general Hilbert space. A linear operator on V can always be represented by multiplication by a matrix. Let v be a column vector with components vi as above. Then Av is the column vector with components (Av)i = aij vj j where aij are the elements of the matrix A. Using the inner product, we can give a simple expression for the elements of A in terms of its action on basis vectors, namely, aij = ei |Aej . 45 This is easily seen by substituting the explicit forms given above for the standard basis vec- tors ei . In fact this lies behind the standard terminology in quantum mechanics that for any wavefunctions ϕ and ψ, ˆ ϕ|Aψ is called a “matrix element” of the operator A ˆ As with the case of operators on H the adjoint or Hermitian conjugate of a matrix A is deﬁned by the property that u|Av = A† u|v (18) for all u and v. It’s not hard to see by substituting the standard basis vectors for u and v that this means A† = A ∗ T . (19) That is, A† is the transpose of the complex conjugate of A — if the elements of A are aij , the elements of A† are a∗ . Hermitian matrices are those for which ji A = A† . (20) In the real case they are nothing other than the symmetric matrices. Following the discussion of Hermitian operators, we can prove that the eigenvalues of a Hermitian matrix are real and the corresponding eigenvectors may be chosen to form an orthonormal basis. In the ﬁnite- dimensional case, however, we have the luxury of not having to specify domains or worry about improper eigenvectors. 4.10 Appendix: the continuous spectrum We have seen that even relatively common observables such as momentum may have improper eigenfunctions. We outline brieﬂy here how such observables are dealt with in quantum me- chanics, although we will skim over all of the technical details. ˆ In general, we suppose that an observable O has a set of proper eigensolutions ˆ Oϕn = λn ϕn where n runs over a discrete index set, sometime ﬁnite and sometimes inﬁnite. The proper ˆ ˆ eigenvalues form the discrete or point spectrum of O. In addition, O may also have improper eigensolutions ˆ Oϕλ = λϕλ ˆ for λ in some continuous subset of the real line, forming the continuous spectrum of O. These improper eigenfunctions are not normalisable, ϕλ = ∞, and are therefore not properly speaking in the Hilbert space H, but they can be approximated by normalisable states. For example, normalised states of the form 2 h ϕp0 (x) = Ceip0 x/¯ − x 46 ˆ are approximate eigenfunctions of the momentum operator p if p0 is real and > 0 is small. Even though improper eigenfunctions are not in the Hilbert space proper we assume in quantum mechanics that we can use them to represent proper states by integrating over the continuous label λ. In particular, we assume that any proper state ψ can be represented in the form ψ= cn ϕn + c(λ)ϕλ dλ ˆ where the integration is over the range of the continuous spectrum of O. We further assume that the improper states ϕλ can be scaled so that if χ= an ϕ n + a(λ)ϕλ dλ is any second state, then χ|ψ = a∗ c n + n a(λ)∗ c(λ) dλ n and in particular ψ|ψ = |cn |2 + |c(λ)|2 dλ. n Notice that for any normalised state ψ, |cn |2 + |c(λ)|2 dλ = 1 n and we can then interpret the parts on the left as probabilities. We amend the ﬁfth postulate to say that if we make a measurement of the observable O we obtain a proper eigenvalue λn with probability pn = |cn |2 ˆ and we obtain a result in the subset I of the continuous spectrum of O with probability p(O ∈ I) = |c(λ)|2 dλ. I ˆ If we make a series of measurements O on quantum systems, all prepared in an identical state ψ, then the average outcome is the expectation value of O, O = λn |cn |2 + λ|c(λ)|2 dλ = ψ| λn c n ϕn + λc(λ)ϕλ dλ = ˆ ψ|Oψ , neatly generalising our previous results. We can think of our standard representation of the state as a function of position as a special ˆ ˆ case of all this. Consider the case of one dimension. If we replace O by x, then the expressions above become our standard expressions for overlaps and so on with c(λ) replaced by ψ(x). In particular, the second postulate eﬀectively becomes a special case of the ﬁfth postulate. 47 4.11 Appendix: compatible observables We conclude our treatment of the general structure for quantum mechanics by discussing a special case that plays an important role in the treatment of symmetries, angular momentum and higher-dimensionsal problems. Two observables A and B are compatible if the corresponding operators commute, ˆ ˆ [A, B] = 0. In this case the Heisenberg uncertainty relation gives no constraint and we will indeed ﬁnd that it is possible to construct states for which we simultaneously know A and B with certainty. These states are ones for which ˆ Aψ = aψ and ˆ Bψ = bψ for real numbers a and b. We say that such states are simultaneous eigenfunctions of the ˆ ˆ operators A and B. We have discussed how eigenfunctions of operators corresponding to observables are con- veniently used as orthonormal bases for Hilbert space. There is a generalisation that can be applies to compatible observables. ˆ ˆ Simultaneous diagonalisation: If A and B and commute and at least one of them has a discrete spectrum then we may construct an orthonormal basis ϕn , n = 1, 2, · · · of simultaneous eigenfunctions ˆ Aϕn = an ϕn ˆ Bϕn = bn ϕn . ˆ We begin by assuming that A has an orthonormal basis χn , n = 1, 2, · · · , with ˆ Aχn = an χn and a1 ≤ a 2 ≤ a 3 ≤ · · · . Notice that ˆˆ ˆˆ ˆ ABχn = B Aχn = an Bχn , ˆ ˆ so for each n either Bϕn vanishes or it is an eigenfunction of A with eigenvalue an . The eigenvalue an is either degenerate or nondegenerate. 48 If it is nondegenerate then we conclude that ˆ Bχn = bn χn for some bn and χn is already an eigenfunction. We let ϕn = χ n in that case. If an is degenerate then we have a ﬁght on our hands. Let a = an = · · · = an+dn −1 ˆ so the eigenvalue is dn -fold degenerate. For n ≤ k ≤ n + dn − 1, Bχk either vanishes or is an ˆ eigenvector of A and either way we conclude that ˆ Bχk = linear combination of χn , · · · , χn+dn −1 n+dn −1 = Bik χi , i=n where ˆ Bik = χi |Bχk . The coeﬃcients Bik for n ≤ i, k ≤ n + dn − 1 form a dn × dn Hermitian matrix. Let this matrix have eigenvalues bj and eigenvectors cj 1 . . , . cj n with j = 1, 2, · · · , dn and deﬁne dn ϕn+j−1 = cj χn+i−1 . i i=1 ˆ Then ϕn+j−1 is an eigenfunction of A with eigenvalue ˆ Aϕn+j−1 = an ϕn+j−1 ˆ and an eigenfunction of B with eigenvalue ˆ Bϕn+j−1 = bj ϕn+j−1 . ˆ Up to a relabeling of eigenvalues of B, this is exactly what we set out to prove. 49 In practice when faced with a basis of simultaneous eigenfunctions we often use a multiple index instead of n. For example we might list the eigenfunctions as ˆ Aϕlm = al ϕlm , with a1 < a2 < · · · and ˆ Bϕlm = blm ϕlm , with m = 1 · · · dl . In situations where the two eigenvalues suﬃce to break the degeneracy of the eigenfunctions it is also common to label them with the eigenvalues themselves, as in ˆ Aϕab = aϕab and ˆ Bϕab = bϕab . 4.12 Appendix: bra-ket notation In the 1950’s, Paul Dirac suggested a notation for states and operators that has since become very popular. It is useful because it helps us to “think quantum-mechanically”, although it is derided in certain quarters circles for encouraging nonrigorous thinking. Even those who prefer not to use it in their own work need to be familiar with its basics, however, since so much of writing about quantum mechanics is done in terms of it. Bra-ket notation takes advantage of the structure ϕ|ψ of the inner product between two states ψ and ϕ. Instead of writing a state as a function ψ(x) of position we write it as a “ket” |ψ . The idea is that |ψ represents a “state vector” in the Hilbert space H. This state vector can be represented concretely by a wavefunction ψ(x) but might equally be represented by the components cn in an arbitrary orthonormal basis. We regard the ψ part simply as a label and in fact can replace it by any symbol we think might be representative of the state. Common alternatives are |α , |n , |λ , or even things like |↑ and |+ . The important point is that whether we represent a state concretely by a wavefunction ψ(x) or a set of components cn , we can form linear combinations such as α|ψ + β|ϕ and kets are elements of a vector space. 50 An important element in bra-ket notation is that for every ket |ψ we deﬁne a corresponding “bra” ψ|. This is deﬁned to be an object which acts linearly on kets to give the complex number ψ| · |ϕ = ψ|ϕ . (Bra-ket, get it?) The relation between a ket and its bra |ψ → ψ| is the Hilbert-space analog of the relation between a column vector and a row vector v1 . ∗ ∗ v = . → v † = (v1 · · · vn ), . vn using the adjoint operation. A very powerful aspect of bra-ket notation is that we can also use to make sense of combi- nations such as |ψ ϕ| as operators. If |χ is any third state deﬁne the operator |ψ ϕ| acting on |χ to be |ψ ϕ| · |χ = |ψ ϕ|χ = ( ϕ|χ ) |ψ . This has use for example in stating that if there is an orthonormal basis consisting of |ϕ n ’s, then |ϕn ϕn | = I. n This is known as a resolution of the identity and is a compact way of expressing the fact that the states |ϕn form a complete set. To see this we note that if the |ϕn ’s are complete then any vector |ψ can be expanded in the form |ψ = cn |ϕn n = ϕn |ψ |ϕn n = |ϕn ϕn |ψ n = |ϕn ϕn | |ψ . n Because in bra-ket notation the label inside the ket can be quite arbitrary and might not have meaning as a state, when we write matrix elements we keep the operator outside the ket and write, for example, ˆ ψ|A|ϕ 51 where in normal or “default” notation we would write ˆ ψ|Aϕ . ˆ This done so that an expression such as 1|A|2 can be given a sensible interpretation, whereas ˆ 1|A2 would be confusing. A slightly annoying aspect of all this is that we have no self- contained means of representing ˆ Aψ|ϕ in bra-ket notation. We must write it instead in terms of the Hermitian conjugate operator ˆ ψ|A† |ϕ and before using bra-ket notation we are therefore required to have established and be familiar with the basic properties of Hermitian conjugates. 5 The harmonic oscillator ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡¡¡¡¡ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¢ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ F=−kx ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡¡¡¡¡¡¡¡¡ ¢¡¡¡¡¡¡¡¡¡¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡¢ ¢¢¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡¢ ¢¡¡¡¡¡¡¡¡¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¡¡¡¡¡¡¡¡ ¢ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¢¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ x ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ Figure 8: In the mechanical oscillator a spring exerts a restoring force F = −kx in opposition 1 to a displacement x. The corresponding potential is V (x) = 2 kx2 . We now ﬁnd eigenvalues and eigenvectors of the harmonic oscillator Hamiltonian ˆ p2 ˆ 1 H= + kˆ2 . x 2m 2 This is the Hamiltonian for a one-dimensional particle of mass m, constrained by a spring with spring constant k. Its importance extends far beyond this simple mechanical model, however. Its eigensolutions form the basis for our understanding of a good deal of modern physics. They provide the foundation for understanding vibrations in crystals (“phonons”), quantum optics (lasers), molecular vibrations and much more. While we will not be directly concerned here 52 with these more esoteric applications of the model, it is nice to know that they are understood using essentially the same analysis we are going to develop here. There are two ways to “solve” the harmonic oscillator in quantum mechanics. The ﬁrst ap- o proach, which we can think of as the wave-mechanical picture, is to write down the Schr¨dinger equation and solve it as a second order diﬀerential equation in terms of known special functions. This would carry on the sort of approach we used in Chapter 3 and is an option available to us no matter what one-dimensional potential we are faced with. It is a perfectly good way of doing things but we will use a second approach, which we might think of as the matrix-mechanical option. This way of doing things takes advantage of the special properties of the harmonic oscillator to solve it using only operator algebra, replacing ordinary calculus with the calculus of commutators. It hardly refers to diﬀerential equations. We prefer it because it is very elegant and is a good way to get into insight the quantum-mechanical way of doing things. In practical terms the techniques learned here can also be used to solve other oscillator-type problems, such as in quantum optics, which are less naturally formulated in terms of diﬀerential equations. 5.1 Overview and some deﬁnitions The main ingredient we need for the operator approach is that the Hamiltonian can be written ˆ ˆ as a quadratic expression in operators x and p which have the commutator x ˆ h [ˆ, p] = i¯ . In order to simplify the manipulation involved we deﬁned the rescaled operators ˆ Q = (mk)1/4 x ˆ and ˆ P = (mk)−1/4 p. ˆ Then, letting ω = k/m denote the classical frequency of oscillation, we can write ˆ 1 ˆ ˆ H = ω Q2 + P 2 (21) 2 ˆ ˆ where the operators Q and P have the commutation relation ˆ ˆ h [Q, P ] = i¯ . (22) Let us deﬁne 1 ˆ ˆ 1 ˆ ˆ a= √ ˆ Q + iP and a† = √ ˆ Q − iP h 2¯ h 2¯ and ˆ ˆˆ N = a† a. ˆ For reasons that will become obvious we call a the annihilation operator, its Hermitian conjugate ˆ † a the creation operator and N ˆ the number operator. 53 Note that ˆ 1 ˆ ˆ ˆ ˆ N = Q − iP Q + iP h 2¯ 1 ˆ ˆˆ ˆˆ ˆ = Q2 + i(QP − P Q) + P 2 h 2¯ 1 ˆ ˆ 1 = Q2 + P 2 − h 2¯ 2 Hˆ 1 = − . ¯ hω 2 This means that we can write the Hamiltonian in the form ˆ ˆ 1 hω H= N+ ¯ 2 ˆ ˆ and ﬁnding the eigensolutions of N is the same as ﬁnding the eigensolutions of H. In fact, if ˆ N ϕn = nϕn ˆ ˆ are the eigensolutions of N , then the eigensolutions of H are ˆ Hϕn = En ϕn , with 1 En = n + ¯ hω. 2 We will be able to prove the following. ˆ Claim: The eigenvalues of N are the positive integers n = 0, 1, 2, · · ·. We will furthermore be able to give a simple recipe for constructing the corresponding eigen- functions and in so doing we will have given a complete solution of the harmonic-oscillator problem. The upshot of all this is that the energy levels of the harmonic oscillator form a regularly- spaced ladder, starting at the so-called zero-point energy E0 = 1 hω — the minimum energy 2 ¯ ¯ allowed for a harmonically conﬁned particle — with a spacing hω between neighbouring levels. This contrasts with the inﬁnite well, where we found ever increasing gaps as the quantum number n increased, and with the hydrogen spectrum, where levels get closer as the quantum number increases. 5.2 Technicalities The main ingredients we need in order to demonstrate the claims of the previous section are provided by the following observations. 54 En V(x) Figure 9: The energy levels of the harmonic oscillator form an equally-spaced ladder, with ¯ neighbouring levels separated by the ﬁxed amount hω. ˆ • N = a† a is a self-adjoint semipositive operator and therefore its eigenvalues n must be ˆˆ real numbers such that n ≥ 0. • [ˆ, a† ] = 1. a ˆ ˆ • If N ϕn = nϕn , then, ˆˆ a N aϕn = (n − 1)ˆϕn (23) and ˆˆ N a† ϕn = (n + 1)ˆ† ϕn . a (24) The ﬁrst point is covered by the discussion in section 4.6. The second comes simply from inserting the deﬁnition of the creation and annihilation operators into the commutator: 1 ˆ ˆ ˆ ˆ [ˆ, a† ] = a ˆ [Q + i P , Q − i P ] h 2¯ 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = [Q, Q] − i[Q, P ] + i[P , Q] + [P , P ] h 2¯ 1 = h h (0 − i(i¯ ) + i(−i¯ ) + 0) h 2¯ = 1. The third is the most interesting and to a certain extent gives the crux of the solution. ˆ Let ϕn be a proper eigenfunction of N . Then, using a† a = aa† + [ˆ† , a], ˆ ˆ ˆˆ a ˆ 55 we get ˆˆ N aϕn = a† aaϕn ˆ ˆˆ = aa† a + [ˆ† , a]ˆ ϕn ˆˆ ˆ a ˆ a = ˆˆ ˆ aN − a ϕn a ˆ = nˆϕn − aϕn , from which we get (23). A similar calculation (exercise!) gives (24). 5.3 The main result ˆ Now suppose we are supplied with a (normalised) proper eigensolution N ϕn = nϕn of N . We ˆ will postpone until the next section the question of whether such a solution exists or whether it is nondegenerate if it does. We will show here that if such a solution exists, then it is necessarily a member of an inﬁnite sequence ϕ0 , ϕ 1 , · · · , ϕ n , · · · , ˆ labelled by the nonnegative integers n ≥ 0. In particular every eigenvalue of N is a nonnegative integer. The key point is (23). This tells us that either ˆ aϕn = 0 or 1 ϕn−1 = ˆ aϕn ˆ aϕn ˆ is a second (normalised) eigenfunction of N with eigenvalue n − 1. If the latter is the case we ˆ apply a once again and carrying on in this way we can generate a sequence of eigenfunctions ˆ a ˆ a · · · ϕn−2 ←− ϕn−1 ←− ϕn with decreasing eigenvalues ··· < n − 2 < n − 1 < n ˆ of N . Either this sequence terminates because aϕn0 = 0 for some n0 = n − k or the sequence is ˆ inﬁnite. We can immediately exclude the latter because in that case negative eigenvalues would ˆ eventually be generated whereas in the previous section we pointed out that N is semipositive deﬁnite. The sequence must therefore terminate. We have ˆ aϕn0 = 0. This is equivalent to 2 0 = ˆ aϕn0 56 = ˆ a aϕn0 |ˆϕn0 = ϕn0 |ˆ† aϕn0 aˆ = ˆ ϕ n0 | N ϕ n0 = n0 (since ϕn0 |ϕn0 = 1). Therefore the sequence terminates by having the eigenvalue n0 = n − k vanish. This means that all of the members of the sequence ˆ a ˆ a ˆ a ϕ0 ←− ϕ1 ←− · · · ←− ϕn generated by applying the annihilation operator to ϕn , including ϕn itself, have nonnegative integer eigenvalues. Not only that, but by applying the creation operator a† to ϕn we can can extend the sequence ˆ upwards a† ˆ a† ˆ ϕn −→ ϕn+1 −→ ϕn+2 · · · . We can show that 1 ϕn+1 = †ϕ a† ϕn ˆ ˆ a n is a normalised eigenfunction with eigenvalue n + 1 (exercise!) and since 2 a † ϕn ˆ =n+1>0 the sequence never terminates when extended in this direction. To summarise, we have shown that starting from any proper eigensolution we can generate a sequence with eigenvalues n running over the nonnegative integers. As a byproduct of this discussion we also have the identities 0 if n = 0, ˆ aϕn = √ (25) n ϕn−1 if n > 0 and √ a† ϕn = ˆ n + 1 ϕn+1 (26) which will prove useful later. It will also be useful to note that given ϕ 0 we can generate the rest of the sequence by repeatedly using (26) and this gives 1 1 2 1 n ϕn = √ a† ϕn−1 = ˆ a† ϕn−2 = · · · = √ a† ϕ0 (27) n n(n − 1) n! for a general eigenfunction. 57 5.4 Concrete solutions The ﬁnal link in the calculation is to show that a starting solution from which we can deduce the inﬁnite sequence in the previous section actually exists and to investigate its degree of degeneracy. To do this we must specify a bit more concretely how the creation and annihilation operators, or equivalently the position and momentum operators, act on wavefunctions. Having done this, we will be able to write concrete formulas for the eigenfunctions. This step will depend on the particular nature of the oscillator problem, and might change depending on the underlying physics. ˆ ˆ We will concentrate almost entirely on the basic mechanical oscillator, where Q and P act on functions of one variable according to ˆ Qψ(x) = (mk)1/4 xψ(x) and ˆ P ψ(x) = −i¯ (mk)−1/4 ψ (x). h Given these deﬁnitions we must ﬁnd a seed solution ϕn (x) to get things going. The simplest case to consider seems to be n = 0. In that case we have the equation ˆ aϕ0 (x) = 0 to solve, which is a ﬁrst order diﬀerential equation and simpler then the second order equation obtained in the general case. In order to avoid being overwhelmed by physical constants when we do this, let us deﬁne a new coordinate mω ξ = h−1/2 (mk)1/4 x = ¯ x, ¯ h Then it is easy to verify that √ √ d ˆ Q= hξ ¯ and ˆ P = −i h , ¯ dξ while 1 d 1 d a= √ ξ+ ˆ and a= √ ξ− ˆ . 2 dξ 2 dξ If in addition we deﬁne rescaled wavefunctions by mω −1/4 ψn (ξ) = ϕn (x), ¯ h the normalisation condition ∞ ∞ |ϕn (x)|2 dx = |ψn (ξ)|2 dξ = 1 −∞ −∞ remains simple. (Alternatively, we could be lazy and simply claim to choose units of length, ¯ time and mass so that h = m = ω = 1. This would lead to the same sort of simpliﬁcation.) With these conventions the seed equation is 1 d aψ0 (ξ) = √ ξ + ˆ ψ0 (ξ) = 0. 2 dξ 58 This is a ﬁrst-order separable ordinary diﬀerential equation whose solution 2 /2 ψ0 (ξ) = Ce−ξ , where C is an integration constant, is easily found. We choose the constant C so that ∞ √ 1= |ψn (ξ)|2 dξ = |C|2 π −∞ and the choice C = π −1/4 gives the ground state 2 /2 ψ0 (ξ) = π −1/4 e−ξ . Notice that we have found a solution, so it exists, and it is unique up to a multiplicative constant, so the ground state is nondegenerate. From the discussion in the previous section, we can extend these attributes to the other members of the sequence (ψ0 (ξ), ψ1 (ξ), · · · , ψn (ξ), · · ·). We can also write explicit expressions for them. For example, from (26), ψ1 (ξ) = a† ψ0 (ξ) ˆ 1 d 2 = √ ξ− π −1/4 e−ξ /2 2 dξ 2 /2 = 2−1/2 π −1/4 (2ξ) e−ξ . The next (normalised) member of the sequence is 1 ψ2 (ξ) = √ a† ψ1 (ξ) ˆ 2 2 /2 = 2−3/2 π −1/4 (4ξ 2 − 2) e−ξ . Carrying on like this we ﬁnd in general the form π −1/4 2 ψn (ξ) = √ Hn (ξ)e−ξ /2 2n n! where Hn (ξ) is a polynomial in ξ of degree n. The polynomials Hn (ξ) are known as the Hermite polynomials. Using (27) we can write n π −1/4 d 2 /2 ψn (ξ) = √ ξ− e−ξ 2 n n! dξ for a general eigenfunction, so in fact the Hermite polynomials can be simply deﬁned by the condition n −ξ 2 /2 d 2 Hn (ξ)e = ξ− e−ξ /2 . (28) dξ 59 The calculations above have shown that H0 (ξ) = 1 H1 (ξ) = 2ξ H2 (ξ) = 4ξ 2 − 2 are the ﬁrst three Hermite polynomials. By manipulating these expressions, we can come up with alternative ways of generating the Hermite polynomials and a couple are worth mentioning in particular. One can show that the Hermite polynomials obey the recursion relation Hn+1 (ξ) = 2ξHn (ξ) − Hn (ξ) and, together with the starting condition H0 (ξ) = 1 this is enough to deﬁne them. Another alternative is provided by the recursion relation Hn+1 (ξ) = 2ξHn (ξ) − 2nHn−1 (ξ). Details of these and other properties of Hermite polynomials are covered in the problems. There are many more properties of Hermite polynomial than can be gone into here. The Hermite polynomials have cropped up in several areas in applied maths and the standard reference books will provide much more detail. It is not necessary to know systematically what these additional features are, but it is useful to keep in mind that they exist and to be ready to delve into the detailed references when necessary. All of these additional properties can be derived straightforwardly from (28) which we obtained as a simple consequence of applying creation operators to the ground state. They o could also have been derived by writing out the Schr¨dinger equation as a diﬀerential equation and solving it using the Frobenious method. You will ﬁnd both approaches in the standard textbooks, but the one adopted here has the advantage of needing less computation and being more general in its application. The upshot is that using creation and annihilation operators we are in a position to solve almost any problem involving harmonic oscillators eﬃciently and elegantly. 5.5 Using creation and annihilation operators to get results As a ﬁnal example of the power of creation and annihilation operators, let us compute the expectation value 1 1 x2 = √ Q2 = Q2 mk mω ˆ ˆ for a particle in the state ϕn . It is useful to note that Q and P are expressed in terms of the creation and annihilation operators according to ˆ ¯ h † ˆ ¯ h † Q= a ˆ (ˆ + a) and P =i a ˆ (ˆ − a). 2 2 60 Then, for the state ϕn , ¯ h Q = ϕn |(ˆ† + a)ϕn a ˆ 2 h √ ¯ √ = n + 1 ϕn |ϕn+1 + n ϕn |ϕn−1 2 = 0, where we have used the fact that as distinct eigenfunctions of a Hermitian operator the states ϕn form an orthonormal set. We can similarly show that P = 0. We also ﬁnd ¯ h Q2 = ϕn |(ˆ† + a)2 ϕn a ˆ 2 ¯ h 2 = ϕn |(a† + a† a + aa† + a2 )ϕn ˆ ˆ ˆˆ ˆ 2 ¯ h = ϕn |(ˆ† a + aa† )ϕn a ˆ ˆˆ 2 ¯ h = (2n + 1) 2 and so 1 h ¯ x2 = n + 2 mω It is not hard to get expectation values of other powers and of combinations of position and momentum in this way. Had we tried to evaluate this expectation value using the explicit form of the eigenfunctions we would have ended up with an integral of the form ¯ h 1 ∞ 2 x2 = √ n n! π ξ 2 [Hn (ξ)]2 e−ξ dξ, mω 2 −∞ which looks like a lot more work. In fact, using the operator approach we never even had to write the eigenfunctions explicitly. It is quite often the case with harmonic oscillators that we can calculate without having to deal with the gory details of Hermite polynomials, or even to explicitly acknowledge their existence. 6 Angular momentum The need to deal with angular momentum in quantum mechanics arises when we try to solve three-dimensional problems with spherical symmetry. Consider the Hamiltonian ˆ h2 ¯ 2 H=− + V (r), 2M 61 where V (r) is a central potential and we write the Laplacian in spherical polar coordinates 2 1 ∂ 2∂ 1 ∂ ∂ ∂2 = r + 2 2 sin θ sin θ + 2 . r2 ∂r ∂r r sin θ ∂θ ∂θ ∂φ We use M for the mass because we want to reserve the symbol m for something else. It will be useful to separate out the angular part of the Laplacian by writing 2 1 ∂ 2∂ 1 = 2 ∂r r − 2 Λ2 r ∂r r where 1 ∂ ∂ ∂2 Λ2 = − sin θ sin θ + 2 sin2 θ ∂θ ∂θ ∂φ (and yes, the minus signs are meant to be there). In the method of separation of variables we will look for eigenfunctions of the form ψ(r, θ, φ) = R(r)Y (θ, φ). o Then the Schr¨dinger equation ˆ 2 2M (E − V (r)) Hψ = Eψ ⇒ (RY ) + RY = 0, h2 ¯ becomes Y d 2 dR R 2 2M (E − V (r)) r − 2Λ Y + RY = 0, r 2 dr dr r h2 ¯ which decouples in the form 1 d 2 dR 2M (E − V (r)) r + R Λ2 Y r2 dr dr h2 ¯ = . 1 Y R r2 On the left of this equation is a function of r. On the right is a function of (θ, φ). The only way we can get them to balance is if each is a constant, λ say. Therefore we look for solutions of Λ2 Y (θ, φ) = λY (θ, φ). Notice that this has the form of an eigenvalue equation. Once we have determined an eigenvalue λ we can return to the radial part of the problem and solve 1 d 2 dR 2M (E − V (r)) λ r + 2 R = 2 R. r2 dr dr h ¯ r This can also be written in the form 1 d 2 dR 2M (E − Veﬀ (r)) r + R = 0, r2 dr dr h2 ¯ 62 where h2 λ ¯ Veﬀ (r) = + V (r). 2M r2 o This radial equation looks a bit like a one-dimensional Scr¨dinger equation with an eﬀective potential Veﬀ (r) where, in addition to the central potential V (r) we have a centripital potential h2 λ/2M r2 in which ¯ L2 = h 2 λ ¯ plays the role of angular momentum squared. When we treat angular momentum quantum- mechanically this is exactly the conclusion we will reach. We will show that the eigensolutions of Λ2 are of the form Λ2 Y (θ, φ) = l(l + 1)Y (θ, φ) with l = 0, 1, 2, · · · called the angular momentum quantum number. These solutions are (2l + 1)-fold degenerate with the degenerate eigenfunctions for a given l naturally labelled by a quantum number m running from −l to l. We write the eigenfunctions in the form Ylm (θ, φ), m = −l, −l + 1, · · · , l − 1, l. They are normalised according to the convention |Ylm |2 dΩ = |Ylm (θ, φ)|2 sin θdθdφ = 1, S2 where the expression at left will be our shorthand for an integral over the angular coordinates with the usual limits and solid angle element dΩ = sin θdθdφ and S 2 indicating that we integrate over the two-dimensional sphere. These functions are called spherical harmonics. They crop up all over applied maths. In traditional ﬁelds they are usually described by solving diﬀerential equations. As with the harmonic oscillator, however, in using quantum mechanics we have the option of tackling them using operator methods. These are far more elegant and powerful, and being able to adopt this point of view is a tremendous beneﬁt of studying quantum mechanics. 6.1 Deﬁnitions and commutation relations In classical mechanics, the angular momentum of a particle passing through the point x with momentum p is deﬁned to be the cross product L = x × p. In components this reads Lx = ypz − zpy Ly = zpx − xpz Lz = xpy − ypx . 63 In quantum mechanics, we deﬁne the obvious operator versions ˆ ¯ h ∂ ∂ ˆ ˆ Lx = y p z − z p y = y −z i ∂z ∂y ˆ ¯ h ∂ ∂ ˆ p Ly = z px − xˆz = z −x i ∂x ∂z ˆ ¯ h ∂ ∂ p ˆ Lz = xˆy − y px . = x −y . i ∂y ∂x ˆ Since x commutes with py and so on, there are no ordering ambiguities in these equations. We also deﬁne the operator ˆ ˆx ˆy ˆz L2 = L2 + L2 + L2 , representing the square magnitude of angular momentum. Having introduced these new operators, the ﬁrst thing we should do is sort out their com- mutation relations. These are very important and worth remembering: ˆ ˆ hˆ [Lx , Ly ] = i¯ Lz ˆ ˆ hˆ [Ly , Lz ] = i¯ Lx ˆ ˆ hˆ [Lz , Lx ] = i¯ Ly (notice that these are related to each other by cyclic permutation of x, y and z) and ˆ ˆ [Li , L2 ] = 0, i = x, y, z. Deriving these is a useful bit of practice in manipulating commutators. We have ˆ ˆ ˆ ˆ ˆ p [Lx , Ly ] = [y pz − z py , z px − xˆz ] p p ˆ p = y[ˆz , z]ˆx + x[z, pz ]ˆy hp hp = y(−i¯ )ˆx + x(i¯ )ˆy h p ˆ = i¯ (xˆy − y px ) hˆ = i¯ Lz ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ and, using the identity [A, B C] = [A, B]C + B[A, C] from section 4.5, ˆ ˆ ˆ ˆ ˆ ˆ [ Lx , L2 ] = [ Lx , L2 + L2 + L2 ] x y z ˆ ˆy ˆ ˆz = [ Lx , L2 ] + [ Lx , L2 ] ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = Ly [ Lx , Ly ] + [ Lx , Ly ] Ly + Lz [ Lx , Lz ] + [ Lx , Lz ] Lz h ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = i¯ (Ly Lz + Lz Ly − Lz Ly − Ly Lz ) = 0 64 and similarly for their cyclic permutations. ˆ ˆ The fact that L2 commutes with Lz (or any of the other components) is fundamental to the theory of angular momentum. In general we say that any two observables for which the corresponding operators commute are compatible. It means that we can ﬁnd eigensolutions that are simultaneously eigensolutions for the two operators. 6.2 ˆ Looking for simultaneous eigenfunctions of L2 and Lz ˆ ˆ The compatibility of L2 and Lz means that it is natural to look for functions Yλµ such that ˆ L2 Yλµ = λYλµ and ˆ Lz Yλµ = µYλµ . ˆ ˆ ˆ The fact that we single out Lz over Lx and Ly here is purely a matter of convention. From the ˆ ˆz ˆy ˆz observation that L2 − L2 = L2 + L2 is semipositive deﬁnite we have the constraint λ − µ2 ≥ 0. That is, λ ≥ µ2 ≥ 0. (29) This will be a useful constraint that will play a similar role to the one that the positivity of Nˆ played in the case of the harmonic oscillator. Now deﬁne the ladder operators ˆ ˆ ˆ L± = Lx ± i Ly and observe that ˆ+ ˆ L† = L− . From the commutation relations in the previous section we can quite easily show that ˆ ˆ hˆ [Lz , L± ] = ±¯ L± ˆ ˆ [L2 , L± ] = 0. Notice now that ˆ ˆ ˆ ˆ ˆ ˆ Lz L± Yλµ = L± Lz Yλµ + [Lz , L± ]Yλµ ˆ ¯ˆ = µL± Yλµ ± hL± Yλµ ¯ ˆ = (µ ± h)L± Yλµ . and ˆ ˆ ˆ ˆ L2 L± Yλµ = L± L2 Yλµ ˆ = λL± Yλµ . 65 ˆ ˆ ˆ ˆ Therefore either L± Yλµ = 0 or L± Yλµ is a new simultaneous eigenvector of L± and L2 with ¯ eigenvalues µ ± h and λ respectively. We can investigate which is the case by evaluating the norm ˆ ˆ± ˆ ˆ ˆ L± Yλµ 2 = Yλµ |L† L± Yλµ = Yλµ |L L± Yλµ . We have ˆ ˆ ˆ ˆ ˆ ˆ L− L+ = (Lx − iLy )(Lx + iLy ) ˆx ˆ ˆ ˆy = L2 + i[Lx , Ly ] + L2 ˆ ˆz ¯ ˆ = L2 − L2 − h Lz . This gives ˆ L+ Yλµ 2 = λ − µ2 − hµ ¯ ¯ = λ − µ(µ + h) (30) and we similarly have ˆ L− Yλµ 2 ¯ = λ − µ(µ − h) (31) (assuming that the eigenfunction Yλµ has been normalised). ˆ ˆ As we did for the harmonic oscillator, we can use L− and L+ to generate new eigenfunctions ˆ ˆ ˆ of Lz and L2 with respectively decreasing and increasing eigenvalues of Lz and a ﬁxed eigenvalue ˆ 2 λ of L . If the sequence carried on indeﬁnitely in either direction we would eventually violate the constraint (29). It must therefore terminate in both directions and we have a ﬁnite sequence of eigenfunctions ˆ L− ˆ L− ˆ L− ˆ L+ ˆ L+ ˆ L+ Yλµmin ←− · · · ←− Yλ,µ−¯ ←− Yλµ −→ Yλ,µ+¯ −→ · · · −→ Yλµmax h h ˆ with Lz -eigenvalues ¯ ¯ µmin < · · · < µ − h < µ < µ + h < · · · < µmax . From (30) and (31) the terminating values are respectively determined by ¯ λ = µmax (µmax + h) and ¯ λ = µmin (µmin − h). ¯ We know also that µmax can be reached from µmin by an integer multiple of h so h µmax = µmin + k¯ for some integer k ≥ 0. From ¯ ¯ µmax (µmax + h) = λ = µmin (µmin − h) 66 we then get k µmax = ¯ h 2 and k k λ= + 1 h2 . ¯ 2 2 Let us denote k l= . 2 ˆ Then the eigenvalues of L2 are constrained to be of the form h λ = l(l + 1)¯ (32) where 2l is a nonnegative integer and once a single simultaneous eigenfunction is found we can ˆ construct from it a sequence containing 2l + 1 members in which λ is ﬁxed the Lz -eigenvalue is h µ = m¯ (33) where m runs over the range m = −l, −l + 1, · · · , l − 1, l. (34) We call l the angular momentum quantum number and m the azimuthal quantum number or magnetic quantum number (the latter because it plays an important role when atoms are placed in magnetic ﬁelds). The standard convention is to label the eigenfunctions using the quantum numbers l and m rather than the eigenvalues λ and µ. From now on we will therefore write the eigenfunctions in the form Ylm rather than as Yλµ . It is useful to give explicit formulas for the generation of new eigenfunctions from a seed solution. The main thing to sort out is the normalisation. Recall from (30) and (31) that ˆ L± Ylm 2 = λ − µ(µ ± h) = (l(l + 1) − m(m ± 1))¯ 2 . ¯ h This means that we can write 0 if m = ±l, 1ˆ L± Ylm = ¯ h l(l + 1) − m(m ± 1) Yl,m±1 otherwise. 6.3 Concrete solutions: spherical harmonics We are now in a position that is familiar from the harmonic oscillator. We have shown that if we are given one simultaneous eigenfunction then the eigenvalues are constrained by (32) and (33) and we can generate from it a sequence 2l + 1 eigenfunctions in which l is ﬁxed and m runs between −l and l in unit steps. In doing this we have used only the commutation relations 67 between the components of angular momentum. To determine whether such solutions exist, and what their forms are if they do, we will start to use more explicitly the expressions we have given for the angular momentum operators as diﬀerential operators. It will help a great deal to express them in terms of polar coordinates on the sphere. By manipulating the deﬁnitions already given in terms of cartesian coordinates it is possible to show that ˆ ∂ ∂ Lx = i¯ sin φ + cos φ cot θ h ∂θ ∂φ ˆ ∂ ∂ h Ly = −i¯ cos φ − sin φ cot θ ∂θ ∂φ ˆ ∂ h Lz = −i¯ ∂φ and ˆ 1 ∂ ∂ 1 ∂2 L2 = −¯ 2 h sin θ + . sin θ ∂θ ∂θ sin2 θ ∂φ2 Notice in particular that ˆ L2 = h 2 Λ 2 , ¯ so in solving the angular momentum problem we will have gone a long way towards solving the o Schr¨dinger equation for central potentials. It is also worth recording that ˆ ∂ ∂ L± = he±iφ ± + i cot θ ¯ . ∂θ ∂φ We will assume in this section that these operators act on functions ψ(θ, φ) on the sphere, on which we deﬁne the inner product ϕ|ψ = ϕ∗ (θ, φ)ψ(θ, φ)dΩ. S2 So even if we are motivated by a problem in three dimensions in this section we will ignore the radial coordinate and restrict our attention to the polar coordinates (θ, φ). ˆ We begin by asking what sorts of functions can be eigenfunctions of Lz . These satisfy ˆ h Lz Ylm (θ, φ) = m¯ Ylm (θ, φ) ∂Ylm ⇒ = imYlm ∂φ ∂ −imφ ⇒ (e Ylm ) = 0 ∂φ ⇒ eimφ Ylm = Θ(θ) ⇒ Ylm (θ, φ) = eimφ Θ(θ) 68 where Θ(θ) is some (as yet unknown) function of θ alone. If Ylm (θ, φ) is to be a single-valued function on the sphere it must satisfy the boundary condition Y (θ, φ + 2π) = Y (θ, φ). This implies that m must be an integer. From (34) we conclude in turn that l must be an integer. This is more restrictive than the conclusions we reached in the previous section based on the commutation relations alone. There 2l might have been an odd integer or, in the parlance of quantum mechanics, l might have been a half-integer. The present case where angular momentum operators act on functions in the sphere (or more generally on functions on three-dimensional space) is known as the case of orbital angular momentum. We say that orbital angular momentum must have integer angular momentum. Nature is not wasteful. If half-integer angular momentum is allowed in principle by the commutation relations it might be no surprise that it pops up somewhere in quantum theory. It turns out that elementary particles have a sort of internal angular momentum called spin for which the operators have the commutation relations we used in the previous section but which cannot be written in the same way as diﬀerential operators. For certain elementary particles (the electron for example) this spin angular momentum is half-integer. The fact that we did not exclude this possibility in the previous section was not therefore an oversight. In any case we will be concerned exclusively with orbital angular momentum in this module so let’s continue with that discussion. For a given (integer) value of l, we know that the top eigenfunction with m = l must be of the form Yll (θ, φ) = Θ(θ)eilφ . We know furthermore that ˆ 0 = L+ Yll (θ, φ) ∂ ∂ = heiφ ¯ + i cot θ Θ(θ)eilφ ∂θ ∂φ = hei(l+1)φ (Θ (θ) − l cot θΘ(θ)) . ¯ So Θ (θ) = l cot θΘ(θ) which can be solved as a separable ﬁrst-order diﬀerential equation to give ln Θ = l cot θdθ = l ln | sin θ| + const. or Θ = C sinl θ, where C is an integration constant we will choose so that the eigenfunction is normalised. It’s not particularly interesting to compute so we will simply quote the result, (−1)l (2l + 1)! l ilφ Yll (θ, φ) = sin θe , 2l l! 4π 69 where the (−1)l is a matter of convention. The other eigenfunctions could in principle be obtained by successively applying the ladder ˆ operator L− to this. We will state without proof the form of a general eigenfunction. For m ≥ 0, 2l + 1 (l − m)! Ylm (θ, φ) = (−1)m Plm (cos θ)eimφ , 4π (l + m)! where Plm (cos θ) is a special function known as the associated Legendre function. It can be calculated from the Legendre polynomial Pl (u) using the expression dm Plm (cos θ) = sinm θ Pl (u), where u = cos θ. dum The spherical harmonics with m < 0 are deﬁned by Yl,−m (θ, φ) = (−1)m [Ylm (θ, φ)]∗ (the (−1)m is once again a matter of convention). It is also useful to note the inversion symmetry Ylm (π − θ, φ + π) = (−1)l Ylm (θ, φ). Notice that (π −θ, φ+π) is the point on the sphere antipodal to (θ, φ). If we have wavefunctions of the form ψ(r, θ, φ) = R(r)Ylm (θ, φ), then this means they have the symmetry ψ(−x) = (−1)l ψ(x). The functions Ylm (θ, φ) are collectively called the spherical harmonics. They form an or- thonormal set, [Ylm (θ, φ)]∗ Yl m (θ, φ)dΩ = δll δmm , S2 and they are complete, in the sense that any square-integrable function on the sphere can be written as an expansion ∞ ∞ ψ(θ, φ) = clm Ylm (θ, φ), l=0 m=−∞ where clm = [Ylm (θ, φ)]∗ ψ(θ, φ)dΩ. S2 In this context they are important not just in quantum mechanics but in other areas of applied maths where we try to represent functions of orientation or solve equations with spherical symmetry. 6.4 Appendix: angular momentum operators in sperical coordinates Here we show how the angular momentum operators can be written in terms of spherical polar ˆ ˆ ˆ ˆ coordinates. We work in terms of the vector of operators L = (Lx , Ly , Lz ) and write ˆ ¯ h Lψ = x × ψ. i 70 Write the gradient using spherical polar coordinates ∂ψ 1 ∂ψ 1 ∂ψ ψ= er + eθ + eφ , ∂r r ∂θ r sin θ ∂φ where er , eθ and eφ are respectively the unit vectors in the radial, polar and azimuthal direc- tions. Writing x = rer the cross product can be computed as er eθ eφ r 0 0 x× ψ = ∂ψ 1 ∂ψ 1 ∂ψ ∂r r ∂θ r sin θ ∂φ 1 ∂ψ ∂ψ = − eθ + eφ . sin θ ∂φ ∂θ Then ˆ ˆ Lx ψ = ex · Lψ ¯ h ∂ψ 1 ∂ψ = ex · e φ − ex · e θ i ∂θ sin θ ∂φ ¯ h ∂ψ 1 ∂ψ = (− sin φ) − cos φ cos θ i ∂θ sin θ ∂φ ∂ ∂ h = i¯ sin φ + cos φ cot θ ψ ∂θ ∂φ ˆ ˆ and the expressions claimed for the other components Ly and Lz can be shown similarly. For ˆ L2 we smiply square and add, giving ˆ ˆ ˆ ∂2 ∂ ∂2 ∂2 L2 + L2 + L2 = −¯ 2 x y z h + cot + cot2 θ 2 + 2 ∂θ2 ∂θ ∂φ ∂φ ∂2 ∂ ∂2 = −¯ 2 h + cot + cosec2 θ 2 ∂θ2 ∂θ ∂φ 2 h ¯ ∂ ∂ ∂2 = − 2 sin θ sin θ + 2 = h 2 Λ2 ¯ sin θ ∂θ ∂θ ∂φ as required. 71 7 Higher-dimensional problems o We will now look at how we might tackle the Schr¨dinger equation h2 ¯ 2 − + V (x) ψ(x) = Eψ(x) 2M in higher dimensions. In general this is a diﬃcult, even intractable problem, but we are lucky that many important examples have enough symmetry that they are completely solvable. Our ultimate goal in this module is to be able to solve Hydrogen, but we will begin by tackling slightly more general problems. Our approach here will be to use the method of separation of variables, which we have al- ready partially covered in the previous chapter. The three-dimensional problem with rotational symmetry is the most important one for obvious reasons, but it is informative to begin with problems separable in cartesian coordinates. 7.1 Separation in cartesian coordinates The method of separation of variables is simplest in cartesian coordinates. It works when we have a potential of the form V (x, y, z) = u(x) + v(y) + w(z). This is obviously a fairly special condition but one important problem for which we have this is the three-dimensional harmonic oscillator 1 1 V (x, y, z) = kr2 = k(x2 + y 2 + z 2 ). 2 2 Let’s illustrate what happens in this case. We look for eigenfunctions of the form ψ(x, y, z) = X(x)Y (y)Z(z). o On substituting into the Schr¨dinger equation in which the Laplacian is written in cartesian coordinates 2 ∂2 ∂2 ∂2 = + 2+ 2 ∂x2 ∂y ∂z we get an equation h2 ¯ − (X Y Z + XY Z + XY Z ) + (u(x) + v(y) + w(z)) XY Z = EXY Z 2M which on dividing by XY Z can be separated on the left into a function of x plus a function of y plus a function of z, h2 ¯ h2 ¯ h2 ¯ − X + u(x)X − Y + v(y)Y − Z + w(z)Z 2M + 2M + 2M = E. X Y Z 72 The only way a function of x, a function of y and a function of z can add to give a constant is if they are each individually constant. We therefore deduce that h2 ¯ − X + u(x)X 2M = Ex X 2 ¯ h − Y + v(y)Y 2M = Ey Y 2 ¯ h − Z + w(z)Z 2M = Ez Z where Ex + Ey + Ez = E. We have therefore replaced the three-dimensional problem with three separate one-dimensional problems. In the case of the three-dimensional harmonic oscillator we have already solved the one- dimensional problems in Chapter 5. We ﬁnd that the separation constants must be of the form 1 1 1 Ex = n + ¯ hω, Ey = m + hω ¯ and Ez = l + ¯ hω, 2 2 2 where n, m and l are independent quantum numbers, each running from 0 to ∞. The corre- sponding eigenfunctions are X(x) = ϕn (x), Y (y) = ϕm (y), and Z(z) = ϕl (z). The three-dimensional eigenfunctions ψnml (x, y, z) = ϕn (x)ϕm (y)ϕl (z) are therefore labelled by three separate quantum numbers and the corresponding energies are 1 1 1 Enml = n+ ¯ hω + m + ¯ hω + l + ¯ hω 2 2 2 3 = n+m+l+ ¯ hω. 2 Notice that these levels are degenerate, a possibility we that was excluded in one-dimensional h problems. For example, there are six ways to get the energy E = (2 + 3/2)¯ ω = (7/2)¯ ω: we h could have (n, m, l) = (2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0), (1, 0, 1) or (0, 1, 1) and each of these has a diﬀerent eigenfunction ψnml (x, y, z). Any linear combination of them such as 1 ψ(x, y, z) = √ (ψ110 (x, y, z) + ψ101 (x, y, z) + ψ011 (x, y, z)) 3 h will also be an eigenfunction. The energy level (7/2)¯ ω therefore deﬁnes a 6-dimensional vector space of eigenfunctions. 73 7.2 Central potentials in two dimensions In two dimensions the Laplacian is expressed in polar coordinates (r, θ) as 2 1 ∂ ∂ 1 ∂2 = r + 2 2. r ∂r ∂r r ∂θ For central potentials V (r) we look for solutions of the form ψ(r, θ) = R(r)Θ(θ). Plugging in and separating into functions of r and functions of θ gives 1 d dR 2M (E − V (r)) r + R Θ r dr dr h2 ¯ = . 1 Θ R r A function of r can only equal a function of θ if they are both equal to the same constant, λ say. The resulting angular equation Θ (θ) = λΘ(θ) has solutions subject to the boundary condition Θ(θ + 2π) = Θ(θ) which are of the (unnormalised) form Θ(θ) = eimθ where m is an integer. These angular functions are the two-dimensional equivalent of the spherical harmonics. Compare the two dimensional equation λ = m2 with its three-dimensional equivalent λ = l(l + 1). Here we are more interested in the radial equation 1 d dR 2M (E − V (r)) m2 r + R= R r dr dr h2 ¯ r or 1 2M (E − V (r)) m2 R (r) + R (r) + − 2 R(r) = 0 r h2 ¯ r o As in the three-dimensional case, this is similar to the one-dimensional Schr¨dinger equation with an eﬀective potential m2 h 2 ¯ Veﬀ (r) = V (r) + . 2M r2 74 The solution, of course, depends on the details of the potential V (r). Let’s try a particular example. Example A particle in a two-dimensional circular cavity. This is a two-dimensional analog of the particle-in-a-box problem we solved in Chapter 3. We might deﬁne the central potential 0 for r < a, V (r) = ∞ for r > a. o Alternatively we write out the Sch¨dinger equation for a free particle in two dimensions and impose the boundary condition ψ(a, θ) = 0. Denoting 2M E k2 = and x = kr h2 ¯ the radial equation can be written as d2 R 1 dR m2 + + 1− 2 R=0 dx2 x dx r and this is recognised as Bessel’s equation. The general solution is R(r) = AJm (kr) + BYm (kr), where A and B are constants and Jm (x) and Ym (x) are Bessel functions in the usual notation. To avoid singularities in the wavefunction at r = 0 we demand B = 0. The boundary condition at r = a then gives Jm (ka) = 0. For each m the zeros of the Bessel function — xnm with n = 1, 2, · · · — are tabulated in reference books or readily available from numerical packages. We therefore get a solution to o the Schr¨dinger equation if xnm k = knm = a and we get the corresponding energy levels h2 knm ¯ 2 h 2 x2 ¯ nm Enm = = . 2M 2M a2 The corresponding eigenfunctions are of the form ψnm (r, θ) = CJm (knm x)eimθ , where C is a normalisation constant. 75 76 ¯ h2 (35) u(r) u (r) + 2M (E − Veﬀ (r)) Substituting this into the radial equation leads to the equation u(r) = rR(r). Deﬁne the function lem on the half-line r > 0 with hard-wall boundary conditions ar r = 0. Figure 10: The three-dimensional radial equation can be mapped onto a one-dimensional prob- r=0 ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢¡¡¡¡¡¡¡¡¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡¡¡¡¡¡¡¡¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢¡¡¡¡¡¡¡¡¡¢ r ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢¡¡¡¡¡¡¡¡¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ ¢ eff ¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¢ V (r) ¢¡¡¡¡¡¡¡¡¡ ¢ ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡¡¡¡¡ ¢ We can make this analogy more precise as follows. 2M r 2 Veﬀ (r) = V (r) + . h l(l + 1)¯ 2 potential We have already remarked that this looks like a one-dimensional equation with the eﬀective r ¯ h2 r2 R (r) + R (r) + − R = 0. 2 2M (E − V (r)) l(l + 1) or r 2 dr dr ¯ h2 r2 r + − R=0 1 d 2 dR 2M (E − V (r)) l(l + 1) where the Ylm ’s are the spherical harmonics and R(r) satisﬁes the radial equation ψ(r, θ, φ) = R(r)Ylm (θ, φ) equation are of the form o We have already established in the previous chapter that separated solutions of the Schr¨dinger Central potentials in three dimensions 7.3 which is precisely the equation we had in one dimension. The boundary conditions are novel however. Notice that u(r) R(r) = r is singular at r = 0 unless u(r) → 0 as r → 0. Imposing this boundary condition, and solving (35) on the half-line 0 < r < ∞ is equivalent to solving the problem of a particle moving in one dimension under the inﬂuence of the potential Veﬀ (r) for r > 0, v(r) = ∞ for r < 0. Alternatively it is the problem of a particle moving on the half-line 0 < r < ∞ under the potential Veﬀ (r) and with a hard wall at r = 0 (Figure 10). The normalisation conditions are also the natural ones we would have for a one-dimensional particle ∞ 2 ψ = |ψ|2 r2 drdΩ 0 S2 ∞ = |R(r)|2 r2 dr · |Ylm (θ, φ)|2 dΩ 0 S2 ∞ = |u(r)|2 dr. 0 In addition, the probability that the particle is somewhere in the shell deﬁned by a < r < b is b P (r ∈ (a, b)) = |u(r)|2 dr a so |u(r)|2 is a kind of radial probability density. Example Spherical solutions of the three-dimensional harmonic oscillator. If 1 V (r) = kr2 2 then the spherically symmetric eigenstates with l = 0 lead to the equation 2M 1 u + 2 E − kr2 u = 0 h ¯ 2 with the boundary conditions u → 0 as r → 0, ∞. o Except for the boundary condition at r = 0, we recognise here the Schr¨dinger equation for the harmonic oscillator, which we have already solved. 77 Imposing u → 0 as r → 0 means that we select the solutions of the harmonic oscillator with odd values of the quantum number n, for which the eigenfunctions are odd. We have explicitly √ 2 mω 1/4 −ξ2 /2 un (r) = √ e Hn (ξ), n = 1, 3, 5, · · · , 2n n! π¯ h √ where ξ = r mω/¯ . There is an extra 2 because the normalisation condition is diﬀerent to h the one used in Chapter 5 in that it involves integrating over the half-line only. In particular the ground-state wavefunction is 1 ψ0 = u1 (r)Y00 (θ, φ) r 1 mω 1/4 −mωr2 /(2¯ ) h mω 1 = e 2r √ r π¯h ¯ h 4π mω 3/4 −mωr2 /(2¯ ) h = e , h π¯ a simple Gaussian. Let’s verify that it is normalised: ∞ 2 ψ0 = |ψ0 |2 r2 drdΩ 0 S2 ∞ = 4π |ψ0 |2 r2 dr 0 mω 3/2 ∞ 2 /¯ h = 4π e−mωr r2 dr h π¯ 0 4 ∞ 2 = √ e−ξ ξ 2 dξ π 0 = 1. The corresponding ground-state energy is 1 3 E0 = 1 + ¯ ¯ hω = hω. 2 2 Notice also that using r 2 = x2 + y 2 + z 2 in the exponential allows us to write ψ0 (x, y, z) = ϕ0 (x)ϕ0 (y)ϕ0 (z) where ϕ0 (x) is the ground state of the one-dimensional harmonic oscillator as calculated in Chapter 5. The ground state therefore also coincides with the eigenfunction denoted by ψ000 (x, y, z) in section 7.1. In that case we found the same energy but explained it diﬀerently — as the sum of the ground state energies for the x, y and z degrees of freedom 1 1 1 3 E0 = hω + hω + hω = hω. ¯ ¯ ¯ ¯ 2 2 2 2 The end result is the same however. 78 7.4 Hydrogen We will adopt a model for Hydrogen in which an electron of mass me and charge −e orbits a ﬁxed nucleus of charge Ze and where the force of attraction is derived from the Coulomb potential Ze2 V (r) = − . [4πε0 ]r For Hydrogen proper Z = 1, but it is simple to allow for the possibilities Z = 2, 3, · · · which would describe ions He+ , Li++ and so on, which are “hydrogen-like ions”. The model of a ﬁxed centre is a bit of a simpliﬁcation because, while heavy compared to the electron, nuclei do have ﬁnite masses and can move around. It turns out that a slight modiﬁcation of the calculation we are about to perform allows this eﬀect to be incorporated without approximation, but a description of how this is done would be too much of a diversion so we will simply assert a centre of inﬁnite mass. The factor in square brackets is present if we use SI units but is absent in cgs units, SI units are easier to compare with experiments but cgs units are better in theoretical work because they lead to simpler equations with fewer factors of 4π and suchlike. When we deal with atomic problems it is usual in fact to make an even stronger simpliﬁcation. In natural units we express mass in units of the electron mass me , distances in units of the Bohr radius a0 and energy in units of the quantity E0 = e2 /([4πε0 ]a0 ) deﬁned in the ﬁrst chapter. This is equivalent to saying that we choose units in which h = e2 /[4πε0 ] = me = 1. ¯ o It means that we can simplify the Schr¨dinger equation from h2 ¯ 2 Ze2 − ψ− ψ = Eψ 2me [4πε0 ]r to 1 2 Z − ψ− ψ = Eψ. 2 r In natural units the Bohr energies are 1 En = − . 2n2 Whenever we get an answer like this, appearing to give a physical answer as a dimensionless quantity, in order to express the result in more familiar units we should simply remember that the dimensionless number gives us the answer as a multiple of some fundamental unit, E0 in the case of energy, a0 in the case of distance etc. As we did for a general central potential in three dimensions, we look for an eigenfunction of the form u(r) ψ(r, θ, φ) = Ylm (θ, φ). r 79 The radial equation for u(r) is Z l(l + 1) u (r) + 2 E + − u(r) = 0. r r2 Introduce the new energy-controlling parameter Z n= √ −2E (remember bound states have E < 0 and we hope to get the answer E = −1/(2n2 ) with integer n for Z = 1) and the new radial coordinate 2Z ρ= r. n Then the radial equation is d2 u 1 n l(l + 1) 2 + − + − u = 0. dρ 4 ρ ρ2 We will perform one more transformation to turn this equation into a standard form. To justify a little why the transformation is chosen as it is, lets us consider a couple of limits. The limit ρ → 0. In this limit the leading terms in the radial equation are d2 u l(l + 1) − u = 0, dρ2 ρ2 for which the solution behaves as u ∼ ρl+1 (there is also a solution ρ−l that we eliminate because it diverges). The limit ρ → ∞. In this limit the leading terms in the radial equation are d2 u 1 − u = 0, dρ2 4 for which the solution behaves as u ∼ e−ρ/2 . In this case we discard an exponentially growing solution if u is to be square-integrable. We make a transformation which incorporates both of these limits. We deﬁne w(ρ) by u = ρl+1 e−ρ/2 w. When the radial equation is worked out in terms of w we get ρw (ρ) + (2(l + 1) − ρ)w (ρ) + (n − l − 1)w(ρ) = 0. 80 This equation is “well-known” and is called Laguerre’s equation. We can write simple solutions of this equation where n and l are integers. For example, w=1 for n = l + 1 and l = 0, 1, 2, · · · and w =2−ρ for n = 2 and l = 0. We will ﬁnd that the most general solution which leads to a square-integrable wavefunction occurs when n and l are integers and w(ρ) is a polynomial of degree n − l − 1. Look for a solution in series form ∞ w(ρ) = ak ρk . k=0 The individual terms in Laguerre’s equation are ∞ ∞ ρw (ρ) = k(k − 1)ak ρk−1 = (k + 1)kak+1 ρk k=0 k=0 ∞ ∞ (2l + 2)w (ρ) = 2(l + 1)kak ρk−1 = 2(l + 1)(k + 1)ak+1 ρk k=0 k=0 ∞ −ρw (ρ) = −kak ρk k=0 ∞ (n − l − 1)w(ρ) = (n − l − 1)ak ρk . k=0 Adding up and equating coeﬃcients gives [k + 2(l + 1)](k + 1)ak+1 + (n − l − 1 − k)ak = 0 (36) or ak+1 1+l+k−n = ak (k + 1)[k + 2(l + 1)] Notice that ak+1 1 ak k for ﬁxed l and n as k → ∞, which is the ratio we ﬁnd in the Taylor series of e ρ . From this one can show that w(ρ) const. × eρ unless the series terminates. This would lead to a u = e−ρ/2 w const. × eρ/2 that diverges as ρ → ∞ and is nonintegrable We therefore arrive at the conclusion that the only solutions leading to square-integrable wavefunctions are those for which the series terminates. This happens when k = n − l − 1. 81 From this we deduce that n is an integer which is strictly greater than l. We think of the terminating value of k as a quantum number and denote it by N . By convention this is called the radial quantum number and counts the zeros in the radial part of the wavefunction. We ﬁnd solutions with N = 0, 1, 2, · · · and we denote n = 1 + l + N, which may take the values n = 1, 2, 3, · · · as the principle quantum number. In summary we have found that the eigenvalues are Z2 En = − 2n2 where n is a positive integer. These eigenvalues are generally quite degenerate. For a ﬁxed n we can write eigenfunctions ψnlm for all values of l and m for which l = 1, 2, · · · , n − 1 and − l ≤ m ≤ l. For the ground state with n = 1 there is a single such state, with l = m = 0 For n = 2 we may take l = 0 or 1 and these have 1- and 3-fold degenerate spherical harmonics respectively. So the ﬁrst excited state is 4-fold degenerate. The degeneracy increases steadily with n in this way. For given values of n and l, we can easily generate the polynomial w(ρ) using (36). We will not discuss in great detail how the properties of these polynomials are developed but we will quote some useful results. The eigenfunctions can be written ψ(ρ, θ, φ) = ρl e−ρ/2 L2l+1 (ρ)Ylm (θ, φ) n+l where dq Lq (ρ) = p Lp (ρ) dρq and the Legendre polynomials Lp (ρ) are generated by dp p −ρ Lp (ρ) = eρ ρe . dρp Example: The wavefunction for states with n = l + 1 is of the form ψnlm (ρ, θ, φ) = const. × ρn−1 e−ρ/2 Ylm (θ, φ) = const. × e−ρ/2+(n−1) ln ρ Ylm (θ, φ). Along a radial line, this is greatest when d ρ 0 = − + (n − 1) ln ρ dρ 2 n−1 1 = − ρ 2 82 or ρ = 2(n − 1). In the proper radial coordinate this is n n(n − 1) r= ρ= . 2Z Z When n is large, this tells us that the wavefunction is greatest on a shell of radius r ≈ n 2 /Z or, in more general units, the faction n2 a0 /Z of the Bohr radius. Therefore the answer has elements of the Bohr model in it but of course presents a much more sophisticated and complete picture. These solutions are of such fundamental importance in atomic theory that special notation, called spectroscopic notation is in common usage. Letters are used to denote the lower values of l; S for l = 0, P for l = 1, D for l = 2 and other letters for larger l. We might write for example ψ2S = const. × (2 − ρ)ρe−ρ/2 for a state with l = 0 and n = 2. The ﬁrst few states are enumerated in the table below. n Possible angular degeneracies total degeneracy momenta and states 1 S ψ1S 1 1 2 S ψ2S 1 P ψ2P 3 4 3 S ψ3S 1 P ψ3P 3 D ψ3D 5 9 For each n the S-states are spherically symmetric states depending only on r. The P states are triply degenerate since the angular part of the wavefunction can be chosen to be Y 1,−1 (θ, φ), Y10 (θ, φ) or Y11 (θ, φ) — or an arbitrary linear combination of all three. We are free to choose cos θ ψ2P = const. × ρe−ρ/2 × sin θ cos φ sin θ sin φ, z = const. × e−ρ/2 × x y, 83 for example. Five states could be labelled 3D, each of the form, ψ3D = const. × ρ2 e−ρ/2 × Y2m (θ, φ) and these could be written as an exponential of the radial coordinate multiplied by a quadratic form in (x, y, z). The degeneracies and shape of these lower eigenfunctions play a very important role in chemistry. 84