Physics 6320/7320 August 2009
Quantum Mechanics Fundamentals
(Ref. Zettili §§3.1-3.5, Shankar Ch.4) Math background: Shankar Ch.1
Introduction. Quantum mechanics is a very successful theory: whenever its predictions have been tested by experiment, they have always been found to be correct. Quantum mechanics is the most fundamental theory of nature. It is essential for understanding most current research in physics, particularly condensed matter, molecular, atomic, nuclear, and elementary-particle physics. Furthermore, classical Newtonian mechanics can be derived from quantum mechanics as a special case. Nevertheless there is a considerable amount of variation in the way diﬀerent physicists teach and learn quantum mechanics compared to some other theoretical subjects such as classical mechanics and classical electrodynamics. Aside from purely subjective diﬀerences in how we like to talk about physics, there are three kinds of issues which should be mentioned here: (a) Relativity. The theory we will learn in this course is primarily the nonrelativistic theory. You will notice right away that time is treated as fundamentally diﬀerent from space. This is the standard theory which is used for most applications, with perhaps some modiﬁcation for relativistic eﬀects where needed. However, quantum mechanics and special relativity are combined successfully in quantum electrodynamics (electroweak interaction) and quantum chromodynamics (strong interaction). The current research frontier is the combination of quantum mechanics and general relativity (quantum theory of gravity) which is still an open question. (b) Measurement theory. Wavefunction collapse is a feature of standard quantum theory that has been widely misunderstood, leading to famous paradoxes such as “Schr¨dinger’s Cat.” Uno fortunately this has caused unnecessary concern about the foundation of quantum mechanics, and encouraged those who would like to discredit all of modern science. Fortunately recent research has shown that the process of measurement can be understood on the same basis as all other quantum processes, without a separate assumption of instantaneous collapse. After presenting the standard theory, we will give a brief outline of these developments, including the idea of decoherence. (c) The fundamental postulates. Most textbooks give a set of fundamental postulates, from which all quantum mechanics should be derived deductively. However there is no consensus on exactly which postulates are fundamental, which equations should be assumed, and which derived. Also it is not always clear which postulates are part of the mathematical formalism we have constructed, and which are actually assumptions about the physical world. We will start with a standard set of ﬁve formal postulates similar to those presented by Zettili in Chapter 3 or Shankar in Chapter 4. Once we have set up the standard theory and presented some examples, we will return to this subject, and discuss diﬀerent approaches. In particular it is possible to determine all the needed formal postulates from the single physical assumption that there are incompatible observables.
The formal postulates. We begin by listing ﬁve postulates which will set up the basic formalism. [Zettili §3.2] (a) States are represented by vectors. (b) Observables are represented by operators. (c) Possible values are given by eigenvalues. (d) Probabilities are given by expansion coeﬃcients. (e) Time development is given by the hamiltonian operator. Note Shankar has postulates I, II, III, IV; they are basically the same, except (c) and (d) are combined into III. Sakurai does not have a formal list, but discusses these things in Ch.1 §§1-4. (a) States are represented by vectors. The state of a system is represented by a vector |Ψ in a linear vector space S . S is sometimes called a Hilbert space, and |Ψ is sometimes called a ket vector, following Dirac, or sometimes a wavefunction. I prefer to call |Ψ a state vector , and S the state vector space. The main points about S are (1) for any two state vectors there is a complex-valued inner product φ | ψ , and (2) any linear combination of state vectors is also a state vector; that is, if | φ1 , | φ2 ∈ S and |Ψ = c1 | φ1 + c2 | φ2 , where c1 , c2 are any complex numbers, then it follows that |Ψ ∈ S . (b) Observables are represented by operators. Any physical observable A is represented by a linear hermitian operator A on S . Two crucial properties of hermitian operators are that their eigenvalues are real, and their eigenvectors span the space. Thus if A is an observable, any state vector can be written as a linear combination of eigenvectors of A. Also, since eigenvectors with diﬀerent eigenvalues are known to be orthogonal, this means we can ﬁnd an orthonormal basis consisting entirely of eigenvectors: |Ψ =
cn | an
where A | an = an | an
an | am
= δmn .
(c) Possible values are given by eigenvalues. All possible values of the observable A are given by the eigenvalues of the corresponding operator. Thus if the measurement gives the result a, the state of the system is known to be the eigenstate | a . If two observables A, B can be simultaneously measured, the corresponding operators must commute: AB | a, b = BA | a, b = ab | a, b . If the operators do not commute, AB = BA, there are no simultaneous eigenstates | a, b , and the observables are said to be incompatible: if one is measured the other is undetermined. (d) Probabilities are given by expansion coeﬃcients. If the system state |Ψ is not an eigenstate of an observable A, the result of a measurement of A is not predictable, except in terms of probabilities. Considering the expansion (1), which we know must exist, the probability that a measurement of A will give result an is given by Pn = |αn |2 . Note that if |Ψ is normalized ( Ψ = 1) then the probabilities sum to one, as they should. Ψ|Ψ =
c∗ cn am | an m
Pn = 1 .
(e) Time development is given by the hamiltonian operator. The time dependence of the state vector | Ψ(t) is given by the Schr¨dinger equation o i¯ h d | ψ(t) = H | ψ(t) . dt (3)
Here H is the hamiltonian operator, which corresponds to the total energy of the system.