VIEWS: 35 PAGES: 24 POSTED ON: 4/11/2011
The Interaction of Light and Matter Screen Metal plate Time Wire Wave-Particle Duality Today, we know (have shown) that elementary particles (e.g., protons, electrons, neutrons), atoms, and molecules exhibit wave-like properties. This revolutionary understanding of the behavior of matter was first proposed in 1924 by the French physicist Louis de Broglie in his PhD thesis. Recall that, in 1905, Einstein used Planck’s idea that the energy of an electromagnetic wave is quantized in the manner whereby a quantum of energy is carried by a photon, to explain Louis de Broglie, the photoelectric effect. 1892-1987 Recall that from the Theory of Special Relativity, the energy and momentum of a photon are related by as had been verified in 1922 by Compton through the Compton effect. Wave-Particle Duality Purely from symmetry arguments, de Broglie proposed that all matter (from elementary particles to people, planets, stars, and entire galaxies) also exhibit wave-like properties such that their frequency and wavelength are given by In this view, the wave-particle duality applies to everything in the physical world: - everything exhibits its wave properties in its propagation - everything manifests its particle nature in its interactions Wave-Particle Duality In 1927, Clinton J. Davisson and Lester H. Germer directed a beam of electrons on a highly polished single crystal of nickel. The lines of atoms on the surface form a sort of diffraction grating for the electron waves with a spacing equal to the lattice spacing of the nickel crystal of 0.215 nm, about the size of an atom. Clinton J Davisson (1881-1958; left) and The electron beam they used had an energy of 54 eV, Lester H. Germer(1896-1971; right) corresponding to a de Broglie wavelength of 0.167 nm. The first-order (m=1) maximum should therefore occur at = sin-1 (/d) = 51o, in agreement with their measurements. Wave-Particle Duality Modern experiments such as that performed in 1989 by A. Tonomura and his colleagues at the Hitachi Advanced Laboratory and Gakushuin University in Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the double-slit experiment for light. Screen Metal plate Time Wire To produce an interference pattern, each electron must pass through both slits, thus revealing the electron’s wave-like properties. Wave-Particle Duality Similar experiments have been performed on other elementary particles as well as atoms. Wave-Particle Duality de Broglie reasoned that the quantization of orbital angular momentum in Bohr’s model of the atom is simply a manifestation of the wave-like nature of the electron. The circumference of an electron’s orbit must be equal to an integral number of wavelengths for the electron to undergo constructive interference. Otherwise, the electron will find itself out of phase and suffer destructive interference. Based on this consideration, one can show that the electron can only have angular momenta given by . Assignment question Probability Waves Quantum mechanics describe particles in terms of probability waves. Consider a particle that comprises the following probability wave, Ψ, a sine wave with a precise wavelength propagating along the x-direction Probability wave Ψ : (x,t) = 0 ei(kx-t) x where k = 2 / =2ν The momentum, p = h / , of a particle described by such a wave is known precisely as the wavelength is known precisely. The probability of finding the particle at a given location x is given by P(x) = * = [0 ei(kx-t)] [0 e-i(kx-t)] = |0| which is a constant independent of x or t. Thus, the particle can be found with equal probability at any point along the x-direction: its position is perfectly uncertain; i.e., a sinusoidal wave has no beginning or end. Probability Waves Consider now a particle that has a probability wave, Ψ, that is equal to the addition of several sine waves with different wavelengths Probability wave Ψ : x The position of such a particle can be determined with a greater certainty because P(x) = * is large only for a narrow range of locations. On the other hand, because Ψ is now a combination of waves of various wavelengths, the particle’s momentum, p = h / , is less certain. This is nature’s intrinsic tradeoff: the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δp, is inversely related. As one decreases, the other must increase. This fundamental inability of a particle to simultaneously have a well-defined position and a well-defined momentum is a direct result of the wave-particle duality of nature. Probability Waves and the Two-Slit Experiment Let us see how we can use the idea of particles as probability waves to explain the two-slit experiment for electrons. To produce a sharp interference pattern, we need to precisely control the wavelength, and therefore momentum, of the electron beam. Otherwise, no interference pattern is seen; i.e., if the incident wave has a range of wavelengths, constructive or destructive interference will occur at different locations at different wavelengths and hence the interference pattern smear out. If we precisely control the momentum and hence wavelength but not the position of the electrons, they can pass through either slits and interfere with each other. Heisenberg’s Uncertainty Principle In 1927, the German physicist Werner Heisenberg presented the theoretical framework for the inherent “fuzziness” of nature, showing that the uncertainty in a particle’s position, Δx, and the uncertainty in its momentum, Δt, is related by He also showed that the uncertainty of energy measurement, ΔE, and the time interval over which this measurement is taken, Δt, is Werner Heisenberg, 1901-1976 related by As the time available for an energy measurement increases, the inherent uncertainty in the result decreases. Thus, making more precise measurements require longer observing times. Quantum Mechanical Tunneling Because of the inherent uncertainty between a particle’s momentum and position, a particle can penetrate a barrier even though the particle’s energy is lower than the barrier potential. If you know the energy of the particle precisely (and it is smaller than the barrier potential), you do not know the position of the particle precisely and so it can be on either side of the barrier (provided the barrier width is sufficiently small). So, a (small) fraction of particles on one side of a barrier can tunnel through to the other side, an effect called quantum mechanical tunneling. Probability wave Ψ : x Quantum Mechanical Tunneling A corresponding effect is seen in a classical wave such as a water of light wave. When such a wave enters a medium through which it cannot propagate, its amplitude decays with distance in this medium: the wave becomes evanescent (fades away). If the barrier width is sufficiently small, the amplitude of the wave may not decay away completely before reaching the other side of the barrier, where the wave can once again propagate. Quantum Mechanical Tunneling and Stellar Nuclear Fusion Quantum mechanical tunneling plays an essential role in stellar nuclear fusion. The first step in converting hydrogen to helium is the fusion of two protons to produce deuterium (proton + neutron nucleus) along with a positron and a γ-ray photon. For two protons to overcome their Coulomb repulsion and approach each other sufficiently close for their strong nuclear force to bind them together, their required kinetic energy corresponds to a gas temperature of ~1010 K. Quantum Mechanical Tunneling and Stellar Nuclear Fusion The central temperature of the Sun is only 1.57 × 107 K. Even taking into consideration the fact that a significant number of particles have speeds well in excess of the average speed (i.e., particle speeds are distributed according to the Maxwell-Boltzmann distribution), not enough protons can overcome their Coulomb repulsion to produce the Sun’s observed luminosity. Quantum mechanical tunneling helps to overcome the Coulomb repulsion between protons for nuclear fusion to proceed. Heisenberg’s Uncertainty Principle and the Atom Heisenberg’s uncertainty principle implies that electrons cannot have well defined orbits as prescribed in Bohr’s model of the atom. Rather, the electron orbits must be imagined as fuzzy clouds of probability, with the clouds being mode “dense” in regions where the electron is more likely to be found. One of the predictions of such a model is that spectral lines cannot be infinitely sharp but must have a certain width (natural linewidth), as is indeed observed. Bohr’s model Quantum mechanical model for the hydrogen atom for the hydrogen atom Schrödinger’s Equation Motivated to find a proper wave equation for the electron, in 1926 the Austrian physicist Erwin Schrödinger formulated the wave equation now known as Schrödinger’s equation that describes how the Erwin Schrödinger, quantum state of a physical system changes in time. 1887-1961 The Schrödinger equation can be solved analytically for the hydrogen atom, giving exactly the same set of allowed energies as those obtained by Bohr. In addition to the principal quantum number n, Schrödinger found that two other quantum numbers, and , are required for a complete description of the electron orbitals such that the angular momentum where an is the orbital quantum number and n is the principle quantum number. For historical reasons related to how spectral lines were first designated, = 0, 1, 2, 3, 4, 5, etc. are referred to as s, p, d, g, f, h etc. Schrödinger’s Equation The spin magnetic quantum number, , yields the projection of the orbital angular momentum in a specified direction (z-axis). The z-component of the angular momentum vector, Lz, can only have values , with equal to any of the integers between and inclusive. Thus, the angular momentum vector can point in different directions. E.g., for n = 2, = 0, 1 and = −1, 0, +1. Quantum States of the Hydrogen Atom Quantum numbers and energies for the ground (n = 1) and first excited state (n = 2) of the hydrogen atom. In the absence of any preferred direction in space (e.g., as defined by an electric or magnetic field), different orbitals with the same principal quantum number n and therefore the same energy are said to be degenerate. The Zeeman Effect An electron in an atom will feel the effect of a magnetic field: the magnitude of this effect depends on the electron’s orbital motion (i.e., magnitude and orientation of the electron’s orbital angular momentum through the magnetic quantum number lll ) and magnetic field strength B. Electron orbitals with the same n and but different values therefore have (slightly) different energies. The splitting of spectral lines in the presence of a magnetic field is called the Zeeman effect. In the example shown, the three frequencies of the split line are given by The Zeeman Effect The Zeeman effect provides the only direct measure of magnetic field strengths in astrophysics. (There are several indirect methods to estimate magnetic field strengths). slit spatial dimension along slit λ Electron Spin More complicated splitting patterns of spectral lines by magnetic fields are sometimes seen, usually involving even number of unequally spaced spectral lines. This effect is called the anomalous Zeeman effect. In 1925, George Uhlenbeck, Samuel Goudsmit, and Ralph Kronig suggested a fourth quantum number that describes the spin of the electron. The electron spin is not a classical top-like rotation (although this is often drawn for visualization purposes) but a purely quantum effect. Each electron orbital (or quantum state) is therefore associated with four quantum numbers. How many electrons can occupy the same quantum state? In 1925, based on empirical knowledge of the properties of atoms, the Austrian theoretical physicist Wolfgang Pauli realized that no two electrons can share the same four quantum numbers, a rule of nature now known as the Pauli exclusion principle. In 1930, the Indian astrophysicist Subrahmanyan Chandrasekhar used the Pauli exclusion principle to show that degenerate electron pressure supports white dwarfs against collapse due to Wolfgang Pauli, 1869- 1955 their own gravity. Electron Spin In 1928, the English physicist Paul Dirac combined Schrödinger’s equation with Einstein’s theory of special relativity to produce a relativistic wave equation for the electron. Dirac’s solution - naturally included the spin of the electron - naturally explained Pauli’s exclusion principle as being applicable to all particles with spin of an odd integer times (such as electrons, protons, and neutrons) known collectively as fermions - particles (such as photons) that have an integral spin do not obey Pauli’s exclusion principle and are known as bosons - predicted the existence of antiparticles (identical to their corresponding particles except for their opposite electrical charges and magnetic moments) Dirac showed that the electron’s spin angular momentum S is a vector of constant magnitude with a projection along the z-axis of , with ms = ±½. The Complex Spectra of Atoms In summary, an electron in an atom is described by four quantum numbers - principal quantum number, n - orbital quantum number, - magnetic quantum number, = … - spin quantum number, ms = ±½ To further complicate matters, the nucleus also has a spin quantum number! In an atom/ion with a single electron, the only interaction is between the electron and the nucleus. The spectrum of such an atom/ion is hydrogen-like. In a multielectron atom, electrons not only interact with the nucleus but also with each other through their spins and orbits angular momenta; i.e., spin-spin interactions, orbit-orbit interactions, and spin-orbit interactions. The spectrum of multi-electron atoms is therefore much more complicated. To learn more, you may enroll in my course on the Interstellar Medium.