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PHYSICS 430 Lecture Notes on Quantum Mechanics

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PHYSICS 430 Lecture Notes on Quantum Mechanics 1 PHYSICS 430 Lecture Notes on Quantum Mechanics J. Greensite Physics and Astronomy Department San Francisco State University Fall 2003 Copyright (C) 2003 J. Greensite 2 CONTENTS Part I - Fundamentals 1. The Classical State Newton’s Laws and the Principle of Least Action. The Euler-Lagrange equations and Hamilton’s equations. Classical mechanics in a nutshell. The classical state. 2. Historical Origins of Quantum Mechanics Black-body radiation, the photoelectric effect, the Compton effect. Heisenberg’s microscope. The Bohr atom. De Broglie waves. 3. The Wave-like Behaviour of Electrons Electron diffraction simplified: the ”double-slit” experiment and its consequences. The wave equation for De Broglie waves, and the Born interpretation. Why an electron is not a wave. 4. The Quantum State How does the electron get from A to B? Representation of the De Broglie ”wave” as a point moving on the surface of a unit sphere. Functions as vectors, wavefunc- tions as unit vectors in Hilbert space. Bra-ket notation. The Dirac delta function. Expectation value < x > and Uncertainty ∆x in electron position. 5. Dynamics of the Quantum State Ehrenfest’s principle. Schrodinger’s wave equation. The momentum and Hamil- tonian operators. Time-independent Schrodinger equation. The free particle and the gaussian wavepacket. Phase velocity and group velocity. Motion of a particle in a closed tube. 6. Energy and Uncertainty Expectation value of energy, uncertainty of momentum. The Heisenberg Uncer- tainty Principle. Why the Hydrogen atom is stable. The energies of a particle in a closed tube. 7. Operators and Observations Probabilities from inner products. Operators and observables, Hermitian opera- tors. States of ”zero uncertainty”; the eigenvalue equation. Commutators, and the generalized Uncertainty Principle. 3 Part II - Exact Solutions 8. Rectangular Potentials Qualitative behavior of energy eigenstates. Why bound state energies are dis- crete, why unbound eigenstates oscillate. Step potentials. The finite square well. Tunnelling, resonance, and the Ramsauer effect. 9. The Harmonic Oscillator Motivation: the most important example in physics. Raising and lowering opera- tors; algebraic solution for the energy eigenvalues. Hermite polynomials. 10. Two Dimensions, Symmetry, and Degeneracy The Parity operator in one dimension. The particle in a square. The two- dimensional harmonic oscillator. The quantum corral. 11. The Spectrum of Angular Momentum Motion in 3 dimensions. Angular momentum operators, and their commutation relations. Raising and lower operators; algebraic solution for the angular momentum eigenvalues. Spherical harmonics. The rigid rotator, and the particle in a spherical box. 12. The Hydrogen Atom Series solution for energy eigenstates. The scale of the world. Part III - Aspects of Spin 13. Electron Spin Evidence for electron spin: the Zeeman effect. Matrix representation of spin angular momentum; Pauli spin matrices. Spin-orbit coupling as motivation to add angular momentum. 14. The Addition of Angular Momentum The general method. Atomic fine structure. 15. Identical Particles and the Periodic Table Bosons, Fermions, and the Pauli Exclusion Principle. The Hartree approximation for many-electron atoms. The Periodic Table. 16. Live Wires and Dead Stars The Kronig-Penney model of electron band structure. The free electron gas and the fermi energy. Degeneracy pressure, and the radii of neutron stars. 4 Part IV - Approximation Methods 17. Time-Independent Perturbation Theory 18. Time-Dependent Perturbation Theory Adiabatic, harmonic, and ”sudden” perturbations. 19. The WKB and Rayleigh-Ritz Approximations Wavefunctions of ”nearly classical” systems. Classical physics as a stationary phase condition. The variational approach for estimating the ground state of a system. 20. Scattering Theory Partial waves. The Born approximation Part V - Advanced Topics 21. Quantum Mechanics as Linear Algebra Review of vectors and matrices. Linear algebra in bra-ket notation. Linear algebra and Hilbert space. The x and p representations. The harmonic oscillator, square well, and angular momentum representations. Canonical quantization. Poisson brackets and commutators. 22. The EPR Paradox and Bell’s Theorem Entangled States. The EPR ”paradox”. Faster than light? Bell’s Theorem. 23. The Problem of Measurement Mixtures and pure states. The problem of measurement. Bohr and von Neu- mann interpretations. Bohm’s ”guiding waves”. The Many-Universe formulation. Decoherence and ”consistent histories” approaches. 24. Feynman Path-Integral Quantization The action approach to quantum theory. From Schrodinger equation to Feynman path integral. Propagators. Functional Derivatives. Classical physics as a stationary phase condition. 25. A Glimpse of Quantum Field Theory Particles as excited states of quantized fields. The quantization of sound. The quantization of light. Casimir’s effect, and the structure of the Vacuum. 5 APOLOGIA These are my lecture notes for Physics 430 and 431, written a number of years ago. They are still a bit incomplete: Chapters 19 and 20 remain to be written, and Chapter 23 is unfinished. Perhaps this year I will get around to it. It is likely that there are still many misprints scattered here and there in the text, and I will be grateful if these are brought to my attention. 6 Chapter 1 The Classical State In the first quarter of this century, it was discovered that the laws of motion formulated by Galileo, Newton, Lagrange, Hamilton, Maxwell, and many others, were inadequate to explain a wide range of phenomena involving electrons, atoms, and light. After a great deal of effort, a new theory (together with a new law of motion) emerged in 1924. That theory is known as quantum mechanics, and it is now the basic framework for understanding atomic, nuclear, and subnuclear physics, as well as condensed-matter (or ”solid-state”) physics. The laws of motion (due to Galileo, Newton,...) which preceded quantum theory are referred to as classical mechanics. Although classical mechanics is now regarded as only an approximation to quan- tum mechanics, it is still true that much of the structure of the quantum theory is inherited from the classical theory that it replaced. So we begin with a lightning review of classical mechanics, whose formulation begins (but does not end!) with Newton’s law F = ma. 1.1 Baseball, F = ma, and the Principle of Least Action Take a baseball and throw it straight up in the air. After a fraction of a second, or perhaps a few seconds, the baseball will return to your hand. Denote the height of the baseball, as a function of time, as x(t); this is the trajectory of the baseball. If we make a plot of x as a function of t, then any trajectory has the form of a parabola (in a uniform gravitational field, neglecting air resistance), and there are an infinite number of possible trajectories. Which one of these trajectories the baseball actually follows is determined by the momentum of the baseball at the moment it leaves your hand. However, if we require that the baseball returns to your hand exactly ∆t seconds after leaving your hand, then there is only one trajectory that the ball can follow. For a baseball moving in a uniform gravitational field it is a simple exercise to determine 7 8 CHAPTER 1. THE CLASSICAL STATE this trajectory exactly, but we would like to develop a method which can be applied to a particle moving in any potential field V (x). So let us begin with Newton’s law F = ma, which is actually a second-order differential equation d2x dV m = (1.1) dt2 − dx It is useful to reexpress this second-order equation as a pair of first-order equations dx p = dt m dp dV = (1.2) dt − dx where m is the mass and p is the momentum of the baseball. We want to find the solution of these equations such that x(t0) = Xin and x(t0 + ∆t) = Xf , where Xin and Xf are, respectively, the (initial) height of your hand when the baseball leaves it, and the (final) height of your hand when you catch the ball.1 With the advent of the computer, it is often easier to solve equations of motion numerically, rather than struggle to find an analytic solution which may or may not exist (particularly when the equations are non-linear). Although the object of this section is not really to develop numerical methods for solving problems in baseball, we will, for the moment, proceed as though it were. To make the problem suitable for a computer, divide the time interval ∆t into N smaller time intervals of duration ! = ∆t/N, and denote, for n = 0, 1, ..., N, tn t0 + n! , x ≡= x(t ) , p = p(t ) , n n n n (1.3) x0 = Xin , xN = Xf An approximation to a continuous trajectory x(t) is given by the set of points xn connected by straight lines, as shown in Fig. [1.1]. We can likewise approximate{ } derivatives by finite differences, i.e. dx x(t ) x(t ) x x n+1 − n = n+1 − n dt → ! ! ! "t=tn dp p(t ) p(t ) p p n+1 − n = n+1 − n dt → ! ! ! "t=tn d2x 1 dx dx ! dt2 " → ! ! dt " − ! dt " t=tn t=tn t=tn−1 1 (xn+1 xn) (xn xn 1) − − − (1.4) → ! ) ! − ! * 1We will allow these positions to be different, in general, since you might move your hand to another position while the ball is in flight. 1.1. BASEBALL, F = MA, AND THE PRINCIPLE OF LEAST ACTION 9 and integrals by sums N 1 t0+∆t − dt f(t) !f(tn) (1.5) t0 → + n,=0 where f(t) is any function of time. As we know from elementary calculus, the right hand side of (1.4) and (1.5) equals the left hand side in the limit that ! 0, which is also known as the continuum limit. → We can now approximate the laws of motion, by replacing time-derivatives in (1.2) by the corresponding finite differences, and find pn xn+1 = xn + ! - m .
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