PHYSICS 430 Lecture Notes on Quantum Mechanics

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PHYSICS 430 Lecture Notes on Quantum Mechanics

J. Greensite Physics and Astronomy Department

San Francisco State University Fall 2003

Copyright (C) 2003 J. Greensite

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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, wavefunctions 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.

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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.

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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.

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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.

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