## Sakurai Homework Solutions Chapter 2

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## Quantum Mechanics I (521-001, 16808)

Fall 2007 Mondays and Wednesdays from 17:30 to 18:45 in room 184

Students may ask me questions about quantum mechanics whenever they see me.

My cell and office phone numbers are 205 5448 and 277 5318.

The course is designed for graduate students in physics. Students should have be familiar with complex numbers, linear algebra, vector calculus, and the Fourier transform, and know something about differential equations.

The principal topics will be:

Basic Concepts and Principles

Dynamics

Angular Momentum

Symmetries and Conserved Quantities

The Hydrogen Atom

Approximation Methods

The textbook is *Modern Quantum Mechanics* (Revised Edition) by J. J. Sakurai (Addison Wesley Longman, 1994). Try to get the latest printing.

The bookstore will carry it but possibly at a high price and in an old printing, so I suggest ordering it elsewhere, such as **Amazon** who charge $115 with free shipping for a new copy. They also list **book-sellers** who charge $55 and up for a used copy and $60 and up for a new one.

A good supplementary textbook is * Lectures on Quantum Mechanics* by Gordon Baym. **Amazon** sells it new for $55.42 and lists **book-sellers** who charge $24.95 and up for used and new copies.

The grader is Mr. Stefan Maier; his e-mail address is ** smaier1@unm.edu**.

Lecture notes on chapter 1 of Sakurai.

Notes on the polarized-light demo that shows why quantum mechanics needs complex numbers.

Notes on matrix algebra. Wiki's description of the singular-value decomposition.

Notes on space-time translations.

Notes on birefringent crystals.

Notes on chiral molecules and polarized light.

Notes on Schroedinger's equation.

Uses of the uncertainty principle.

Dirac's delta function.

The Schroedinger & Heisenberg Pictures

Ehrenfest's theorem.

Bohr frequencies.

Notes on chapter 2 of Sakurai.

Notes on harmonic oscillators and coherent states.

Supersymmetric Quantum Mechanics

Harmonic Oscillators Are Ubiquitous

The Virial Theorem

Feynman's Path Integral

Path Integrals and the WKB Approximation

Path Integrals and Ground States

Particle in an Electromagnetic Field

Notes on chapter 3 of Sakurai.

Notes on Rotations

Notes on the Lie Algebra of the Rotation Group

Orbital Angular Momentum in Spherical Coordinates

Central Potentials

The Two-Body Problem

The Hydrogen Atom

The Hydrogen Atom in a Magnetic Field

Adding Angular Momenta

The 2-D Harmonic Oscillator

The 3-D Isotropic Harmonic Oscillator

First-Order Perturbation Theory and the Linear Stark Effect

Higher-Order Non-Degenerate Perturbation Theory

The Quadratic Stark Effect on the Ground State of Hydrogen

Higher-Order Perturbation Theory for a Degenerate Level

A More-Direct version of Degenerate Perturbation Theory

Isospin

The Variational Method

Fine Structure and the Spin-Orbit Effect

The Lorentz Group and the Dirac Matrices

Pages 261-273 of Dirac's Book

The Dirac Equation and the Magnetic Moment of the Electron

Spin Is Angular Momentum

Dirac's Hydrogen Atom

Invariances of the Dirac Equation

The Interaction Picture

The Time-Energy Uncertainty Principle, Fermi's Golden Rule, and Detailed Balancing

The Cubic Equation

Light and Atoms

Spontaneous Emission: Lifetime of the 2p state of atomic hydrogen

Ionization of atomic hydrogen

Classical currents make coherent states

Solutions to the non-HW problems of chapter 1.

In the following homework assignments, the "chapters" are those of the textbook by Sakurai.

Homework problems due Wednesday, September 5th: Problems 1-6 of chapter 1. Due to the two-day extension, this date is firm.

Feel free to ask for hints in class. Solutions to the problems of the first homework assignment.

Second homework assignment due Monday, September 17th: Problems 8, 10, 13, 14, 19, 21, & 23 of chapter 1.

Solutions to the problems of the second homework assignment.

Third homework assignment due Monday, October 1st: Problems 26, 30, & 32 of chapter 1 and 1, 3, 11, & 13 of chapter 2.

Solutions to the problems of the third homework assignment.

Fourth homework assignment due Wednesday, October 17th: Problems 15, 18, 19, & 36 of chapter 2 and 1, 2, 3, 4 of chapter 3.

Hint on problem 3.4: look at problem 1.7a.

Solutions to the problems of the fourth homework assignment.

_{y}in terms of J

_{+}and J

_{-}) and problem 3.20. Finally, do problem 7.1 and parts (a, b, c, & d) of problem 7.2 of Cohen-Tannoudji and this four-part solar-neutrino problem. Hints for C-T's problem 7.2. More hints for C-T's problem 7.2.

Solutions to the problems of the fifth homework assignment.

Sixth homework assignment due Monday, November 19th: Problems 3.15 (hint: use the formulas for the Y

_{1}

^{m}'s on page 451), 3.18 (hint), 5.1, 5.4, 5.8, & 5.18 of Sakurai and two special problems: special problem 6.1 & special problem 6.2. Hint for problem 6.2. Having been asked, I now defer the due date until Monday, November 19th. I plan to discuss isospin and the variational method on Monday, the 12th.

Solutions to the sixth set of homework problems.

Seventh homework assignment due Monday, December 3d: Special problem 7.1 : Compute the spin-orbit splitting of the 2p

_{3/2}and 2p

_{1/2}states of hydrogen. Special problem 7.2: Compute the total splitting between these two levels by using Dirac's theory as summarized by Charles G. Darwin's formula. Hint: Equation (98) of these notes may help. Do extra-credit special problem 7.3. Hint: The notes on solving cubic equations may help. Do special problem 7.4. Do problems 5.7, 5.13, & 5.30 of Sakurai.

Solutions to the seventh set of homework problems.

Eighth homework assignment: Special problem 8.1: Using equations (1, 2, 3, 4, & 7) of the notes on light and atoms, show that in the absense of charges and currents, the vector potential A satisfies the wave equation (12) of those same notes. Special problem 8.2: For atomic hydrogen, compute the matrix element <1, 0, 0| z | 2, 1, 0> of the 3rd component z of the position operator x between the ground state <1, 0, 0| and the 2p state | 2, 1, 0 > with m = 0. Make sure you get the answer I got in my notes on the 2p state of hydrogen. Special problem 8.3. Special problems 8.2 & 8.3 are due on Wednesday, December 12th; the other one (8.1) is due on Friday, December 12th, by 4 pm in Stefan Meier's mail or e-mail box.

Solutions to the last set of homework problems.

The video files of the lectures are 300 to 400 MB long despite the intrinsic compression of the wmv format. It is best to download them to your computer before trying to watch them.

The audio & video of lecture of 20 August was lost by an unnamed computer professional.

Video of lecture of 22 August

Video of lecture of 27 August

First video of lecture of 29 August

Second video of lecture 29 August

Video of lecture of Wednesday, 5 September 2007.

Video of lecture of Monday, 10 September 2007.

Video of lecture of Wednesday, 12 September 2007. The audio was lost by an unnamed computer professional.

Video of lecture of Monday, September 17th

Video of lecture of Wednesday, September 19th

Video of lecture of Monday, September 24th

Video of lecture of Wednesday, September 26th

Video of lecture of Monday, October 1st

Video of lecture of Wednesday, October 3d

Video of lecture of Monday, October 8th

Video of lecture of Wednesday, October 10th

Video of lecture of Monday, October 15th

Video of lecture of Wednesday, October 17th

Video of lecture of Monday, October 22d

Video of lecture of Wednesday, October 24th

Video of lecture of Monday, October 29th

Video of Hallowe'en lecture

Video of lecture of Monday, November 5th

Video of lecture of Wednesday, November 7th

Video of lecture of Monday, November 12th, on Clebsch-Gordan coefficients, isospin, and the variational method.

Video of lecture of Wednesday, November 14th, on the variational method and on spin-orbit coupling.

Video of lecture of Monday, November 19th, on the Dirac equation

Video of lecture of Wednesday, November 21st, on the application of the Dirac equation to the hydrogen atom

Video of lecture of Monday, November 26th, on the interaction picture, the Dyson expansion, and the time-energy uncertainty principle

Video of lecture of Wednesday, 28 November, on applications of time-dependent perturbation theory, including detailed balancing

Video of lecture of Monday, December 3d, on the interaction of photons with atoms

Video of lecture of Wednesday, December 5th, on the absorption and emission of photons by atoms

Video of lecture of Wednesday, December 12th, on the lifetime of the 2p state of atomic hydrogen and on the ionization of hydrogen-like atoms

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