Shankar – Principles of Quantum Mechanics

Here are my notes and solutions to various problems in R. Shankar’s textbook Principles of Quantum Mechanics, Second Edition. Obviously I can’t offer any guarantee that all the solutions are actually correct, but I’ve given them my best shot.

Shankar’s book covers quantum mechanics at a level somewhat higher than that of Griffiths’s Introduction to Quantum Mechanics, and it’s my hope that it will provide the necessary background for tackling quantum field theory.

Although I’ve found the book very well written and for the most part easy to follow (Shankar’s style is similar to that of Griffiths, so is quite informal), some of the older printings of the book have a large number of typos in them, so do beware. Some of the typos are just misspelled or omitted words in the text, but some are also in the equations.

These solutions are the only ones that I’ve worked out so far, so please don’t ask me to post “the rest of the chapters” as I haven’t worked on those yet. I will get to them eventually.

An errata list is available here, but this seems to apply only to later, corrected printings, so if you have an older printing, you may have to fend for yourself in detecting typos.

Chapter 1 – Mathematical Introduction

1.1.1, 1.1.2, 1.1.3, 1.1.4, 1.1.5, 1.3.1, 1.3.2, 1.3.3, 1.3.4, 1.4.1, 1.4.2, 1.6.1, 1.6.2, 1.6.3, 1.6.4, 1.6.5, 1.6.6, 1.7.1, 1.7.2, 1.8.1, 1.8.2, 1.8.3, 1.8.4, 1.8.5, 1.8.6, 1.8.7, 1.8.8, 1.8.9, 1.8.10, 1.8.11, 1.8.12, 1.9.1, 1.9.2, 1.9.3, 1.10.1, 1.10.2, 1.10.3, 1.10.4

Hadamard’s lemmaBaker-Campbell-Hausdorff formulaDifferential operators – matrix elements and hermiticityDifferential operator – eigenvalues and eigenstatesDirac delta function as limit of a gaussian integral

Chapter 2 – Review of Classical Mechanics

2.1.1, 2.1.2, 2.1.3, 2.3.1, 2.5.1, 2.5.2, 2.5.3, 2.5.4, 2.7.1, 2.7.2, 2.7.3, 2.7.4, 2.7.5, 2.7.6, 2.7.72.7.8 (1-3), 2.7.8 (4), 2.7.9, 2.8.1, 2.8.2, 2.8.3, 2.8.4, 2.8.5, 2.8.6, 2.8.7

Electromagnetic LagrangianHamiltonian for the electromagnetic force

Chapter 4 – The Postulates – a General Discussion

4.2.1, 4.2.2, 4.2.3

Schrödinger equation and propagators, Time-dependent propagators

Chapter 5 – Simple Problems in One Dimension

5.1.1, 5.1.2, 5.1.3, 5.2.1, 5.2.2(a)5.2.2(b), 5.2.3, 5.2.45.2.55.2.6 (even), 5.2.6 (odd), 5.3.1, 5.3.2, 5.3.3, 5.3.4, 5.4.2(a), 5.4.2(b)

Free particle: Gaussian wave packet

Chapter 6 – The Classical Limit

Classical limit of quantum mechanics; Ehrenfest’s theorem

Chapter 7 – The Harmonic Oscillator

7.3.1, 7.3.2, 7.3.3, 7.3.4, 7.3.57.3.6, 7.3.7, 7.4.1, 7.4.2, 7.4.37.4.4, 7.4.5, 7.4.6, 7.4.7. 7.4.87.4.9, 7.4.10, 7.5.1, 7.5.2, 7.5.3, 7.5.4

Zero-point energy from uncertainty principle

Chapter 8 – The Path Integral Formulation of Quantum Theory

8.6.1, 8.6.2, 8.6.3, 8.6.4

Introduction & free particle propagatorFree particle propagator from a complete path integralEquivalent to the Schrödinger equation

Chapter 9 – The Heisenberg Uncertainty Relations

9.4.1, 9.4.2, 9.4.3, Uncertainty principle

Chapter 10 – Systems With N Degrees of Freedom

10.1.1, 10.1.2, 10.1.3, 10.2.1, 10.2.2, 10.2.3, 10.3.1, 10.3.2, 10.3.3, 10.3.410.3.510.3.6

Chapter 11 – Symmetries and their Consequences

11.2.1, 11.2.2, 11.2.3, 11.4.1, 11.4.2, 11.4.3, 11.4.4

Translation operator from passive transformationsCorrespondence between classical and quantum transformationsTranslational invariance and conservation of momentumTime translation and conservation of energyTime reversal, antiunitary operators and Wigner’s theorem

Chapter 12 – Rotation Invariance and Angular Momentum

12.1.1, 12.2.1, 12.2.2, 12.2.3, 12.2.4, 12.3.1, 12.3.2, 12.3.3, 12.3.4, 12.3.5, 12.3.612.3.7 (1)-(5), 12.3.7 (6)-(7), 12.3.7 (8)-(10), 12.3.8, 12.4.1, 12.4.2, 12.4.3, 12.4.4, 12.5.1, 12.5.2, 12.5.3, 12.5.4, 12.5.5, 12.5.6, 12.5.7, 12.5.8, 12.5.9, 12.5.10, 12.5.11, 12.5.12, 12.5.13, 12.5.14