Essentials of Quantum Mechanics and Relativity

299 Pages·2020·15.166 MB·English

by  Shangwu Qian

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Essentials of Quantum Mechanics and Relativity
Qian Shangwu
This edition is published by World Scientific Publishing Company Pte Ltd by arrangement with
Science Press, Beijing, China.
All rights reserved. No reproduction and distribution without permission.
ESSENTIALS OF QUANTUM MECHANICS AND RELATIVITY
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Preface
Thisbookabidesbytheless-is-moreprincipleandadoptsanapproachthat
is easy to understand. It introduces in a concise manner the essentials of
quantum mechanics and relativity, and along the way some related basic
mathematical knowledge.
This book contains two independent parts: the first is the essentials of
quantum mechanics. It is my lecture notes of the course4310.001,givenin
the1990fallsemester(14weeks)forseniorstudentsofPhysicsDepartment
of the University of North Texas (UNT). The second part is the essentials
ofrelativity,whichismylecturenotesofthe course4900,giveninthe1991
spring semester (14 weeks) for graduate students of Physics Department
of UNT. Both lectures received the highest student evaluations from the
students of Physics Department of UNT.
I am greatly indebted to Mr. Zhu Zhaoxuan ( ) for the arrange-
ment and preparationof the manuscripts based on my lecture notes of the
course 4310.001. Thanks very much for the financial support coming from
the School of Physics, Peking University.
This book is written mainly for the readers in the training and pro-
vides necessaryelementaryknowledgeofthe two pillarsofmodernphysics:
quantum mechanics and relativity.
Qian Shangwu
School of Physics, Peking University
v
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Contents
Preface v
Quantum Mechanics 1
1. Preliminaries 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 What is QM?. . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Why do we need QM? . . . . . . . . . . . . . . . . 4
1.1.3 How do QM explain atomic phenomena? . . . . . 5
1.2 Review of some basic concepts of CM. . . . . . . . . . . . 6
1.2.1 Generalized coordinates . . . . . . . . . . . . . . . 6
1.2.2 The Lagrange equations . . . . . . . . . . . . . . . 6
1.2.3 Hamilton’s equations . . . . . . . . . . . . . . . . 8
1.2.4 Poisson brackets . . . . . . . . . . . . . . . . . . . 9
1.3 Historical review: Experiments and theories . . . . . . . . 10
1.3.1 The work of Planck: Black body radiation . . . . 11
1.3.2 The work of Einstein: Photoelectric effect . . . . . 13
1.3.3 The work of Bohr: Hydrogen atom. . . . . . . . . 14
1.3.4 The work of Compton: Compton effect and the
dual nature of light . . . . . . . . . . . . . . . . . 16
1.3.5 The work of de Broglie: Matter waves . . . . . . . 20
1.3.6 The work of Schro¨dinger: Wave equation . . . . . 21
1.3.7 The work of Born: Probability interpretation . . . 25
1.3.8 The work of Heisenberg: Uncertainty
principle . . . . . . . . . . . . . . . . . . . . . . . 31
vii
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viii Essentials of Quantum Mechanics and Relativity
2. Postulates and One-Dimensional Problems 35
2.1 Postulates of quantum mechanics . . . . . . . . . . . . . . 35
2.1.1 Postulate 1: Quantum state . . . . . . . . . . . . 35
2.1.2 Postulate 2: Physical observable . . . . . . . . . . 38
2.1.3 Postulate 3: Time evolution of quantum
state . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.4 Postulate 4: Measurement . . . . . . . . . . . . . 44
2.1.5 Postulate 5: Expectation value . . . . . . . . . . . 44
2.2 Preliminary applications on one-dimensional
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 One-dimensional free particle . . . . . . . . . . . . 46
2.2.2 Linear box . . . . . . . . . . . . . . . . . . . . . . 47
2.2.3 Potential barrier of infinite width . . . . . . . . . 49
2.2.4 Tunneling effect . . . . . . . . . . . . . . . . . . . 51
2.2.5 Linear harmonic oscillator . . . . . . . . . . . . . 53
3. Matrix Mechanics 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 Matrix mechanics . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Transformation of representations . . . . . . . . . . . . . . 70
3.4.1 Transformation of basis . . . . . . . . . . . . . . . 70
3.4.2 Transformation of a state vector . . . . . . . . . . 71
3.4.3 Transformation of an operator . . . . . . . . . . . 71
3.5 Coordinate representation and momentum
representation . . . . . . . . . . . . . . . . . . . . . . . . . 71
4. Central Forces and Angular Momentum 75
4.1 Spherically symmetric potential . . . . . . . . . . . . . . . 75
4.1.1 Angular momentum . . . . . . . . . . . . . . . . . 75
4.1.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . 76
4.1.3 Separation of variables . . . . . . . . . . . . . . . 76
4.1.4 Solution of the angular equation . . . . . . . . . . 77
4.1.5 Associated Legendre equation: Regular point
and indicial equation . . . . . . . . . . . . . . . . 78
4.1.6 Spherical harmonics . . . . . . . . . . . . . . . . . 79
4.1.7 Solution of the Schro¨dinger equation . . . . . . . . 80
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Contents ix
4.2 Radial equation for the Coulomb potential . . . . . . . . . 81
4.2.1 Radial part of the wavefunction: Principle
quantum number. . . . . . . . . . . . . . . . . . . 81
4.2.2 Eigenstates of the hydrogen atom . . . . . . . . . 84
5. Summary, Problems and Solutions 87
5.1 A brief summary of this course . . . . . . . . . . . . . . . 87
5.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Quiz 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Quiz 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Final examination . . . . . . . . . . . . . . . . . . . . . . 97
5.6 Solutions of problems . . . . . . . . . . . . . . . . . . . . . 98
5.7 Solutions of Quiz 1 . . . . . . . . . . . . . . . . . . . . . . 111
5.8 Solutions of Quiz 2 . . . . . . . . . . . . . . . . . . . . . . 112
5.9 Solution of final examination . . . . . . . . . . . . . . . . 113
Relativity 117
6. Preliminaries 119
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1.1 What is the meaning of relativity in physics? . . . 119
6.1.2 How relativity governs the form of all
physical laws? . . . . . . . . . . . . . . . . . . . . 121
6.1.3 Why do we say quantum mechanics and special
relativity are the two pillars of modern
physics? . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Vectors and tensors . . . . . . . . . . . . . . . . . . . . . . 121
6.2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.3 Permutation symbol erst . . . . . . . . . . . . . . 123
6.2.4 The e–δ identity . . . . . . . . . . . . . . . . . . . 124
6.3 Principle of Galilean relativity . . . . . . . . . . . . . . . . 125
6.4 Essentials of electrodynamics . . . . . . . . . . . . . . . . 126
6.4.1 Electrostatics and magnetostatics . . . . . . . . . 126
6.4.2 Maxwell equations . . . . . . . . . . . . . . . . . . 130