CONTENTS
CHAPTER 1 Historical Review 1
Blackbody Radiation and Planck’s Formula 1
The Photoelectric Effect 6
The Bohr Theory of the Atom 7
de Broglie’s Hypothesis 10
Quiz 11
CHAPTER 2 Basic Developments 13
The Schrödinger Equation 13
Solving the Schrödinger Equation 18
The Probability Interpretation and Normalization 24
Expansion of the Wavefunction and Finding
Coefficients 35
The Phase of a Wavefunction 44
Operators in Quantum Mechanics 46
Momentum and the Uncertainty Principle 54
The Conservation of Probability 59
Quiz 63
CHAPTER 3 The Time Independent Schrödinger Equation 65
The Free Particle 66
Bound States and 1-D Scattering 74
Parity 88
Ehrenfest Theorem 95
Quiz 96
CHAPTER 4 An Introduction to State Space 99
Basic Definitions 99
Hilbert Space Definitions 100
Quiz 110
CHAPTER 5 The Mathematical Structure of Quantum
Mechanics I 111
Linear Vector Spaces 111
Basis Vectors 122
Expanding a Vector in Terms of a Basis 124
Orthonormal Sets and the Gram-Schmidt
Procedure 124
Dirac Algebra with Bras and Kets 125
Finding the Expansion Coefficients in the
Representation of Bras and Kets 127
Quiz 129
CHAPTER 6 The Mathematical Structure of Quantum
Mechanics II 131
The Representation of an Operator 133
Eigenvalues and Eigenvectors 142
The Hermitian Conjugate of an Operator 152
The Commutator 167
Quiz 172
CHAPTER 7 The Mathematical Structure of Quantum
Mechanics III 175
Change of Basis and Unitary Transformations 175
The Generalized Uncertainty Relation 185
Projection Operators 188
Functions of Operators 193
Generalization to Continuous Spaces 194
Quiz 203
CHAPTER 8 The Foundations of Quantum Mechanics 205
The Postulates of Quantum Mechanics 205
Spectral Decomposition 209
Projective Measurements 211
The Completeness Relation 212
Completely Specifying a State with a CSCO 220
The Heisenberg versus Schrödinger Pictures 221
Describing Composite Systems in Quantum
Mechanics 222
The Matrix Representation of a Tensor Product 223
The Tensor Product of State Vectors 224
The Density Operator 226
The Density Operator for a Completely
Mixed State 229
A Brief Introduction to the Bloch Vector 237
Quiz 239
CHAPTER 9 The Harmonic Oscillator 241
The Solution of the Harmonic Oscillator in the
Position Representation 241
The Operator Method for the Harmonic Oscillator 250
Number States of the Harmonic Oscillator 253
More on the Action of the Raising and Lowering
Operators 256
Quiz 258
CHAPTER 10 Angular Momentum 259
The Commutation Relations of
Angular Momentum 260
The Uncertainty Relations for
Angular Momentum 262
Generalized Angular Momentum and
the Ladder Operators 262
Matrix Representations of Angular Momentum 272
Coordinate Representation of Orbital Angular
Momentum and the Spherical Harmonics 283
Quiz 293
CHAPTER 11 Spin-1/2 Systems 295
The Stern-Gerlach Experiment 296
The Basis States for Spin-1/2 Systems 298
Using the Ladder Operators to Construct Sx, Sy 300
Unitary Transformations for Spin-1/2 Systems 308
The Outer Product Representation of the Spin
Operators 310
The Pauli Matrices 312
The Time Evolution of Spin-1/2 States 317
The Density Operator for Spin-1/2 Systems 328
Quiz 329
CHAPTER 12 Quantum Mechanics in Three Dimensions 331
The 2-D Square Well 332
An Overview of a Particle in a Central Potential 341
An Overview of the Hydrogen Atom 342
Quiz 356
Final Exam 357
Answers to Quiz and Exam Questions 363
References 385
Index 387
PREFACE
Quantum mechanics, which by its very nature is highly mathematical (and therefore
extremely abstract), is one of the most difficult areas of physics to master. In these
pages we hope to help pierce the veil of obscurity by demonstrating, with explicit
examples, how to do quantum mechanics. This book is divided into three main
parts.
After a brief historical review, we cover the basics of quantum theory from the
perspective of wave mechanics. This includes a discussion of the wavefunction,
the probability interpretation, operators, and the Schrödinger equation. We then
consider simple one-dimensional scattering and bound state problems.
In the second part of the book we cover the mathematical foundations needed to
do quantum mechanics from a more modern perspective. We review the necessary
elements of matrix mechanics and linear algebra, such as finding eigenvalues and
eigenvectors, computing the trace of a matrix, and finding out if a matrix is Hermitian
or unitary. We then cover Dirac notation and Hilbert spaces. The postulates
of quantum mechanics are then formalized and illustrated with examples. In the
chapters that cover these topics, we attempt to “demystify” quantum mechanics by
providing a large number of solved examples.
The final part of the book provides an illustration of the mathematical foundations
of quantum theory with three important cases that are typically taught in a first
semester course: angular momentum and spin, the harmonic oscillator, and an
introduction to the physics of the hydrogen atom. Other topics covered at some
level with examples include the density operator, the Bloch vector, and two-state
to cover advanced topics from non-relativistic quantum theory such as scattering,
systems.
identical particles, addition of angular momentum, higher Z atoms, and the WKB
approximation.
There is no getting around the mathematical background necessary to learn
quantum mechanics. The reader should know calculus, how to solve ordinary and
partial differential equations, and have some exposure to matrices/linear algebra
and at least a basic working knowledge of complex numbers and vectors. Some
knowledge of basic probability is also helpful. While this mathematical background
is extensive, it is our hope that the book will help “demystify” quantum theory for
those who are interested in self-study or for those from different backgrounds such
as chemistry, computer science, or engineering, who would like to learn something
about quantum mechanics.
Unfortunately, due to the large amount of space that explicitly solved examples
from quantum mechanics require, it is not possible to include everything about the
theory in a volume of this size. As a result we hope to prepare a second volume
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