Quantum Mechanics

Problems and Solutions

K. Kong Wan

• Publisher: Jenny Stanford Publishing
• Year: 2020
• eBook ISBN: 9781000022568
• Topic: Physical Sciences
• Subtopic: Chemistry
• Index: Chemistry

This is a companion volume to K. Kong Wan's textbook Quantum Mechanics: A Fundamental Approach, published in 2019 by Jenny Stanford Publishing. The book contains more than 240 exercises and problems listed at the end of most chapters. This essential manual presents full solutions to all the exercises and problems that are designed to help the reader master the material in the textbook. Mastery of the material in the book would contribute greatly to the understanding of the concepts and formalism of quantum mechanics.

Chapter 1

Structure of Physical Theories

This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables. We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components:

1. Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space.

2. Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system.

3. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest:

• (1) For a deterministic theory like classical mechanics a state should determine the values of all observables.
• (2) For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables.

4. Description of time evolution (dynamics).

Chapter 2

Classical Systems

This chapter sets out some general physical properties of classical systems which are divided into discrete and continuous:

1. Discrete systems These are systems of discrete point particles. The specific structure of classical mechanics is presented with position, linear and angular momenta serving as basic observables.

2. Continuous systems These systems are illustrated by a vibrating string. Continuous systems have different kinds of properties and observables, e.g., wave properties. In particular we have discussed:

• (1) Description of states by solutions of the classical wave equation.
• (2) The concept of eigenfunctions with orthonormality property and their superposition and interference.
• (3) The concept of a complete set of states.

These discussions are given specifically to provide an intuition to help a better understanding of similar properties of quantum systems.

Chapter 3

Probability Theory for Discrete Variables

Q3(1) Prove Theorem 3.4(1).

SQ3(1) Theorem 3.4(1) can be proved using properties PM3.4(1), PM3.4(2) and PM3.4(3) in Definition 3.4(2).

• (1) To prove Eq. (3.25) let E be any event. Then E ∪ ∅ = E and E ∩∅ = ∅. By PM3.4(3) we have,
  $$\begin{array}{l}{ℳ}_p(E\cup \varnothing )={ℳ}_p(E)\\ {ℳ}_p(E\cup \varnothing )={ℳ}_p(E)+{ℳ}_p(\varnothing )\end{array}\}\Rightarrow {ℳ}_p(\varnothing )=0.$$
• (2) To prove Eq. (3.26) we start with ${ℳ}_p\left(S_{am}\right)=1$ by PM3.4(2). Since E ∩ E ^c = ∅ and E ∪ E ^c = S_am we have
  $$\begin{array}{lll}{ℳ}_p\left(S_{am}\right)\hfill & =\hfill & {ℳ}_p\left(E\cup E^c\right)={ℳ}_p(E)+{ℳ}_p\left(E^c\right)=1\hfill \\ \hfill & \Rightarrow \hfill & {ℳ}_p\left(E^c\right)=1-{ℳ}_p(E).\hfill \end{array}$$
• (3) To prove Eq. (3.27) we first observe that E₁ ⊂ E₂ ⇒ E₂ =.(E₂ – E₁) ∪ E₁ Since (E₂ – E₁) and E₁ are disjoint, i.e., (E₂ –E₁) ∩ E₁ = ∅ , we have
  $$\begin{array}{lll}{ℳ}_p\left(E_2\right)\hfill & =\hfill & {ℳ}_p\left(\left(E_2-E_1\right)\cup E_1\right)\hfill \\ \hfill & =\hfill & {ℳ}_p\left(E_2-E_1\right)+{ℳ}_p\left(E_1\right)\ge {ℳ}_p\left(E_1\right)....\hfill \end{array}$$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• 1 Structure of Physical Theories
• 2 Classical Systems
• 3 Probability Theory for Discrete Variables
• 4 Probability Theory for Continuous Variables
• 5 Quantum Mechanical Systems
• 6 Three-Dimensional Real Vectors
• 7 Matrices and Their Relations with Vectors
• 8 Operations on Vectors in E→3
• 9 Special Operators on E→3
• 10 Probability, Selfadjoint Operators, Unit Vectors and the Need for Complexness
• 11 Complex Vectors
• 12 N-Dimensional Complex Vectors
• 13 Operators on N-Dimensional Complex Vectors
• 14 Model Theories Based on Complex Vector Spaces
• 15 Spectral Theory in Terms of Stieltjes Integrals
• 16 Infinite-Dimensional Complex Vectors and Hilbert Spaces
• 17 Operators in a Hilbert Space ℋ→
• 18 Bounded Operators on ℋ→
• 19 Symmetric and Selfadjoint Operators in ℋ→
• 20 Spectral Theory of Selfadjoint Operators in ℋ→
• 21 Spectral Theory of Unitary Operators on ℋ→
• 22 Selfadjoint Operators, Unit Vectors and Probability Distributions
• 23 Physics of Unitary Transformations
• 24 Direct Sums and Tensor Products of Hilbert Spaces and Operators
• 25 Pure States
• 26 Observables and Their Values
• 27 Canonical Quantisation
• 28 States, Observables and Probability Distributions
• 29 Time Evolution
• 30 On States after Measurement
• 31 Pure and Mixed States
• 32 Superselection Rules
• 33 Many-Particle Systems
• 34 Conceptual Issues
• 35 Harmonic and Isotropic Oscillators
• 36 Angular Momenta
• 37 Particles in Static Magnetic Fields
• Bibliography