This book provides a clear understanding of quantum mechanics (QM) by developing it from fundamental postulates in an axiomatic manner, as its central theme. The target audience is physics students at master’s level. It avoids historical developments, which are piecemeal, not logically well knitted, and may lead to misconceptions. Instead, in the present approach all of QM and all its rules are developed logically starting from the fundamental postulates only and no other assumptions.
Specially noteworthy topics have been developed in a smooth contiguous fashion following the central theme. They provide a new approach to understanding QM. In most other texts, these are presented as disjoint separate topics. Since the reader may not be acquainted with advanced mathematical topics like linear vector space, a number of such topics have been presented as “mathematical preliminary.” Standard topics, viz. derivation of uncertainty relations, simple harmonic oscillator by operator method, bound systems in one and three dimensions, angular momentum, hydrogen-like atom, and scattering in one and three dimensions, are woven into the central theme.
Advanced topics like approximation methods, spin and generalized angular momenta, addition of angular momenta, and relativistic quantum mechanics have been reserved for Volume II.
Author(s): Tapan Kumar Das
Series: UNITEXT for Physics
Publisher: Springer
Year: 2023
Language: English
Pages: 314
City: Singapore
Tags: Quantum Mechanics, Hilbert Spaces, Schrödinger Equation, Uncertainty Relation, Harmonic Oscillator, Scattering
Preface
Contents
Common Abbreviations Used Throughout This Book
Special Notations Used Throughout This Book
1 Introduction
1.1 New Experiments and Their Interpretations
1.2 Problems
References
2 Mathematical Preliminary I: Linear Vector Space
2.1 Linear Vector Space
2.1.1 Formal Definition
2.1.2 Subspace
2.1.3 Linear Independence of Vectors
2.1.4 Basis and Dimension
2.2 Scalar (Inner) Product and Inner Product Space
2.2.1 Condition of Linear Independence
2.2.2 Schwarz Inequality
2.2.3 Orthogonality and Normalization
2.3 Operators on a Vector Space
2.3.1 Eigen Value Equation Satisfied by an Operator
2.4 Matrix Representation of Linear Operators
2.5 Closure Relation of a Basis
2.6 Change of Basis
2.7 Dirac's Bra and Ket Notation
2.8 Infinite-Dimensional Vector Spaces
2.9 Hilbert Space
2.10 Problems
References
3 Axiomatic Approach to Quantum Mechanics
3.1 Linear Vector Spaces in Quantum Mechanics
3.2 Fundamental Postulates of Quantum Mechanics
3.3 Coordinate Space Wave Function: Interpretation
3.4 Mathematical Preliminary: Dirac Delta Function
3.5 Normalization
3.6 Problems
References
4 Formulation of Quantum Mechanics: Representations and Pictures
4.1 Position (Coordinate) Representation
4.2 Momentum Representation
4.3 Change of Representation
4.4 Matrix Representation: Matrix Mechanics
4.5 Math-Prelim: Matrix Eigen Value Equation
4.6 Quantum Dynamics—Perspectives: Schrödinger, Heisenberg and Interaction Pictures
4.7 Problems
References
5 General Uncertainty Relation
5.1 Derivation of Uncertainty Relation
5.2 Minimum Uncertainty Product
5.3 Problems
Reference
6 Harmonic Oscillator: Operator Method
6.1 Importance of Simple Harmonic Oscillator
6.2 Energy Eigen Values and Eigen Vectors
6.3 Matrix Elements
6.4 Coordinate Space Wave Function
6.5 Uncertainty Relation
6.6 Problems
References
7 Mathematical Preliminary II: Theory of Second Order Differential Equations
7.1 Second Order Differential Equations
7.1.1 Singularities of the Differential Equation
7.1.2 Linear Dependence of the Solutions
7.1.3 Series Solution: Frobenius Method
7.1.4 Boundary Value Problem: Sturm–Liouville Theory
7.1.5 Connection Between Mathematics and Physics
7.2 Some Standard Differential Equations
7.3 Problems
References
8 Solution of Schrödinger Equation: Boundary and Continuity Conditions in Coordinate Representation
8.1 Conditions on Wave Function
8.2 Eigen Solutions
8.3 Other Properties
8.4 Free Particle Wave Function
8.5 Wave Packet and Its Motion
8.6 Ehrenfest's Theorem
8.7 Problems
References
9 One-Dimensional Potentials
9.1 A Particle in a Rigid Box
9.2 A Particle in a Finite Square Well
9.3 General Procedure for Bound States
9.4 A Particle in a Harmonic Oscillator Well
9.5 Wave Packet in a Harmonic Oscillator Well
9.6 Potential with a Dirac Delta Function
9.7 Quasi-bound State in a δ-Function Barrier
9.8 Problems
References
10 Three-Dimensional Problem: Spherically Symmetric Potential and Orbital Angular Momentum
10.1 Connection with Orbital Angular Momentum
10.2 Eigen Solution of Orbital Angular Momentum
10.3 Radial Equation
10.4 Problems
References
11 Hydrogen-type Atoms: Two Bodies with Mutual Force
11.1 Two Mutually Interacting Particles: Reduction to One-Body Schrödinger Equation
11.2 Relative Motion of One-Electron H-Type Atoms
11.3 Problems
References
12 Particle in a 3-D Well
12.1 Spherically Symmetric Hole with Rigid Walls
12.2 Spherically Symmetric Hole with Permeable Walls
12.3 A Particle in a Cylindrical Hole with Rigid Walls
12.4 3-D Spherically Symmetric Harmonic Oscillator
12.5 Problems
References
13 Scattering in One Dimension
13.1 A Free Particle Encountering an Infinitely Rigid Wall
13.2 Penetration Through a Finite Square Barrier
13.3 Scattering of a Free Particle by a δ-Barrier
13.4 Problems
Reference
14 Scattering in Three Dimension
14.1 Kinematics for Scattering
14.2 Scattering Cross-Section
14.3 Schrödinger Equation
14.4 Spherically Symmetric Potential: Method of Partial Waves
14.4.1 Optical Theorem
14.4.2 Phase Shifts
14.4.3 Relation Between Sign of Phase Shift (δl) and the Nature (Attractive or Repulsive) of Potential
14.4.4 Ramsauer–Townsend Effect
14.5 Scattering by a Perfectly Rigid Sphere
14.6 Coulomb Scattering
14.7 Green's Function in Scattering Theory
14.8 Born Approximation
14.9 Resonance Scattering
14.10 Problems
References
Appendix Orthogonality: Physical and Mathematical
Reference
Index
