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Table of Contents
Intro
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: Schrd̲inger, 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 Schrd̲inger 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
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: Schrd̲inger, 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 Schrd̲inger 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