Quantum Mechanics - Non-Relativistic and Relativistic Theory

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
About the Author
SECTION I: Non-Relativistic Theory
1. Quantum Mechanics Basic Concepts
1.1. Inadequacies of Classical Mechanics
1.2. Wave Function
1.3. Wave Function Statistical Interpretation
1.4. Uncertainty of Two Types of Measurements
1.5. Superposition Principle Generalized Formulation
1.6. Operators of Physical Quantities
1.6.1. Expectation Value (Observable) and Operator of a Physical Quantity
1.6.2. Properties of Operators
1.7. Linear Self-Adjoint (Hermitian) Operators
1.7.1. Translation Operator
1.8. Eigenfunction and Eigenvalue
1.8.1. Conclusion
1.9. Properties of Eigenfunctions of Hermitian Operators
1.10. Theorem on the Commutation of Operators and Their Physical Application
1.11. Heisenberg Uncertainty Relations for Arbitrary Observables
1.12. Limiting Transition from Quantum Mechanics to Classical Mechanics
2. Schrödinger Equation
2.1. Stationary States
2.1.1. Particle in an Infinite Deep Potential Well
2.1.2. A Particle in an Infinitely High Potential Well
2.1.3. Coordinate Representation Delta Potential
2.2. Time-Dependent Operators
2.2.1. Classical Equation of Motion
2.2.2. Quantum-Mechanical Poisson Bracket and Quantum Correspondence Principle
2.2.3. Quantum Mechanical Equation of Motion
2.2.4. Postulates of Quantum Mechanics
2.2.5. Velocity and Acceleration of a Charged Particle in an Electromagnetic Field
2.2.6. Probability Density and Probability Current Density
2.2.7. Current Density of a Charged Particle in an Electromagnetic Field
2.2.8. Change with Time of a Wave Packet
3. Momentum Operator
3.1. Translation Operator
3.2. Momentum Operator
3.3. Heisenberg Uncertainty Relation
3.4. Momentum Representation
3.4.1. Momentum Representation of Particle in Triangular Potential Well
3.4.2. Momentum Representation of Particle in Delta Potential Well
3.4.3. Momentum Representation of an Operator in Matrix Form
3.5. Particle Hamiltonian in a Potential Field
3.5.1. Hamilton Function Operator and Ehrenfest Theorem
3.6. Angular Momentum Operator
3.6.1. Infinitesimal Rotation Operator
3.6.2. Angular Momentum Operator
3.6.3. Commutation Relations of Angular Momentum Operators
3.6.4. Eigenvalue and Eigenfunction of z-Component Angular Momentum Operator
3.7. Square of Angular Momentum Operator
3.7.1. Square of Angular Momentum Operator Commutation Relations
3.7.2. Square of Angular Momentum Operator Eigenvalue in The Dirac Representation
3.8. Square of Angular Momentum Operator Eigenstates
3.8.1. Legendre Polynomials
3.8.1.1. Asymptotic Legendre Polynomials
3.8.2. Angular Momentum Eigenstates
3.8.3. Dirac Representation Eigenstates
3.8.4. Matrix Representation and Finite Rotations Eigenstates
4. Total Angular Momentum
4.1. Infinitesimal Symmetry Transformation Generator
4.2. Total Angular Momentum Justification
4.3. Addition of Two Angular Momenta
4.3.1. Clebsch-Gordan Coefficients
4.3.1.1. Other Representation of Clebsch-Gordan Coefficients
4.3.1.2. Clebsch-Gordan Coefficients Recursion Relations
4.3.2. Triangular Rule
4.4. Spherical Spinors
4.4.1. Spinor Rotation
4.4.2. Spin Density
4.5. Spin of a System of Two Particles
4.6. Rotation Operator
4.6.1. Finite Rotation Operator About Some Angle Along Some Axis
4.6.2. Finite Rotation Operator for Spinor One-Half
4.6.3. Finite Rotation Operator for Spinor One-Half General Case
4.6.4. Rotation Operator Matrix
4.6.4.1. Spherical Harmonics Connection
4.7. Irreducible Tensor Operators
4.7.1. Wigner-Eckart Theorem
5. One-Dimensional Motion General Principles
5.1. One-Dimensional Motion General Principles
5.2. Potential Well
5.3. Particle in a One-Dimensional Finite Square Well Potential
5.4. Potential Barrier
5.5. Particle in a Square Potential Barrier
6. Schrödinger Equation
6.1. Linear Harmonic Equation
6.2. Harmonic Oscillator Eigenstates and Eigenvalues
6.2.1. Hermite Polynomial and Harmonic Oscillator Eigenfunction
6.2.1.1. Hermite Polynomials
6.2.1.2. Hermite Polynomials Integral Representation
6.2.1.3. Harmonic Oscillator Eigenfunction and Normalization Condition
6.2.1.4. Hermite Polynomials Orthogonality Condition
6.3. Motion in a Central Field
6.3.1. Radial Schrödinger Equation
6.3.2. Radial Wave Function Qualitative Investigation
6.3.3. Continuous Spectra Radial Wave Functions
6.3.3.1. Jost Function
6.3.4. Delta Potential Radial Solution
6.4. Motion in a Coulombic Field
6.4.1. Hydrogen Atom (Spherical Coordinates)
6.4.2. Eigenvalue and Eigenfunction
6.4.2.1. Hydrogen Atom’s Wave Function
6.4.2.2. Laguerre Polynomials Integral Representation
6.4.2.3. Eigenvalue and Degeneracy
6.4.3. Hydrogen Atom (Parabolic Coordinates)
6.4.3.1. Energy Levels
6.4.3.2. Wave Functions
6.4.4. Spherical Oscillator (Spherical Coordinates)
6.4.5. Particle in an Infinite Deep Spherical Symmetric Potential Well
6.4.6. Kepler Problem in Two Dimensions
7. Representation Theory
7.1. Matrix Wave Functions and Operator Representation
7.2. Properties of Matrices
7.3. Rule on Matrix Operations
7.4. Action of an Operator on a Wave Function
7.5. Mean Value of an Operator
7.6. Eigenstate and Eigenvalue Problem
7.7. Unitary Transformation in State Vector Space
7.7.1. Unitary Matrix
7.7.2. Matrix Element of a Transformation Operator
7.7.3. Invariance of the Trace of a Matrix Under Unitary Transformations
7.8. Schrödinger and Heisenberg Representations
7.9. Interaction Representation
7.10. Energy Representation
7.10.1. Evolution Operator
7.10.2. Oscillator in the Energy Representation
7.10.2.1. Matrix Element of the Oscillator Coordinate
7.10.2.2. Hamiltonian Operator Eigenvalue
7.10.2.3. Harmonic Oscillator Ground-State Eigenfunction
7.10.2.4. Quantization of Operators
8. Quantum Mechanics Approximate Methods
8.1. Variational Principle
8.1.1. Ritz Method
8.2. Case of the Hydrogen Atom
8.3. Perturbation Theory
8.3.1. Stationary Perturbation Theory – Non-Degenerate Level Case
8.4. Perturbation Theory – Case of a Degenerate Level
8.4.1. The Stark Effect
8.4.1.1. Hydrogen Atom
8.4.2. Stark Effect (Spherical Coordinates)
8.4.3. Stark Effect (Parabolic Coordinates)
8.5. Time-Dependent Perturbation Theory
8.5.1. Transition Probability
8.5.2. Adiabatic Approximation
8.6. Time-Independent Perturbation
8.7. Time and Energy Uncertainty Relation
8.8. Density of Final State
8.8.1. Transition Rate
8.9. Transition Probability-Continuous Spectrum
8.9.1. Harmonic Perturbation
8.10. Transition in a Continuous Spectrum Due to a Constant Perturbation
9. Many-Particle System
9.1. System of Indistinguishable Particles
9.2. Interacting System of Particles
9.3. System of Two Electrons
9.3.1. Exchange Interaction
9.3.2. Two Electrons in an Infinite Square Potential Well – Heisenberg Exchange Interaction
10. Approximate Method for the Helium Atom
10.1. The State of the Helium Atom
10.2. Self-Consistent Field Method
11. Approximate Method for the Hydrogen Molecule
11.1. Vibrational and Rotational Levels of Diatomic Molecules
12. Scattering Theory
12.1. Scattering Cross Section and Elastic Scattering Amplitude
12.1.1. Relation Between the Laboratory and Center-of-Mass Systems
12.2. Method of Partial Waves
12.3. S-Scattering of Slow Particles
12.4. Resonance Scattering
12.5. The Unitary Scattering Conditions
12.5.1. Optical Theorems
12.6. Time-Reversal Symmetry
12.6.1. Inversion Operator and Reciprocity Theorem
12.7. Schrödinger Equation Green’s Function
12.8. Born Approximation
12.8.1. Scattering of Fast Charged Particles on Atoms
12.8.1.1. Scattering Amplitude in Momentum Representation
12.8.2. Perturbation Theory Method Approach for Born Approximation
12.8.2.1. Phase Shift
12.8.2.2. Spherical Potential Well
12.8.2.3. Coulomb Interaction and Rutherford’s Formula
12.8.2.4. Lippman Schwinger Equation, 1D Delta Potential
12.9. Elastic and Inelastic Collisions
12.9.1. Fast and Slow Particle Total Cross Section
12.10. Wentzel-Kramer-Brillouin (WKB) Method
12.10.1. Motion in a Central Symmetric Field
12.11. Scattering of Indistinguishable Particles
13. Polaron Theory
13.1. Lee-Low-Pines (LLP) Technique
13.1.1. Lee-Low-Pines (LLP) Bulk Polaron
13.1.2. Lee-Low-Pines (LLP) Surface and Slow Moving Polaron
13.1.3. Lee-Low-Pines (LLP) Surface and Fast Moving Polaron
13.2. Polaron in a Quantum Wire
13.3. Polaronic Exciton and Haken Exciton
SECTION II: Relativistic Theory
14. Case of an Electron
14.1. Spin Operators
14.1.1. Spin and Spin Operator Commutation Relations
14.1.2. Pauli Matrices
14.1.3. Derivation of Pauli Matrices
14.2. Spinors
14.2.1. Lorentz Transformation and Spinor Transformation
14.2.2. Arbitrary Spinor Transformation
15. Klein-Gordon Equation
15.1. Probability and Charge Densities
15.2. Motion in an Electromagnetic Field
15.3. Spinless Charge Particle in a Coulombic Field
15.4. Non-Relativistic Limiting Equation
16. Dirac Equation
17. Probability and Current Densities
18. Electron Spin in the Dirac Theory
19. Free Electron State with Defined Momentum-Positronium Motion
19.1. Stationary Dirac Equation
19.1.1. Dirac Hypothesis-Hole Theory
20. Dirac Equation
20.1. Electron Motion in an External Electromagnetic Field
20.1.1. Quasi-Relativistic Approximation-Pauli Equation
20.1.2. Second-Order Relativistic Correction
20.1.2.1. Spin-Orbital Interaction
20.1.2.2. Fine Structure Levels
20.1.2.3. Fine Structure Effect
20.2. Bound Electronic States in a Coulombic Field
21. Motion in a Magnetic Field
21.1. Landau Levels
21.2. Spin Precession in a Magnetic Field
21.3. Theory of the Zeeman Effect
21.3.1. Russell-Saunders Coupling
21.3.2. Weak Field Limiting Case – Zeeman Effect
21.3.3. Strong Field for Exceedingly Small Spin-Orbit Interaction – Paschen-Back Effect
21.3.4. Landau Case
21.4. Atomic Paramagnetism and Diamagnetism
SECTION III: Appendix: Special Functions
22. Gamma Functions
22.1. First Kind Euler Integral-Beta Function
22.2. Gamma Function (Second Kind Euler Integral)
22.3. Gamma Function Analytic Continuation
22.4. Hankel Integral Representations
22.5. Reflection or Complementary Formula
23. Confluent Hypergeometric Functions
23.1. Classical Gauss Confluent Hypergeometric Function
23.2. Euler Integral Representation: Mellin–Barnes Integral Representation
23.3. Confluent Hypergeometric Function – Kummer Function
24. Cylindrical Functions
24.1. Cylindrical Function of the First Kind
24.2. Neumann Function
24.3. Hankel Functions
24.4. Modified Bessel Function
24.5. Modified Bessel Function with Imaginary Argument
24.6. Bessel Function of the First Kind Integral Formula
24.7. Neumann Function Integral Formula
24.8. Hankel Function Integral Formula
24.9. Airy Function
25. Orthogonal Polynomials
25.1. Orthogonal Polynomials General Properties
25.2. Transforming Confluent Hypergeometric Function into a Polynomial
25.3. Jacobi Polynomials
25.4. Jacobi Polynomial Generating Function
25.5. Gegenbauer Polynomials
25.6. Gegenbauer Polynomial Generating Function
25.7. First Kind Tschebycheff Polynomial
25.8. Generating Function of the First Kind Tschebycheff Polynomial
25.9. Tschebycheff Polynomial of the Second Kind
25.10. Generating Function of the Second Kind Tschebycheff Polynomial
25.11. Legendre Polynomials
25.12. Legendre Polynomial Generating Function
25.13. Legendre Polynomials Integral Representation
25.14. Associated Legendre Polynomials
25.15. Associated Legendre Polynomials Integral Representation
25.16. Spherical Functions
25.17. Laguerre Polynomials
25.18. Associated Laguerre Polynomial Generating Function
25.19. Hermite Polynomials
25.20. Hermite Polynomial Generating Function
References
Index