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by Dr Singh Tyagi
Principles of Quantum Mechanics
Cover
Title Page
Contents
About the Author
Dedication
Foreword
Preface
Chapter 1: Introduction
1.1 Motivation
1.2 The usual (classical) mechanics
1.3 The process of observation
1.4 The new mechanics
Chapter 2: Wave-particle Duality
2.1 Introduction
2.2 Young’s two-slit experiment
2.3 Bragg’s x-ray diffraction
2.4 Photoelectric effect
2.5 Compton effect
2.6 Wave-particle nature of electromagnetic radiations
2.7 Electron/neutron diffraction
2.8 Davisson and Germer electron diffraction experiment
2.9 Wave-particle nature of matter
2.10 What is the real nature of matter and radiations?
Exercises
Solutions
References
Chapter 3: Wave Packets and Uncertainty Principle
3.1 Introduction
3.2 Superposition of waves
3.3 Phase velocity and group velocity
3.4 de Broglie relation
3.5 Measurement and uncertainty principle
3.6 Uncertainty principle: Thought experiments
3.7 Conclusions
Exercises
Solutions
References
Chapter 4: Operators, Eigenstates, Eigenvalues and Schrodinger Equation
4.1 Introduction
4.2 Measurement process as operator operating on the state function/wave function of a particle having definite linear momentum
4.3 Physical interpretation of wave function ψ(r, t)
4.4 Schrodinger equation for a free particle
4.5 Schrodinger equation for a free wave packet
4.6 Schrodinger equation for a particle in a potential
4.7 Expectation value and operators
4.8 Probability current density: Equation of continuity
4.9 Gaussian wave packet and its spread with time
4.10 Wave function in momentum space
4.11 The Ehrenfest theorem
4.12 The uncertainty relations (revisited)
4.13 The (resulting) quantum logic
Exercises
Solutions
References
Chapter 5: One-dimensional Problems
5.1 Introduction
5.2 Time-independent schrodinger equation and stationary states
5.3 Some characteristics of wave functions
5.4 Particle in a one-dimensional potential box
5.5 Potential box with periodic boundary conditions
5.6 The potential step
5.7 Rectangular potential barrier
5.8 Potential well of finite depth
5.9 Kronig–Penney model
Exercises
Solutions
References
Chapter 6: The Linear Harmonic Oscillator
6.1 Introduction
6.2 Classical harmonic oscillator
6.3 Quantum harmonic oscillator
6.4 The Normalized Wave Functions
6.5 The Hermite polynomials
6.6 Parity
6.7 Conclusions
Exercises
Solutions
References
Chapter 7: The Linear Vector Space
7.1 Introduction
7.2 Some characteristics of eigenstates of Hermitian operators
7.3 Dirac bra and ket notations
7.4 More about bra, ket vectors and linear vector space
7.5 Matrix representation of state vectors and operators
7.6 Some special matrices/operators
7.7 Change of basis: Unitary transformation
7.8 Tensor product or direct product of vector spaces
7.9 Outer product operators
Exercises
Solutions
References
Chapter 8: Linear Harmonic Oscillator—Revisited
8.1 Introduction
8.2 The creation and annihilation operators
8.3 Energy eigenstates
8.4 Matrix representation of various operators
8.5 Expectation values of various operators
8.6 The coherent states
8.7 Time evolution of the coherent state and its comparison with classical oscillator
8.8 The Schrodinger and Heisenberg pictures
Exercises
Solutions
References
Chapter 9: Angular Momentum
9.1 Introduction
9.2 Orbital angular momentum operator
9.3 Commutation relations
9.4 Angular momentum operator in spherical polar coordinates
9.5 The eigenvalues and eigenfunctons of L2 and Lz
9.6 Measurement of angular momentum components and the uncertainty relations
9.7 Orbital angular momentum and spatial rotation
Exercises
Solutions
References
Chapter 10: Three-dimensional Systems
10.1 Introduction
10.2 A particle in a cubic potential box
10.3 Cubic box with periodic boundary conditions
10.4 Density of states of free particles (free electron gas in metals)
10.5 Spherically symmetric potentials
10.6 The free particle in spherical polar coordinates
10.7 Schrodinger equation for a two-body system
10.8 The hydrogenic atom
Exercises
Solutions
References
Chapter 11: Angular Momentum—Revisited
11.1 Introduction
11.2 Raising and lowering operators (the ladder operators)
11.3 Eigenvalues and eigenstates of orbital angular momentum operators: Second construction of spherical harmonics
11.4 The constants C+ and C–
11.5 Matrix representation of angular momentum operator corresponding to j = 1
11.6 Matrix representation of angular momentum operator corresponding to j = 1/2
Exercises
Solutions
References
Chapter 12: The Spin
12.1 Introduction
12.2 Orbital angular momentum and magnetic moment
12.3 The electron spin: Spin operators and spin eigenstates
12.4 Total wave function of an electron
12.5 The Stern–Gerlach experiment
12.6 Spin and rotation (spinor transformation)
12.7 A magnetic moment in a uniform magnetic field: The Larmor precession
12.8 Electron spin resonance
Exercises
Solutions
References
Chapter 13: Addition of Angular Momenta
13.1 Introduction
13.2 Addition of two angular momenta
13.3 Recursion relations for the C–G coefficients
13.4 The possible values of j
13.5 Addition of two spin 1/2 angular momenta
13.6 Addition of j = 1 and j = 1/2 angular momenta
Exercises
Solutions
References
Chapter 14: WKB Approximation and Electron Tunnelling
14.1 Introduction
14.2 The essential idea of WKB method
14.3 Development of WKB approximation
14.4 Validity of WKB approximation
14.5 The connection formulae
14.6 Application of WKB technique to barrier penetration
14.7 Cold emission of electrons from metals
14.8 Alpha-decay of nuclei
Exercises
Solutions
References
Chapter 15: Time-independent Perturbation Theory
15.1 Introduction
15.2 Non-degenerate perturbation theory
15.3 Harmonic oscillator subject to perturbing potential
15.4 Degenerate perturbation theory
15.5 The Stark effect
15.6 The fine structure of hydrogen
15.7 The Zeeman effect
Exercises
Solutions
References
Chapter 16: Time-dependent Perturbation Theory
16.1 Introduction
16.2 Time development of states and transition probability
16.3 Constant perturbation
16.4 The adiabatic approximation
Exercises
Solutions
References
Chapter 17: Semi-classical Theory of Radiations
17.1 Introduction
17.2 Interaction of one-electron atom with electromagnetic field
17.3 Harmonic perturbation theory
17.4 Spontaneous emission: Einstein A and B coefficients
17.5 Selection rules for electric dipole transitions
17.6 Lifetime and line-width
Exercises
Solutions
References
Chapter 18: Theory of Scattering
18.1 Introduction
18.2 Scattering experiments and scattering cross-section
18.3 Classical theory of scattering: Rutherford scattering
18.4 Quantum theory of scattering
18.5 Solution of Schrodinger equation for scattering problem: Green’s function
18.6 The Born approximation
18.7 Method of partial waves and phase shifts
Exercises
Solutions
References
Chapter 19: Theory of Measurement in Quantum Mechanics
19.1 Introduction
19.2 Process of measurement: A simple treatment
19.3 Measurement of spin of an atom
19.4 The EPR paradox
19.5 The hidden variables and Bell’s theorem
19.6 Time–evolution of a system: Quantum zeno paradox
Exercises
Solutions
References
Chapter 20: Introduction to Quantum Computing
20.1 Introduction
20.2 Binary number systems
20.3 Classical logic gates
20.4 Turing machine
20.5 Qubits
20.6 Entanglement
20.7 Quantum logic gates
20.8 Quantum computation
Exercises
Solutions
References
Appendices
Appendix A: Early Quantum Mechanics
A.1 Planck’s formula of black body radiations
A.2 Atomic spectra and Bohr’s model of hydrogen atom
A.3 Bohr’s correspondence principle
A.4 The Franck–Hertz experiment
Appendix B: Some Supplementary Topics
B.1 Fourier transform
B.2 Dirac delta function
B.3 Bloch theorem
B.4 The variational method
Exercises
Solutions
Appendix C: Some Mathematical Relations
C.1 Some algebraic relations
C.2 Some trigonometric relations
C.3 Coordinate systems
C.4 Some vector relations
C.5 Some calculus relations
C.6 Some definite integrals
C.7 An important integral
Appendix D: Various Tables
D.1 Table of fundamental physical constants
D.2 Table of conversion factors between units used to express magnitude of energy
D.3 Table of spectrum of electromagnetic radiations
D.4 Range of radiations
D.5 Symbols and values of pre-factors
D.6 The Greek alphabets
Copyright
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