Essentials of Quantum Mechanics and Relativity

Contents
Preface
Quantum Mechanics
	1. Preliminaries
		1.1 Introduction
			1.1.1 What is QM?
			1.1.2 Why do we need QM?
			1.1.3 How do QM explain atomic phenomena?
		1.2 Review of some basic concepts of CM
			1.2.1 Generalized coordinates
			1.2.2 The Lagrange equations
			1.2.3 Hamilton’s equations
			1.2.4 Poisson brackets
		1.3 Historical review: Experiments and theories
			1.3.1 The work of Planck: Black body radiation
			1.3.2 The work of Einstein: Photoelectric effect
			1.3.3 The work of Bohr: Hydrogen atom
			1.3.4 The work of Compton: Compton effect and the dual nature of light
				1.3.4.1 Compton effect
				1.3.4.2 Dual nature of light
			1.3.5 The work of de Broglie: Matter waves
				1.3.5.1 Bohr’s first postulate and the de Broglie hypothesis
				1.3.5.2 The phase velocity of plane wave and group velocity of wave packet
			1.3.6 The work of Schrodinger: Wave equation
			1.3.7 The work of Born: Probability interpretation
			1.3.8 The work of Heisenberg: Uncertainty principle
	2. Postulates and One-Dimensional Problems
		2.1 Postulates of quantum mechanics
			2.1.1 Postulate 1: Quantum state
			2.1.2 Postulate 2: Physical observable
			2.1.3 Postulate 3: Time evolution of quantum state
			2.1.4 Postulate 4: Measurement
			2.1.5 Postulate 5: Expectation value
		2.2 Preliminary applications on one-dimensional problems
			2.2.1 One-dimensional free particle
			2.2.2 Linear box
			2.2.3 Potential barrier of infinite width
			2.2.4 Tunneling effect
			2.2.5 Linear harmonic oscillator
				2.2.5.1 Analytic solution
				2.2.5.2 Algebraic solution
	3. Matrix Mechanics
		3.1 Introduction
		3.2 Matrix algebra
		3.3 Matrix mechanics
		3.4 Transformation of representations
			3.4.1 Transformation of basis
			3.4.2 Transformation of a state vector
			3.4.3 Transformation of an operator
		3.5 Coordinate representation and momentum representation
	4. Central Forces and Angular Momentum
		4.1 Spherically symmetric potential
			4.1.1 Angular momentum
			4.1.2 Hamiltonian
			4.1.3 Separation of variables
			4.1.4 Solution of the angular equation
			4.1.5 Associated Legendre equation: Regular point and indicial equation
			4.1.6 Spherical harmonics
			4.1.7 Solution of the Schr¨odinger equation
		4.2 Radial equation for the Coulomb potential
		4.2.1 Radial part of the wavefunction: Principle quantum number
		4.2.2 Eigenstates of the hydrogen atom
	5. Summary, Problems and Solutions
		5.1 A brief summary of this course
		5.2 Problems
		5.3 Quiz 1
		5.4 Quiz 2
		5.5 Final examination
		5.6 Solutions of problems
		5.7 Solutions of Quiz 1
		5.8 Solutions of Quiz 2
		5.9 Solution of final examination
Relativity
	6. Preliminaries
		6.1 Introduction
			6.1.1 What is the meaning of relativity in physics?
			6.1.2 How relativity governs the form of all physical laws?
			6.1.3 Why do we say quantum mechanics and special relativity are the two pillars of modern physics?
		6.2 Vectors and tensors
			6.2.1 Vectors
			6.2.2 Tensors
			6.2.3 Permutation symbol erst
			6.2.4 The e–δ identity
		6.3 Principle of Galilean relativity
		6.4 Essentials of electrodynamics
			6.4.1 Electrostatics and magnetostatics
				6.4.1.1 Electric charge and electric field
				6.4.1.2 Electric current
				6.4.1.3 Biot–Savart law and Ampere’s law
			6.4.2 Maxwell equations
				6.4.2.1 Faraday’s law
				6.4.2.2 Displacement current
				6.4.2.3 Maxwell equation in vacuum
				6.4.2.4 Energy density and energy flux, momentum density and momentum flux
				6.4.2.5 Wave equation
	7. Special Relativity: Kinematics
		7.1 Attempts to locate the absolute frame: Michelson–Morley experiment
		7.2 Postulates of special relativity and the Lorentz transformation
			7.2.1 Postulates of special relativity
			7.2.2 Lorentz transformation
		7.3 The structure of spacetime
			7.3.1 Comparison
			7.3.2 Spacetime interval
				7.3.2.1 Time-like interval
				7.3.2.2 Light-like interval
				7.3.2.3 Space-like interval
			7.3.3 Proper time and coordinate time
			7.3.4 Light cone
		7.4 Law of causation and the relative character of simultaneity
		7.5 Time dilation and length contraction
			7.5.1 Time dilation
			7.5.2 Length contraction
		7.6 Velocity transformation
	8. Relativistic Electrodynamics
		8.1 Four-dimensional vectors and tensors
		8.2 Four-dimensional current density vector
		8.3 Four-dimensional potential vector
			8.3.1 Scalar potential and vector potential
			8.3.2 Four-dimensional potential vector
		8.4 Field tensor
		8.5 Electric field of a point charge in uniform motion
		8.6 Generalization of Coulomb’s law to Maxwell equations using special relativity
	9. Relativistic Dynamics
		9.1 Equation of motion
		9.2 Relativistic force Kμ: Force four-vector
		9.3 Momentum four-vector
		9.4 Lorentz force
		9.5 Acceleration in relativistic dynamics
		9.6 Quiz
	10. More about Tensors
		10.1 Contravariant and covariant vectors
			10.1.1 Vectors in Euclidean space
			10.1.2 Vectors in four-dimensional Minkowski spacetime
		10.2 Lorentz tensors
			10.2.1 Lowering and raising an index
			10.2.2 Contraction
		10.3 General tensors
		10.4 Metric tensor
		10.5 Christoffel symbol and the covariant derivative of a vector
		10.6 Geodesics and parallel transport
		10.7 Riemann curvature tensor
	11. General Relativity
		11.1 Principle of equivalence
			11.1.1 Weak and strong principle of equivalence
			11.1.2 Description of gravitation
		11.2 Principle of general covariance
		11.3 Principle of minimal coupling
		11.4 Field equations: Relations between metric field and matter field
			11.4.1 Einstein tensor
			11.4.2 Einstein’s field equations
		11.5 Final examination
Appendices of Essentials of Quantum Mechanics
	Appendix A Wave Mechanics or Wave Statistical Mechanics
	Appendix B Alternative Derivation of the Propagator in Polar Coordinates by Feynman’s Physical Interpretation of the Characteristic Function
	Appendix C Separation of Variable Treatment for Solving Time-Dependent Potentials
	Appendix D Supersymmetry and Shape Invariance of the Effective Screened Potential
	Appendix E Knotted Pictures of Entangled States
	Appendix F Knotted Picture of the Whole Complete Quantum Measurement Process of Quantum Teleportation
Appendices of Essentials of Relativity
	Appendix G Einstein’s Field Equations
		G.1 Riemann curvature tensor
			G.1.1 Non-commutativity of covariant derivative
			G.1.2 Curvature tensor and its properties
			G.1.3 Curvature tensor and parallel transport
		G.2 Einstein’s equation of gravitational field
			G.2.1 Process of obtaining Einstein’s equation of gravitational field
			G.2.2 Discussions
			G.2.3 Weak field linear approximation
				G.2.3.1 Linearized Ricci tensor
				G.2.3.2 Coordinate condition
				G.2.3.3 Energy-momentum tensor
				G.2.3.4 Field equation
				G.2.3.5 Newton’s approximation
				G.2.3.6 External spherically symmetric static mass
		G.3 Schwarzschild solution and its consequences
			G.3.1 Schwarzschild solution
				G.3.1.1 General expression of the metric of spherically symmetric static mass
				G.3.1.2 Christoffel symbol
				G.3.1.3 Einstein tensor
				G.3.1.4 Solution of the vacuum field equations
				G.3.1.5 Birkhoff theorem
			G.3.2 Motion of planets: Perihelion precession of Mercury
				G.3.2.1 Differential equation of the orbit of planet
				G.3.2.2 Precession of perihelion of Mercury
			G.3.3 Deflection of light ray in the Schwarzschild field
			G.3.4 Time delay of the return wave of radar signal
			G.3.5 Gravitational redshift
			G.3.6 Schwarzschild geometry
				G.3.6.1 Singularity and pseudo-singularity
				G.3.6.2 Schwarzschild event horizon
	Appendix H A Possible Modification of Einstein’s Theory of General Relativity
	Appendix I A Possible Interpretation on Distance-Dependent Effect of Gravitational Constant in Newton’s Theory of Gravitation
	Appendix J Metric of Rotating Charged Spherical Mass in Vacuum for Vector Graviton Metric Theory of Gravitation
	Appendix K Some Interesting Features for External Region of Spherical Symmetric Mass in New Theory of Gravitation VGM
Bibliography
Index