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by Peter S. Riseborough

TOC

1 Principles of Classical Mechanics
1.1 Lagrangian Mechanics
1.1.1 The Principle of Least Action
1.1.2 The Euler-Lagrange Equations
1.1.3 Generalized Momentum
1.2 Hamiltonian Mechanics
1.2.1 The Hamilton Equations of Motion
1.2.2 Time Evolution of a Physical Quantity
1.2.3 Poisson Brackets
1.3 A Charged Particle in an Electromagnetic Field
1.3.1 The Electromagnetic Field
1.3.2 The Lagrangian for a Classical Charged Particle
1.3.3 The Hamiltonian of a Classical Charged Particle

2 Failure of Classical Mechanics
2.1 Semi-Classical Quantization

3 Principles of Quantum Mechanics
3.1 The Principle of Linear Superposition
3.2 Wave Packets
3.3 Probability, Mean and Deviations
3.4 Operators and Measurements
3.4.1 Operator Equations
3.4.2 Operator Addition
3.4.3 Operator Multiplication
3.4.4 Commutators
3.4.5 Eigenvalue Equations
3.4.6 Adjoint or Hermitean Conjugate Operators
3.4.7 Hermitean Operators
3.4.8 Eigenvalues and Eigenfunctions of Hermitean Operators
3.4.9 Hermitean Operators and Physical Measurements
3.5 Quantization
3.5.1 Relations between Physical Operators
3.5.2 The Correspondence Principle
3.5.3 The Complementarity Principle
3.5.4 Coordinate Representation
3.5.5 Momentum Representation
3.5.6 Commuting Operators and Compatibility
3.5.7 Non-Commuting Operators
3.5.8 The Uncertainty Principle
3.6 The Philosophy of Measurement
3.7 Time Evolution

4 Applications of Quantum Mechanics
4.1 Exact Solutions in One Dimension
4.1.1 Particle Confined in a Deep Potential Well
4.1.2 Time Dependence of a Particle in a Deep Potential Well
4.1.3 Particle Bound in a Shallow Potential Well
4.1.4 Scattering from a Shallow Potential Well
4.1.5 The Threshold Energy for a Bound State
4.1.6 Transmission through a Potential Barrier
4.1.7 The Double Well Potential
4.1.8 The delta function Potential
4.1.9 Bound States of a delta function Potential
4.2 The One-Dimensional Harmonic Oscillator
4.2.1 The Raising and Lowering Operators
4.2.2 The Effect of the Lowering Operator
4.2.3 The Ground State
4.2.4 The Effect of The Raising Operator
4.2.5 The Normalization
4.2.6 The Excited States
4.2.7 Time Development of the Harmonic Oscillator
4.2.8 Hermite Polynomials
4.2.9 The Completeness Condition
4.3 Dual-symmetry
4.4 Bargmann Potentials
4.5 Orbital Angular Momentum
4.5.1 Simultaneous Eigenfunctions
4.5.2 The Raising and Lowering Operators
4.5.3 The Eigenvalues and Degeneracy
4.5.4 The Effect of the Raising Operators
4.5.5 Explicit Expressions for the Eigenfunctions
4.5.6 Legendre Polynomials
4.5.7 Associated Legendre Functions
4.5.8 Spherical Harmonics
4.5.9 The Addition Theorem
4.5.10 Finite-Dimensional Representations
4.5.11 The Laplacian Operator
4.6 Spherically Symmetric Potentials
4.6.1 The Free Particle
4.6.2 The Spherical Square Well
4.6.3 Ladder operators for a free particle
4.6.4 The Rayleigh Equation
4.6.5 The Isotropic Planar Harmonic Oscillator
4.6.6 The Spherical Harmonic Oscillator
4.6.7 The Bound States of the Coulomb Potential
4.6.8 Ladder Operators for the Hydrogen Atom
4.6.9 Rydberg Wave Packets
4.6.10 Laguerre Polynomials
4.7 A Charged Particle in a Magnetic Field
4.7.1 The Degeneracy of the Landau Levels
4.7.2 The Aharanov-Bohm Effect
4.8 The Pauli Spin Matrices
4.9 Transformations and Invariance

5 The Rotating Planar Oscillator

6 Dirac Formulation
6.1 Dirac Notation
6.1.1 Bracket Notation
6.1.2 Operators
6.1.3 Adjoints and Hermitean Operators
6.1.4 Representation of Operators
6.2 Representations
6.3 Gram-Schmidt Orthogonalization

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