Physics 660 Quantum Mechanics I Fall
Lecture Notes
I. Schrodinger Wave Mechanics
1.1. Introduction
1.2. Postulates
1.2.1. States and Wavefunctions
1.2.2. Probability density
1.2.3. Observables and Spectral Decomposition
1.2.4. Time Evolution: The Schrodinger Equation
1.3. Consequences and Physical Interpretation
1.3.2. Free Particle Wave Functions
a. Plane Waves
b. Gaussian Wave Packets
1.3.4. Canonical Commutation Relations
1.3.5. Ehrenfest’s Theorem and Classical Mechanics
1.3.6. Heisenberg Uncertainty Principle
1.3.7. Stationary States, Hermitian Operators, eigenvalues and eigenfunctions
1.3.8. Boundary Conditions on the Wavefunction
1.3.9. Feynman’s Path Integral Formulation of Quantum Mechanics
II. Applications In One-Dimension
2.1. Scattering off a Potential Step
2.1.1. Reflection and Transmission Coefficients
2.1.2. Barrier penetration
2.2.1. Square Well
2.2.2. Simple Harmonic Oscillator
2.2.3. Periodic Potentials and Bloch’s Theorem, the Kronig-Penny Model and
conduction bands in solids
III. Central Potential Problem
3.1. The Two Body Problem, the center of momentum frame
3.2. Spherical Polar Coordinates, Separation of Variables and Spherical Harmonics
3.3. Orbital Angular Momentum and the Spherical Harmonic Eigenfunctions
3.4. The Radial Equation and Bound States
IV. The Abstract Formulation of Quantum Mechanics
4.1. Hilbert Space and Dirac bra-ket notation
4.2. Operators, Eigenvalues and Observables
4.3. The Postulates of Quantum Mechanics
4.4. The Schrodinger, Heisenberg and Interaction Pictures
4.5. Position, Momentum and Energy Representations—Eigenstate bases
4.6. Consequences and Physical Interpretation of the Postulates
4.7. One-Dimensional Simple Harmonic Oscillator
4.8. One-Dimensional Simple Harmonic Oscillator in an external time dependent potential
4.9. The N-Dimensional Isotropic Simple Harmonic Oscillator and SU(N)
4.10. Feynman’s Path Integral Formulation of Quantum Mechanics Revisited