PHY 6645 - Fall 2001
Topics Covered

Mathematical Introduction

Linear vector spaces (S1.1) Inner product spaces (S1.2) Dual spaces (S1.3) Vector subspaces (S1.4) Linear operators (S1.5) Matrix representations of linear operators (S1.6) Hermitian anti-Hermitian, and unitary operators (S1.6) The eigenvalue problem (S1.8) Eigenvalues and eigenvectors of normal operators (M10.1) Functions of operators (S1.9) Infinite-dimensional vector spaces (S1.10, B1.4, handout)

Classical Mechanics (background reading only)

Hamiltonian formulation (S2.5) Possion brackets (S2.7) Canonical transformations (S2.7) Symmetries and conservation laws (S2.8)   The key ideas are summarized in a one-page handout

Postulates of Non-Relativistic Quantum Mechanics

Postulates for one degree of freedom (S4.1) Postulate I: Quantum mechanical states (S4.2) Postulate II: Quantum mechanical operators (S4.2) Postulate III: Quantum mechanical measurements (S4.2, B9.3) Expectation values and uncertainties (S4.2) Systems with N degrees of freedom (S4.2) Postulate IV: Time evolution (S4.3)

One-Dimensional Wave Mechanics

The Schrodinger wave equation The free particle (S5.1) The Gaussian wave packet (S5.1) General properties of wave functions (S5.1, S5.6) General analysis of 1D piecewise-constant potentials (S5.3, M6.3, study guide) Potential steps (S5.4, M6.1) Potential barriers (M6.2) Potential wells (S5.2, M6.4) End of material tested on Exam 1

The Classical Limit

Equations of motion for QM expectation values (S6, B14.1) The limit hbar --> 0 (B14.2) The limit of large quantum numbers (B14.4)

The Harmonic Oscillator

Motivation (S7.1) Solution in the coordinate basis (S7.3, M5.3) Physical properties of the eigensolutions (S7.3, M5.3) Operator solution (S7.4, S7.5) Time evolution (M5.4)

Uncertainty Relations

The Heisenberg uncertainty relations (S9.2, M10.5) Minimum-uncertainty states (S9.3, M10.5) Estimating the ground-state energy (S9.4) Energy-time uncertainty (B12.3)

Systems with More Than One Degree of Freedom

Commutation relations (S7.4) The Hilbert space (S10.1, 10.2) Time evolution of the state vector (S10.1)

Alternative Formulations of Quantum Mechanics

The Schrodinger, Heisenberg, and interaction pictures (M14.2) The density operator (M15.5, B2.2, B2.3)

Symmetry in Quantum Mechanics

Symmetry operators (B3.1, M17.1) Constants of motion and degeneracies Continuous symmetries (B3.1) Spatial displacements (S11.2, B3.4) Space translational invariance (S11.2) Relationship between classical and quantum mechanical symmetries (S11.2) Temporal displacements and time translational invariance (S11.3) Spatial inversion and parity invariance (S11.4, B13.1) Time reversal and time-reversal invariance (S11.5, B13.3)

Rotational Symmetry

Rotational symmetry operators (S12.2, S12.4) Rotational invariance about a single axis (S12.3) End of material tested on Exam 2 The angular momentum eigenproblem in 3D (S12.5) The rotation group SO(3) (M17.3, M17.4) Spatial eigenfunctions of L2 and Lz (S12.5, M11.3, M11.4) General solution of rotationally invariant problems (S12.6) The free particle in spherical coordinates (S12.6) The spherical well potential (M12.3) The isotropic harmonic oscillator (S12.6)

The Hydrogen Atom

Shankar Ch. 13 in its entirety is assigned reading.

Spin

QM description of spin (S14.3) Spin and rotation (S14.3) Spin-1/2 particles (S14.3, B7.4) Spin-1 particles (S14.3, B7.4) Rotation through 2 pi (B7.6, B13.3)

Addition of Angular Momenta

Addition of angular momentum of two spin-1/2 particles (S15.1) General addition of two angular momenta (S15.2, B7.7) Addition of arbitrary L and spin-1/2 S (S15.2, B7.7) End of material tested on Final Exam Scalar, vector, and tensor operators (S15.3) Irreducible tensor operators (S15.3)