PHY 6645 - Fall 2003
Topics Covered

Mathematical Introduction

	Linear vector spaces (Sh1.1)
	Inner product spaces (Sh1.2)
	Dual spaces (Sh1.3)
	Vector subspaces (Sh1.4)
	Linear operators (Sh1.5)
	Matrix representations of linear operators (Sh1.6)
	Hermitian anti-Hermitian, and unitary operators (Sh1.6)
	The eigenvalue problem (Sh1.8)
	Eigenvalues and eigenvectors of normal operators (Me10.1)
	Functions of operators (Sh1.9)
	Infinite-dimensional vector spaces (Sh1.10, B1.4, handout, online book by U. H. Gerlach)

Classical Mechanics (background reading only)

	Hamiltonian formulation (Sh2.5)
	Possion brackets (Sh2.7)
	Canonical transformations (Sh2.7)
	Symmetries and conservation laws (Sh2.8) The key ideas are summarized in a one-page handout

Postulates of Non-Relativistic Quantum Mechanics

	Postulates for one degree of freedom (Sh4.1, handout)
	Postulate I: Quantum mechanical states (Sh4.2)
	Postulate II: Quantum mechanical operators (Sh4.2)
	Postulate III: Quantum mechanical measurements (Sh4.2, Ba9.3, notes by W. de Muynck)
	Expectation values and uncertainties (Sh4.2)
	Postulate IV: Time evolution (Sh4.3) NB: Different QM "pictures", the state (density) operator formulation, and systems with more than one degree of freedom will not be discussed until later in the semester.

One-Dimensional Wave Mechanics

	The Schrodinger wave equation
	The free particle (Sh5.1)
	The Gaussian wave packet (Sh5.1)
	General properties of wave functions (Sh5.1, Sh5.6)
	General analysis of 1D piecewise-constant potentials (Sh5.3, Me6.3, HW 4)
	Potential steps (Sh5.4, Me6.1)
	Potential barriers (Me6.2)
	Potential wells (Sh5.2, Me6.4)

Different Formulations of QM

	Different pictures of quantum mechanics (Sh18.3, Me14.2, Sa2.2) End of material tested on Exam 1
	The state operator formulation of quantum mechanics (Ba2.1-2.3, handout)

The Classical Limit

	Equations of motion for QM expectation values (Sh6, Ba14.1)
	The limit hbar -> 0 (Ba14.2)
	The limit of large quantum numbers (Ba14.4)

The Harmonic Oscillator

	Motivation (Sh7.1)
	Solution in the coordinate basis (Sh7.3, Me5.3)
	Physical properties of the eigensolutions (Sh7.3, Me5.3)
	Operator solution (Sh7.4, S7.5)
	Time evolution (Me5.4)
	Coherent states (Sh21.2, Me10.7)

Uncertainty Relations

	The Heisenberg uncertainty relations (Sh9.2, Me10.5)
	Minimum-uncertainty states (Sh9.3, Me10.5)
	Estimating the ground-state energy (reading assignment: Sh9.4)
	Energy-time uncertainty (Ba12.3)

Systems with More Than One Degree of Freedom

	Generalized postulates and commutation relations (Sh7.4)
	The Hilbert space (Sh10.1, 10.2)
	Time evolution of the state vector (Sh10.1)

Symmetry in Quantum Mechanics

	Symmetry operators (Ba3.1, Me17.1)
	Constants of motion and degeneracies
	Continuous symmetries (Ba3.1)
	Spatial displacements (Sh11.2, Ba3.4)
	Space translational invariance (Sh11.2)
	Temporal displacements and time translational invariance (Sh11.3)
	Spatial inversion and parity invariance (Sh11.4, Ba13.1)
	Time reversal and time-reversal invariance (Sh11.5, Ba13.3)

End of material tested on Exam 2

Rotational Symmetry

	Rotational symmetry operators (Sh12.2, S12.4)
	Rotational invariance about a single axis (Sh12.3)
	The angular momentum eigenproblem in 3D (Sh12.5)
	The rotation group SO(3) (Me17.3, Me17.4)
	Spatial eigenfunctions of L2 and Lz (Sh12.5, Me11.3, Me11.4)
	General solution of rotationally invariant problems (Sh12.6)
	The free particle in spherical coordinates (Sh12.6)
	The spherical well potential (Me12.3)
	The isotropic harmonic oscillator (Sh12.6)

The Hydrogen Atom

Shankar Ch. 13 in its entirety is assigned reading.

Spin

	QM description of spin (Sh14.3)
	Spin and rotation (Sh14.3)
	Spin-1/2 particles (Sh14.3, Ba7.4)
	Spin-1 particles (Sh14.3, Ba7.4)
	Rotation through 2 pi (Ba7.6, Ba13.3)

Addition of Angular Momenta

	Addition of angular momentum of two spin-1/2 particles (Sh15.1)
	General addition of two angular momenta (Sh15.2, Ba7.7)
	Addition of arbitrary L and spin-1/2 S (Sh15.2, Ba7.7) End of material tested on Final Exam
	Scalar, vector, and tensor operators (Sh15.3)
	Irreducible tensor operators (Sh15.3)