PHY 407 Quantum Mechanics I

Semester: Fall
Credit Hours: 4
Prerequisite: PHY 246, or permission of instructor

Quantum-mechanical axioms. Probability densities and currents. Boson representations of the oscillator. Angular momentum including Clebsch-Gordan coupling, spherical tensors, finite rotations, and applications to atoms and nuclei. Simple gauge transformations. Aharonov-Bohm effect. Bell's theorem. The SO(4) treatment of the hydrogen atom.

syllabus

1. Basic assumption in quantum mechanics, waves and particles. Superposition principle.
2. Motion of free wave packets. Group velocity. Spreading of packet.
3. Schroedinger equation. Transition to classical physics. Conservation of probability.
4. Simple well potentials. Infinitely deep well, wells of finite depth. Parity. Bound states, scattering states.
5. Vector spaces and operators on the Hermitian, Unitary operators. Basis transformations. Eigenvalues and Eigenvectors. Diagonalization of operators. Commutativity and its consequences. Differences between finite-dimensional and infinite-dimensional vector spaces. Hilbert spaces. Uncertainty relations as consequences of non-commutativity.
6. Dynamics. Electromagnetic fields. Difference between canonical and physical momentum. Time development of system.
7. Angular momentum components. Commutation relations. Raising and lowering operators. Total angular momentum. Integer and halfinteger spin.
8. One dimensional armonic oscillator. Creation and annihilation of "quanta". Eigenstates. Eigenstates as eigenfunction of Fourier transform operator.
9. Two and three dimensional harmonic oscillator. Operators which change both angular momentum state and energy in predictable ways. Simultaneous eigenstates of energy and angular momentum.
10. Applications: electron in magnetic field. Rotation of spin. Magnetic resonance. Rotation by 360 degrees. Connection between SU2 and O3 electron in magnetic field, connection to harmonic oscillator problem.
11. Central potential, separation of variables. Angular momentum Eigenstates, and connection to one-dimensional problem. Hydrogen atom.

Go to Physics Course Listing