## PHY 307: Quantum Mechanics II

3 credits | Prerequisites: PHY 304

### Course rationale

This elective course is a continuation of Quantum Mechanics 1. The course Quantum Mechanics 1 has laid the foundation of quantum theory. Quantum Mechanics 2 (QM2) will focus on symmetry and approximation methods to solve Schrodinger equation in different cases. QM2 will also introduce basics of relativistic quantum mechanics, and supersymmetric quantum mechanics

### Course content

Review of Quantum Mechanics: Schrodinger equation, Hilbert space, bra-ket, operator, uncertainty relations, Schrodinger equation in 3-dimensions, radial and angular equation, spin; Symmetry and Conservation Laws: symmetry in classical and quantum mechanics, SU(2), SO(3) and SO(4) symmetry, parity, parity selection rule, rotational symmetry, rotational selection rule, time translation, Heisenberg’s picture, time reversal symmetry, symmetry and degeneracy; Identical Particles: bosons and fermions, exchange force, spin, generalized symmetrization principle, atoms, Helium, free electron gas, Fermi surface and energy, band structure of solid; WKB approximation: classical region, tunneling, connection formulas; Time Independent Perturbation Theory: non-degenerate perturbation theory, first order correction, second order correction, degenerate perturbation theory, wave function renormalization; spin-orbit coupling, the Zeeman effect, hyperfine splitting;Time Dependent Perturbation Theory: interaction picture, the Dyson series, transition probability, Fermi’s golden rule, constant perturbation, harmonic perturbation; Variational Method: theory, ground state of Helium, Hydrogen ion, Hydrogen molecule; Hamiltonian With Extreme Time Dependence: sudden approximation, adiabatic approximation, Berry’s phase, Berry’s phase for spin ½, Aharonov-Bohm effect and magnetic monopole; Scattering Theory: transition rates and cross section, T matrix, scattering amplitude, optical theorem, phase shifts, Born approximation, partial wave expansion, unitarity and phase shifts, hard sphere scattering, Coulomb scattering, partial waves and eikonal approximation, rectangular well, rectangular barrier, poles and bound states, resonance scattering, symmetry consideration, electron-atom scattering; Relativistic Quantum Mechanics: natural unit, Klein-Gordon equation, interpretation of negative energies, Dirac equation, free-particle solution, interpretation of negative energies, electromagnetic interactions, angular momentum, parity, charge conjugation, time reversal, CPT; Supersymmetric Quantum Mechanics: bosonic harmonic oscillator, fermionic harmonic oscillator, supersymmetry algebra, the superpotential, shape invariance, exactly solvable examples.

### Course objectives

- Understand the advanced concepts of Quantum Mechanics.
- Understand different approximation methods to obtain wavefunction and energy.
- Use appropriate mathematical formalism to solve different problems of interest.
- Understand the basic concepts of relativistic quantum mechanics.
- Familiarize with the basic notions of supersymmetric quantum mechanics.

### References

- Modern Quantum Mechanics (3rd edition) by J. J. Sakurai, and Jim Napolitano
- Lectures on Quantum Mechanics (2nd edition) by Steven Weinberg
- Introduction to Quantum Mechanics (3rd edition) by David J. Griffiths, and Darrell F. Schroeter
- Principles of Quantum Mechanics (2nd edition) by R. Shankar
- Quantum Mechanics by Albert Messiah
- Quantum Physics (3rd edition) by Stephen Gasiorowicz
- Quantum Mechanics: A Modern Development (2nd edition) by Leslie E. Ballentine