# Quantum Mechanics II

Phys 712 — Spring 2015 |

Department of Physics & Astronomy |

Instructor | Dr. Kevin Beach |

Office: | 206 Lewis Hall |

Email: | kbeach@olemiss.edu |

Website: | https://www.phy.olemiss.edu/~kbeach |

More details provided in the syllabus.

# Assignments

- due Friday, January 30; solution to question 3(b)
- due Wednesday, February 18
- due Friday, March 6; please only submit questions 2 and 3 for grading
- due Friday, April 15

# Lectures

- Review of the syllabus; differences between classical and quantum mechanics in the Lagrangian and Hamiltonian schemes; features of quantum mechanics that break our classical intuition; Dirac notation; linear vector spaces; orthonormality and completeness of states
- Operator formalism; Hilbert space axioms; operators and outer products; self-adjointness; time evolution of a quantum state; basis of energy eigenstates; unitarity of the evolution operator; Schrödinger and Heisenberg pictures; Dyson series; general case of evolution as a time-ordered exponential
- Quick review of 1D wave mechanics; free particles; Fourier transform relation between position and momentum; evolution operator evaluated with Suzuki-Trotter time slices; formal connection to Feynman path integral
- Propagators for free particle, harmonic oscillator, and generic single-particle models; derivations via operators and path integrals—see Sakurai Ch. 2.6
- Lagrangian for a charged particle in a classical EM field; Aharanov-Bohm effect—see Sakurai Ch. 2.7
- Cancelled due to University shutdown
- Review of Assignment 1; pole structure of propagators; interaction picture (as distinct from the Schrödinger and Heisenberg pictures)—see Sakurai Ch. 5.5
- Interaction picture; time dependence of states and operators; expansion coefficients obey a coupled set of first order differential equations; example: Rabi oscillations—see Sakurai Ch. 5.5
- Dyson equation; formulation of time-dependent perturbation theory in terms of the evolution operator in the interaction picture; transition probabilities; kicked harmonic oscillator; harmonic perturbations—see Sakurai Ch. 5.7
- Scattering of free particles from a time-dependent barrier in 1D; sudden approximation; adiabatic approximation—see Sakurai Ch. 5.6
- Importance of gapped versus gapless excitations and level crossings for adiabatic evolution; shaken harmonic oscillator; lattice translation; simultaneous eigenstates of definite energy and crystal momentum; tight-binding model of electron hopping around a ring
- Dr. Luca Bombelli
- Dr. Luca Bombelli
- Dr. Luca Bombelli
- Cancelled due to University shutdown
- Dr. Luca Bombelli
- Continuous and discrete translational symmetry; momentum and crystal momentum eigenstates;
- Cancelled
- Discrete translational symmetry; diagonalization of the tight-binding model via discrete Fourier Transform; block diagonal structure in the case of a nontrivial basis in each unit cell, e.g. graphene; symmetries in systems of quantum spins
- Spin-rotation invariance; spin sectors; addition of spin angular momentum; block structure of the spin Hamiltonian; spin-flip and time-reversal invariance
- Review of symmetry operators, good quantum numbers, selection rules, and block diagonalization; time-reversal operator; anti-unitarity; time reversal of free, spin-half particles
- Time-reversal symmetry and Kramer’s degeneracy; hyperfine splitting of Hydrogen and Deuterium; time reversal in the language of path integrals and propagators
- Many-body quantum states; expanded Hilbert space for a tight-binding model with two or more electrons; indistinguishability; particle exchange symmetry for fermions and bosons; Helium atom
- Helium atom with perturbative treatment of the e–e Coulomb repulsion; excited states; stabilization of para- and orthohelium; effective spin interactions arising from purely Coulombin forces; fermionic creation and annihilation operators
- Second quantization; number operator formalism; states of definite occupation; bilinear Hamiltonians; many-bond wave functions built as products of single-particle wave functions; contractions and Slater Determinants
- Second quantized form of the Hamiltonian for fermions and boson; pair-wise interaction terms
- Generalization of tight-binding electrons to higher spatial dimension, inifinite lattice sites, and continuum propagation; ground state of a gas of noninteracting electrons; fermi sea, fermi wave vector, and fermi energy