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