Phys 503: Advanced Quantum Mechanics, Fall 2023

Topics Covered in the Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Oct. 03	Review of classical mechanics (CM): States and observables in CM, kitenmatic and dynamical aspects of CM; Formulation based of the choice of dynamical eq (e.g. Newton's eqs of motion.); Conservative forces and conservation energy; Hamiltonian formulation of CM, Hamiltonian dynamical systems, Poisson bracket	-
2	Oct. 05	Lagrangian Formulation of CM: Action functional, Hamilton’s least action principle, Euler-Lagrange eqs, Legendre transformation and the Hamiltonian associated with a (nonsingular) Lagrangian, canonical Poisson bracket relations and canonical transformations	-
3	Oct. 10	Generating functions for canonical transformations, classical action function, and Hamilton-Jacobi formulation of CM, application to a simple harmonic oscillator in one dimension	-
4	Oct. 12	Basic idea of classical statistical mechanics, probability density and its local conservation, Liouville’s equation; Mathematical tools for QM: Complex vector and inner-product spaces, norm given by an inner product, unit and orthogonal vectors, orthonormal subsets, span, basis and dimension, closure of a subset, closed and open subsets, Schauder and orthonormal bases, separable inner product spaces	Pages 7-12 of textbook
5	Oct. 17	Cauchy sequences, Hilbert spaces, finite-dimensional inner-product spaces are Hilbert spaces, C^0[0,1] with L^2inner product is not complete, space of square-summable complex sequences l^2and its standard orthonormal basis, space of smooth functions with compact support C0^∞(R), L^2inner product is not an inner product on the space of square-integrable functions, the Hilbert space L^2(R) and a space of equivalence classes of square-integrable functions, C0^∞(R) as a dense subspace of L^2(R).	-
6	Oct. 19	Construction a non-convergent Cauchy sequence in C^0[0,1] with L^2inner product, Linear operators, matrix representation of a linear operator mapping between finite-dimensional vector spaces, matrix representation of vectors and linear operators in orthonormal bases, continuous functions and bounded linear operators, the multiplication (X) and differentiation (D) operators acting in L^2(R), their maximal domain and unboundedness. Examples of bounded operators acting in L^2(R): Multiplication operator by a Gaussian, parity, translations, and dilations	-
7	Oct. 24	Dual space of an inner-product space, the Dirac bra-ket notation; isomorphisms, isometries, and unitary operators, classification of separable Hilbert spaces; projection operators, complete orthogonal sets of projection operators associated with an orthonormal basis of a separable Hilbert space, orthogonal projections	-
8	Oct. 26	Orthogonal direct sum decomposition of an inner-product space induced by an orthogonal projection, symmetric operators, matrix representation of a symmetric operator in an orthonormal basis of a finite-dimensional inner-product space, Hermitian matrices, Hermitian adjoint of a matrix, adjoint of a linear operator acting in a finite-dimensional inner-product space, normal operators acting in a finite dimensional inner-product space and their spectral theorem; Adjoint of a densely-defined linear operator acting in an infinite-dimensional inner-product space, self-adjoint operators	-
9	Oct. 31	Proof of the self-adjointness of the multiplication operator X:L^2(ℝ)→ L^2(ℝ); regular values, resolvent set, the spectrum, point spectrum, continuous spectrum, and residual spectrum of a linear operator acting in an inner-product space, Spectrum of a self-adjoint operator acting in a Hilbert space, the spectrum of X:L^2(ℝ)→ L^2(ℝ); Operators with a discrete spectrum, Spectral Theorem for self-adjoint operators with a discrete spectrum, functions of a self-adjoint operator with a discrete spectrum, unitary operators acting in a Hilbert space, unitary group of a Hilbert space	-
10	Nov. 02	Matrix representation of a unitary operator acting in a finite-dimensional inner-product space, unitary matrices and unitary groups U(n); Using unitary operators mapping a Hilbert space to another for the purpose of inducing a one-to-one correspondence between linear operators acting in these Hilbert space, approximate eigenvalues, identifying the elements of the continuous spectrum of a self-adjoint operator with approximate eigenvalues, approximate eigenvalues and the spectrum of the K:L^2(ℝ)→ L^2(ℝ) operator that maps a function to –i times its derivative; approximate eigenvalues and the spectrum of the X:L^2(ℝ)→ L^2(ℝ) operator and Dirac delta function, Spectral Theorem for general self-adjoint operators; Kinematic structure of Quantum Mechanics (QM): Hilbert space, state vectors, states, projective Hilbert space, observables and von Neumann’s measurement (projection) axiom	-
11	Nov. 07	Measurement of an observable with a discrete spectrum: Probability of outcome of the measurement, expectation value and uncertainty; Measurement of an observable with continuous spectrum: Probability of outcome of the measurement, expectation value; the multiplication operator X:L^2(ℝ)→ L^2(ℝ) and its ‘generalized basis’ expansion
12	Nov. 09	The K:L^2(ℝ)→ L^2(ℝ) operator and its ‘generalized basis’ expansion, Fourier transform as a unitary operator acting in L^2(ℝ); Giving physical meaning to the quantum observables, Dirac’s canonical quantization program, Heisenberg’s uncertainty principle and coherent states	Pages 68-89 of textbook
		Winter Break
		Midterm Exam 1
13	Nov. 21	Heisenberg-Weyl algebra and the uniqueness of its unitary representations (Stone-von Neumann Theorem) as the quantum analog of Darboux’s theorem on the uniqueness of symplectic structures on ℝ^n, Quantum Dynamics: Time-evolution operator and its unitarity, Liouville-von Neumann equation, Dyson series expansion of evolution operator, time-ordered exponentials.	Pages 100-104, 121-123 of textbook
14	Nov. 23	Dynamical invariants with a discrete spectrum and their properties, cyclic and stationary states; time-dependent Hamiltonians with a discrete spectrum and expansion of evolving state vectors in its eigenvectors in the absence of level crossings	-
15	Nov. 28	Adiabatic approximation, dynamical phase, non-Abelian geometric phase, Berry’s phase and connection	-
16	Nov. 30	Necessary and sufficient condition for the exactness of the adiabatic approximation; The Hilbert space-Hamiltonian pairs representing the same quantum system: The relation between state vectors, observables, and the Hamiltonians in different representations of a given quantum system	-
17	Dec. 05	Dynamics in Heisenberg picture, Heisenberg equations, dynamical invariants in the Heisenberg picture; Schrödinger equation in the position representation.	Pages 125-130 of textbook
18	Dec. 07	Polar representation of the solution of the Schrödinger equation in the position representation for a standard Hamiltonian, quantum potential, quantum Hamilton-Jacobi equation, continuity equation for probability density of localization of a particle in configuration space, Bohm’s Causal interpretation of QM, the semiclassical (WKB) approximation.	Pages 374-376 of textbook
19	Dec. 12	Application of WKB approximation in determining the eigenvalues and eigenfunctions of a confining potential, the semiclassical formula for the spectrum and the Bohr-Sommerfeld quantization conditoon; Propagator in position representation	Weinberg’s Lectures on QM Pages 171-177
20	Dec. 14	Propagator for a free particle in position representation and spreading of the wave packets; Solution of the time-independent Schrödinger equation for a piecewise constant barrier/well potential in 1D and the transfer matrix; Bound state solutions as the positive imaginary zeros of the transfer matrix	Sakurai’s Modern QM 1994 Edition, Pages 109-112
21	Dec. 19	Bound states and scattering from a piecewise constant barrier/well potential in 1D for scattering states with energy less than potential at x=+infinity, reflection amplitude and consequences of unitarity	-
22	Dec. 21	Tunneling and scattering properties of a piecewise constant barrier/well potential in 1D, transmission reciprocity and consequences of unitarity; the general setup for the scattering theory of short-range potentials, transfer matrix and reflection and transmission amplitudes	Turk J. Phys. 2000 Review paper posted in Blackboard
		Midterm Exam 2
23	Dec. 26	Proof of the transmission reciprocity in 1D, EM analog of potential scattering in 1D, complex short-range potentials and material with loss and gain, absorption coefficients, the solution of the scattering problem for delta-function potentials in 1D, the case of complex coupling constant, spectral singularities and lasing	Turk J. Phys. 2000 Review paper posted in Blackboard
24	Dec. 28	The scattering matrix, composition property of the transfer matrix and the dynamical formulation of stationary scattering in 1d, application to single and multi-delta-function potentials	Turk J. Phys. 2000 Review paper posted in Blackboard
25	Jan. 02	Time-independent Perturbation Theory: General methodology of perturbation theory, perturbative solution of a transcendental equation, pertubative solution of time-independent Schrödinger equation for a self-adjoint Hamiltonian with a discrete and nondegenerate spectrum.	Pages 357-361 of textbook + Pages 289-294 of Sakurai’s 1994 Edition
26	Jan. 04	Application of time-independent perturbation theory for a two-level system and quartic perturbation of a simple harmonic oscillator; Basic idea of time-dependent perturbation theory	Pages 362-364 of textbook
27	Jan. 09	Time-dependent perturbation theory and its application in computing transition probabilities; Fermi’s golden rule	Pages 366-367 of textbook & Pages 325-333 of Sakurai’s 1994 Edition
28	Jan. 11	Scattering theory in dimensions 2 and 3: Lippmann-Schwinger equation, scattering amplitude, Born series, and n-th order Born approximation	Pages 379-390 of Sakurai’s 1994 Edition

Note: The pages from the textbook listed above may not include some of the material covered in the lectures. Quizzes are 30-50 minutes-long mini exams.