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, C0[0,1] with L2 inner
product is not complete, space of square-summable complex sequences l2 and its standard
orthonormal basis, space of smooth functions with compact support C0∞(R),
L2
inner product is not an inner product
on the space of square-integrable functions, the
Hilbert space L2(R) and a space of equivalence classes of square-integrable functions, C0∞(R)
as a dense subspace of L2(R). |
- |
6 |
Oct. 19 |
Construction a non-convergent Cauchy sequence in C0[0,1]
with L2 inner 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 L2(R), their maximal domain and unboundedness. Examples
of bounded operators acting in L2(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:L2(ℝ)→ L2(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:L2(ℝ)→ L2(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:L2(ℝ)→ L2(ℝ) operator
that maps a function to –i times its derivative; approximate eigenvalues and the spectrum of
the X:L2(ℝ)→ L2(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:L2(ℝ)→ L2(ℝ) and its
‘generalized basis’ expansion |
|
12 |
Nov. 09 |
The K:L2(ℝ)→ L2(ℝ) operator and its
‘generalized basis’ expansion, Fourier transform as a unitary operator acting
in L2(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 ℝn, 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.