Phys 402: Quantum Mechanics II

Spring 2019

Topics Covered in Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 04	Review of the content of the syllabus; review of basic topics covered in Phys. 401, Representations of a quantum system using a Hilbert space-Hamiltonian operator pair (unitary equivalence)	-
2	Feb.06	Solution of the time-independent Schrodinger equation for a delta-function potential; Minimum uncertainty states and the simple harmonic oscillator (SHO), the unique role played by SHO in classical mechanics (behavior of a system near a stable equilibrium), Solution of the time-independent Schrodinger equation for a SHO in the position representation 1: Analysis of the asymptotic form of the solutions	Pages 237-241& 185-192 of Shankar
3	Feb. 11	Construction of the solutions of the time-independent Schrodinger equation for a SHO in the position representation: Hermit polynomial, energy spectrum, and energy eigenfunctions; Ladder operators	Pages 192-196 of Shankar
4	Feb. 13	Algebraic solution of the time-independent Schrodinger equation for a SHO: Construction of the eigenvector of the Hamiltonian and determination of the eigenvalues, the number operator, properties of the raising and lowering operators	Pages 202-207 of Shankar
5	Feb. 18	Representation of the linear operators in the energy basis of a SHO, calculation of the product of position and momentum uncertainties in the energy states of a SHO, Propagator for SHO. Digression: Canonical transformations and their generators, consequences of performing a canonical transformation that maps the Hamiltonian to zero.	Pages 206-209 & 92-97 of Shankar
6	Feb. 20	The classical action function as the generator of the canonical transformation that maps the Hamiltonian to zero: The Hamilton-Jacobi formulation of classical mechanics, application to one-dimensional systems; Polar form of the time-dependent Schrödinger equation in the position representation, Quantum potential and a quantum analog of the Hamilton-Jacobi equation in 1 dimension; probability current density, continuity equation for probability and its local conservation, the Pilot-wave interpretation of QM	-
7	Feb. 25	Polar form of the time-independent Schrödinger equation in the position representation, and the WKB approximation: the WKB wave function, application of the WKB approximation for the determination of the spectrum of a potential with a discrete spectrum (general description, no examples)	Pages 435-438 of Shankar
8	Feb. 27	Quantum mechanics of a particle with configuration space R^N: The Heisenberg algebra hN and the uniqueness of its unitary representations, the Hilbert space L^2(R^N) and the operators representing the position and momentum observables, the position representations of the time-dependent and time-independent Schrödinger equations for standard Hamiltonian operators in N dimensions; eigenvalues and eigenfunctions of the Hamiltonian for a particle in a box; appearance of degeneracies.	Pages 141-142 of Shankar
9	Mar. 04	Symmetry in QM: Review of the concepts of canonical transformations and symmetry transformations in CM; infinitesimal symmetry transformations and their generators; Nöther’s theorem in Hamiltonian formulation of CM; Unitary transformations of the Hilbert space as the quantum mechanical analog of canonical transformations; Winger’s symmetry theorem, linear symmetry transformations and their generators in QM, quantum analog of Nöther’s theorem, the characterization of nontrivial symmetries in terms of degenerate energy eigenvalues	Pages 279-289 of Shankar
10	Mar. 06	Translations and translational symmetry, reflections and parity operator(s)	Pages 289-301 of Shankar & pages 99-105 of Supplementary Material #1
11	Mar. 08	Time-reversal transformation and time-reversal symmetry, Rotations in 2D as a classical canonical transformation, the definition of the rotation operator, its linearity and unitarity, angular momentum operator Lz as the generator of rotations, derivation of the expression for the rotation operator in terms of Lz	Pages 99-105 of Supplementary Material #1 and pages 301-310 of Shankar
Midterm Exam 1	Mar. 15
12	Mar. 18	Commutation relations for position, momentum, and angular momentum operators in 2D, Campbell-Backer-Hausdorff identity, its proof, and application in computing the effect of a finite rotation on the position and momentum operators in 2D, rotational symmetry in 2D: Central potentials, solution of the eigenvalue problem for Lz in 2D and the quantization of angular momentum	Pages 310-315 of Shankar
13	Mar. 20	Time-independent Schrödinger equation for a central potential in 2D, solution for the isotropic infinite-well potential in 2D; Angular momentum in 3D: Commutation relations among Xi, Pi, and Li, Lie algebras, so(3) and su(2)	Pages 315-319 of Shankar
14	Mar. 22	Rotationally invariant quantum systems, the square of total angular momentum and its commutation with Li’s; Construction of an orthonormal set of common eigenvectors of L^2 and Lz.	Pages 319-322 of Shankar
15	Mar. 25	Eigenvalues of L^2 and Lz, and the unitary irreducible representations of the so(3) algebra	Pages 322-324 of Shankar
16	Mar. 27	Half-integer representations and the intrinsic angular momentum (spin), expressing the total angular momentum J=(J_1,J_2,J_3) as the sum of orbital angular momentum and spin, derivation of the matrix representation of Ji in the unitary irreducible representations of so(3)	Pages 324-329 of Shankar
17	Apr. 01	Position representation of the common eigenvectors of L^2 and Lz in spherical coordinate, absence of half-odd-integer representations, Explicit form of the eigenfunctions of L^2 and Lz in terms of the spherical harmonics and associated Legendre polynomials	Pages 333-338 of Shankar
18	Apr. 03	Solution of the time-independent Schrödinger equation for a standard rotationally invariant Hamiltonian operator in 3D (part 1): Derivation of the differential equation for the r-dependent part of the energy eigenfunctions, equivalence to a particle moving in a half-line	Pages 338-340 of Shankar
Spring Break
19	Apr. 15	Derivation of the boundary conditions for the radial part of the energy eigenfunctions of the rotationally invariant standard Hamiltonians at r=0 and r=infinity.	Pages 340-345 of Shankar
20	Apr. 17	The hydrogen atom: Differential equation determining the energy eigenfunctions and its reduction to a 2^nd order linear homogeneous equation with a regular singularity at r=0, application of the Frobenius method, determination of the energy levels of the hydrogen atom.	Pages 353-356 of Shankar
21	Apr 24	the position wave functions for the ground and first excited states of the hydrogen atom, the degeneracy of the energy eigenvalues, The emission and absorption energy and transition between energy levels of the hydrogen atom and the wavelength of the emitted and absorbed electromagnetic waves.	Pages 356-359 & 365-367 of Shankar
Midterm Exam 2	Apr. 26
22	Apr. 29	Spin as internal angular momentum, the Hilbert space for a particle with spin, the total angular momentum and effect of a rotation on the states of a spinning particle, basis in which L^2, L_z, S^2, and S_z are diagonal, the spin-half particles, spinors, Pauli matrices; the matrix form of a finite rotation for spin-1/2 particles, a spin ½ particle in a constant magnetic field	Pages 373-390 of Shankar
23	May 03	Dynamics of a spin ½ particle in a constant magnetic field, effects of placing a hydrogen atom in a constant magnetic field; QM of a Many-Particle system - Mathematical Preliminaries: Tensors, covariant and contravariant tensors, the vector space of tensors, their bases, and tensor product	Pages 385-392 & 397-399 of Shankar
24	May 06	Bases of space of tensors of a given type, the tensor algebra, inner product on the tensor product of N inner-product spaces, the tensor product of linear operators, their linearity, composition, and Hermiticity	-
25	May 08	Classical mechanics of a system of two distinct particles moving in a straight line: States, observables, Hamilton’s equations, transformation to center of mass and relative coordinates and the corresponding conjugate momenta; Quantum mechanics of a system of two distinct particles moving in a straight line: The Hilbert space, state vectors, and observables, the position and momentum operators, canonical commutation relations, the position and momentum kets and bras, the position wave function, the standard Hamiltonian when the particles do not interact and its energy eigenvalues, the standard Hamiltonian when there is only an inter-particle interaction depending on the relative coordinate, the energy spectrum and energy eigenfunctions for a pair of particles with a linear force between them (attached to one another by a spring);	Pages 247-258 of Shankar
26	May 13	Quantum mechanics of an identical pair of particles: Bosons and fermions; the symmetric and antisymmetric tensor product, their orthonormal bases and completeness relation	Pages 260-266 of Shankar
27	May 15	Measurement of observables for an identical pair of particles, the position measurement, position wave function, and probability of localization of pairs of identical bosons and fermions in space, The Pauli Exclusion Principle, formulation of QM for N distinct and identical particles, the permutation group and totally symmetric and antisymmetric tensor product, the Hilbert space for N identical bosons and fermions	Pages 267-273 of Shankar

Note: The pages from the textbook listed above may not include some of the material covered in the lectures.