Phys 402/OEPE 542: Quantum Mechanics II
Sprinf 2018
Topics Covered in Lectures
|
Lecture Number |
Date |
Content |
Corresponding Reading material |
|
1 |
Feb. 06 |
Review of the content of the syllabus;
review of basic topics covered in Phys. 401 |
- |
|
2 |
Feb.13 |
Physical equivalence of Hamiltonians that differ by a
possibly time-dependent multiple of the identity operator, solution of the
time-independent Schrodinger equation for a delta-function potential; The
uncertainty principle, the minimum uncertainty states, and the simple
harmonic oscillator (SHO) as the system having minimum uncertainty states as
an energy eigenstate, the unique role played by SHO in classical mechanics
(behavior of a system near a stable equilibrium.) |
Pages
237-241& 185-186 of Shankar |
|
3 |
Feb. 15 |
Solution of the time-independent Schrodinger equation for a
SHO in the position representation: Hermit polynomial, energy spectrum, and
energy eigenfunctions |
Pages
189-196 of Shankar |
|
4 |
Feb. 20 |
Algebraic solution of the time-independent Schrodinger
equation for a SHO: Ladder operators, construction of the eigenvector and
eigenvalues of the Hamiltonian |
Pages
202-206 of Shankar |
|
5 |
Feb. 22 |
Properties of the raising and lowering operators,
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 |
Pages
206-209 of Shankar |
|
6 |
Feb. 27 |
Propagator for SHO; Pilot-wave interpretation of QM:
Quantum potential and quantum analog of Hamilton-Jacobi equation in 1
dimension; probability current density, continuity equation for probability,
and its local conservation, the Pilot-wave interpretation, application to
stationary states and the semiclassical
approximation |
- |
|
7 |
Mar. 01 |
The WKB wave function, application of the WKB approximation
for the determination of the spectrum of a potential with a discrete spectrum
(general description of the idea, no examples); Quantum mechanics of a
particle with configuration space R^N: The Heisenberg algebra h_N and the uniqueness of its unitary representations,
the Hilbert space L^2(R^N) and the operators representing the position and
momentum observables |
Pages
435-438 & 141-142 of Shankar |
|
8 |
Mar. 06 |
Consequences of inclusion of adding a function of position
to the standard momentum operators and unitary equivalence of the unitary
representations of the Heisenberg algebra h_N, 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 an infinite potential well; appearance of degeneracies. |
- |
|
9 |
Mar. 08 |
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;
symmetry transformations and their generators in QM; a quantum analog of Nöther’s theorem |
Pages
279-289 of Shankar |
|
10 |
Mar. 13 |
Translations and translational symmetry, reflections and
parity operator(s), antilinear symmetries,
time-reversal transformation and time-reversal symmetry |
Pages
289-303 of Shankar & 20-24 of Supplementary Material #1 |
|
11 |
Mar. 15 |
Adjoint of an antilinear operator, properties of the time-reversal
operator; Rotations in 2D as a classical canonical transformation, the definition
of the rotation operator, its linearity and unitarity,
angular momentum operator L_z as the generator of
rotations, derivation of the expression for the rotation operator in terms of
L_z |
Pages 20-24 of Supplementary
Material #1 and 305-310 of Shankar |
|
Midterm
Exam 2 |
Mar. 20 |
|
|
|
12 |
Mar. 21 |
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 L_z in 2D and the quantization of angular momentum. |
Pages
310-315 of Shankar |
|
13 |
Mar. 22 |
Comment on canonical
and unitary transformations and their
difference with active and passive transformation of classical and quantum
systems; time-independent Schrödinger equation for a central potential in 2D,
solution for the isotropic infinite-well potential in 2D |
Pages
315-318 of Shankar |
|
14 |
Mar. 27 |
Angular momentum in 3D:
Commutation relations among X_i, P_i, and L_i, Lie algebras,
so(3) and su(2); the rotationally invariant quantum
systems, the square of total angular momentum and its commutation with L_i’s |
Pages
318-321 of Shankar |
|
15 |
Mar. 29 |
Construction of an
orthonormal set of eigenvectors of L^2 and L_3 using the so(3) algebra
relations |
Pages
321-324 of Shankar |
|
16 |
Apr. 03 |
Interpreting the analysis of
the preceding lecture as the construction of the unitary irreducible
representation of so(3); the 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. 05 |
Position representation of
the common eigenvectors of L_3 and L^2 in spherical coordinate: Spherical
harmonics and associated Legendre polynomials |
Pages
333-338 of Shankar |
|
Spring Break |
|
|
|
|
18 |
Apr. 17 |
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, implications of the self-adjointness and boundary condition at r=0 and r=infinity. |
Pages
338-342 of Shankar |
|
19 |
Apr. 19 |
Solution of the
time-independent Schrödinger equation for a standard rotationally invariant
Hamiltonian operator in 3D (part 2): Boundary condition at r=0; The hydrogen
atom: Differential equation determining the energy eigenfunctions
and its reduction to a 2nd order linear homogeneous equation with
a regular singularity at r=0. |
Pages
342-343 &353-354 of Shankar |
|
20 |
Apr. 24 |
Application of Frobenius method to determine the energy eigenfunctions of the hydrogen atom and its energy
levels, the position wave functions for the ground and first excited states
of the hydrogen atom, the degeneracy of the energy eigenvalues |
Pages
354-359 of Shankar |
|
21 |
Apr. 26 |
The emission and absorption
energy and transition between energy levels of the hydrogen atom and the
wavelength of the emitted and absorbed electromagnetic waves; 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 |
Pages
361-368 & 373-385 of Shankar |
|
22 |
May 03 |
Dynamics of a spin ½
particle in a constant magnetic field, effects of placing a hydrogen atom in
a constant magnetic field |
Pages
385-392 & 397-399 of Shankar |
|
Midterm
Exam 2 |
|
|
|
|
23 |
May 08 |
QM of a Many-Particle
system - Mathematical Preliminaries: Tensors, covariant and contravariant
tensors, the vector space of tensors, their bases, and tensor product; The
tensor algebra |
- |
|
24 |
May 10 |
Inner product on the tensor
product of N inner-product spaces, the tensor product of linear operators,
their linearity, composition, and Hermiticity,
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 |
- |
|
25 |
May 15 |
The position and momentum
operators for a quantum system of two distinct particles moving in a straight
line, the 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 coordinates, the energy spectrum and energy eigenfunctions for a pair of particles with a linear
force between them (attached to one another by a spring); Quantum mechanics
of an identical pair of particles: Bosons and fermions; the symmetric and
antisymmetric tensor product, their orthonormal bases and completeness
relation |
Pages
247-258 & 260-266 of Shankar |
|
26 |
May 17 |
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.