Topics covered: QM2, Spring 2015
- Lecture 1 (Feb 3):
- Basic notation: States, operators, observables
- Measurable quantities: eigenvalues, expectation values
- Hermitian operators: eigenstates as the basis
- Non-commuting operators: uncertainty relation
- Unitary operators: change of basis, transformations of
states and operators
- Time evolution: Time dependent Schroedinger's equation,
unitarity implying Hermitian H
- Time varying H: Dyson series
- Heisenberg vs Schroedinger representation: conservation of
quantities when operators commute with H
- Lecture 2 (Feb 5):
- Position and momentum: < x | alpha > as alpha(x),
momentum as a generator of translation
- Solving a differential equation for < x | p >,
Fourier Transform alpha(x) <-> alpha(p) using < x | p >
- Free particle, and particle in a potential V(x), using
Heisenberg representation, Ehrenfest theorem
- Discretization of momentum operator
- Assignment 1 : Due on Feb 17
- Lecture 3 (Feb 17):
- Electromagnetism: Hamiltonian H = (1/2m)(p-e*A/c)^2+e*Phi
and equations of motion, canonical momentum p vs. kinematical momentum
Pi = m dx/dt = (p-e*A/c)
- Commutation relations between Pi, equation of motion for Pi,
modification of probability and current due to A
- Lorentz force law in QM
- Gauge transformation as a change of basis
- Lecture 4 (Feb 21):
- Scroedinger and Heisenberg pictures: how states, operators and
basis kets evolve
- Propagator as the Kernel of the integral operator, and hence
as the transition amplitude
- Propagator in terms of the action, the composition of propagators,
the path integral formulation
- Calculation of the Aharanov-Bohm effect using path integral
formulation
- Lecture 5 (Feb 24):
- Norm of an operator: Supremum of ||A |alpha> || / || |alpha> ||
- Decoherence: physical and practical / effective (due to measurement
limitations)
- Incoherent mixtures and ensemble averages
- Density matrix: hermitian, positive semidefinite, Tr(rho)=1
- Pure states vs mixed states using Tr(rho^2)
- Ensemble average using Tr(rho A)
- Lecture 6 (Mar 3):
- Examples of density matrices, non-uniqueness
- Time evolution of density matrix
- Density matrix and statistical mechanics: entropy
- Entropy maximization during thermal equilibrium leading to
Maxwell-Boltzmann energy distribution
- Lecture 7 (Mar 5):
- Angular momentum as the generator of rotation, commutation relations
between J from non-commuting rotations about different axes
- Angular momentum generators as operators that obey the SU(2) algebra,
Orbital angular momentum and spin as special cases of angular momentum
- Eigenstates of (J^2, Jz) as basis states, J(+/-) as raising and lowering
operators, Upper and lower bounds on Jz eigenvalues, quantization of
angular momentum in multiples of hbar/2
- |j m > as the basis states, operation of J^2, Jz, J(+/-) on
|j m > , matrix elements of these operators between basis states
- Assignment 2 : Due on Mar 17
- Lecture 8 (Mar 9):
- Addition of angular momenta, alternative bases of
|j1 j2 m1 m2 > and |j1 j2 j m >
- Clebsch-Gordan coefficients: selection rules, recursion relations,
ab-initio calculation using J(+/-)|j m >
- Lecture 9 (Mar 10):
- Effect of rotation on states (in terms of Euler angles)
- Wigner D-matrices and reduced Wigner functions
- Effect of rotation on operators
- Vector operators: commutation with angular momentum
- Spherical tensor operator: definition through rotation,
commutation relations with angular momentum operator
- Selection rules, Wigner-Eckart theorem
- Lecture 10 (Mar 17):
- Assignment 3 : Due on Mar 31
- Lecture 11 (Mar 19):
- Examples of spherical tensors, components of a rank-1
spherical tensor (vector) in terms of its cartesian components
- Constructing spherical tensors of ranks 0, 1, 2 from
a vector, Spherical harmonics as spherical tensors
- Application of Wigner-Eckart theorem: electric dipole
interaction
- Angular distributions of particle decays / atomic transitions
using Wigner matrices
- Lecture 12 (Mar 24):
- Semiclassical (WKB) approximation: conditions for validity,
quantization of bound states, tunnelling
- Patching of WKB solution with the solution of linear potential
problem, Airy functions and their asymptotic behaviour, modified
quantization and tunnelling conditions
- Lecture 13 (Mar 26):
- Variational principle for approximating ground state energy:
conditions for the test function, optimizing the estimation by
variation of parameters
- Time-independent perturbation theory: setting up the problem
in terms of unperturbed eigenstates
- First order correction to eigenvalue
- Conjugate projection operator and first order correction to
eigenvector, why this works only for non-degenerate cases
- Second order corrections to eigenvalues and eigenstates
- Lecture 14 (Mar 31):
- Normalization of perturbed eigenstates
- Convergence of perturbative expansion
- Two-state system: matching perturbative and exact solutions till
second order, radius-of-convergence argument for validity of the
approximation
- Perturbation of simple harmonic oscillator with [b x^2]: contribution
of higher states when the ground state is perturbed
- Hydrogen-like atom in a constant electric field: quadratic
Stark effect, polarizability of the atom
- Midterm 1: On Apr 5, at 10 am
- Lecture 15 (Apr 7):
- Time independent perturbation theory (degenerate):
modification of the expressions for first and second order
corrections to energies and eigenkets
- Diagonalization of V to determine the degenerate eigenkets
to be used in the expansion
- Linear Stark effect for the excited states of electron in H,
symmetry breaking giving rise to degeneracy breaking
- Spin-orbit (L.S) coupling and fine structure in atomic
energy levels
- Lecture 16 (Apr 9):
- General form of Hamiltonian for an electron in an atom:
potential terms due to L.S, L.B, S.B, |B|^2 (x^2+y^2), Sp.Se
- Relative strengths of the Coulomb potential, fine splitting,
Hyperfine splitting
- Atomic level splittings giving rise to Zeeman effect
[(Lz+2Sz)B term smaller than L.S term] and
Paschen-back effect [[(Lz+2Sz)B term dominating over L.S term]
- Lecture 17 (Apr 16):
- Time dependent perturbation theory: H = H0 + V(t),
aim to calculate transition amplitudes between energy eigenkets
of H0.
- Interaction picture: states, operators and their evolution
with time
- Dyson series as a solution to the evolution operator
in interaction picture
- Transition probabilities to first and second order in V(t)
- The case of constant potential turned on at some time t0:
Interpretation of energy-time uncertainty relation in
terms of transition probability to states with a different energy
- Lecture 18 (Apr 20):
- Constant potential turned on at t0: Fermi golden rule
- Assignment 4: Due on Apr 30
- Lecture 19 (Apr 21):
- Potential increasing as Exp[eta t]
- Transition probabilities to non-initial states match Fermi golden rule
- The limit [eta->0] matches time independent perturbation theory
- Second order corrections to the energy of the initial state gives
an imaginary contribution
- Relation of the imaginary contribution to energy with the decay rate
as well as the width of the Fourier power spectrum
- Lecture 20 (Apr 23):
- Harmonic perturbation: absorption and stimulated emission,
principle of detailed balance
- Interaction of atomic states with EM field: absorption
cross section, electric dipole approximation
- Density of final states, box normalization, transition rates
- Lecture 21 (Apr 28):
- When the eigenvalues and eigenstates of [H0+V] are known:
- Sudden approximation: depends only on initial and final Hamiltonian
- Adiabatic approximation: rate of transition depends on
the rate at which the perturbation is varied
- Harmonic oscillator with oscillating center in two extreme
limits: sudden approximation (maximum transition to a high energy state) and
adiabatic approximation (transition only to adjacent energy levels)
- Assignment 5: Due on May 14
- Lecture 22 (May 5):
- Scattering: basic formalism, Lippmann-Schwinger equation with
a vanishingly small quantity [epsilon] as regulator,
solution of the L-S equation with an incoming plane wave,
- calculation of propagator [<x'|1/(E-H0+i*epsilon)|x>],
Green's function
- Local V: correction due to scattered state as Fourier transform of
[V(x)]
- Confined V (i.e. |x'| << r): incoming plane wave and outgoing
spherical wave
- Scattering amplitude [f(k',k)], differential scattering cross section
- Lecture 23 (May 5):
- First order Born approximation, validity
- Born approximation with spherically symmetric potential
- Higher order corrections to Born approximation, transition operator T,
scattering amplitude in terms of [<k'|T|k>]
- Higher order corrections to Born amplitude: propagator in momentum
space, Feynman rules
- Lecture 24 (May 7):
- Spherical wave solutions | n l m > to Schrodinger equation
for free particles
- Projection of | n l m > on position basis and momentum basis
- Spherical potential and Wigner-Eckart theorem: T as scalar operator
- Decomposition of scattering amplitude in terms of partial-wave
amplitudes
- Interpretation of scattering in terms of spherical incoming wave
and outgoing wave (phase-shifted)
- Qualitative understanding of phase shifts with 1-dim and 3-dim potentials
- Lecture 25 (May 14):
- Spherically symmetric potential: equivalence of formalisms with
incoming plane wave and incoming-outgoing spherical waves, cross section
in terms of phase shifts [delta_l]
- Wavefunctions [u_l = r A_l] outside the potential in terms of
[j_l, n_l, delta_l], calculation of [delta_l] by matching boundary conditions
- Hard sphere scattering: phase shifts in terms of [j_l, n_l],
suppression of larger [l] values at small energies
- Hard shpere: S-wave phase shift [delta0 = -kR] and cross section
[4 Pi R^2]
- Final Exam (May 17)
- Lecture 26 (May 18):
- Low energy scattering: only S-wave, phase shift with a spherical finite
repulsive potential, matching wavefunctions inside and outside the
barrier
- Low energy scattering with a spherical finite attractive potential,
conditions for resonance (maximum cross section) and perfect transmission
- Extremely low energy limit: matching linear wavefunction and
long-wavelength sinusoidal wavefunction, scattering length
[a=(-1)/(k cot delta0)], S-wave scattering cross section
- Scattering length as "effective radius of equivalent hard-sphere",
scattering length for repulsive and attractive potentials, large
scattering length and cross section for low-energy bound states
- Assignment 6: Due on May 31
- Lecture 27 (May 19):
- High-energy scattering: cross section [2 Pi R^2], separation of
"real" [f_reflection] and "imaginary" [f_shadow]
- Calculating forward scattering amplitude
- Optical theorem: relating forward scattering amplitude with total
cross section
- Resonant scattering, resonance width, total resonant cross section
- Not covered....
- Eikonal approximation
- Scattering of identical particles
- Time-dependent scattering
- Inelastic scattering
- Scattering on long-range potentials (Coulomb)