Topics covered: QM2, Spring 2016

• Lecture 1 (Feb 9):
  • Basic notation: States, operators, observables, eigenstates
  • Measurable quantities: eigenvalues, expectation values
  • Unitary operators: change of basis, transformations of states and operators, norms of states and operators
• Lecture 2 (Feb 12):
  • Diagnostic Quiz 0: in class
  • Position operator and its eigenstates, projections on position eigenstates as wavefunctions: < x | alpha > = alpha(x)
  • Translation operator T(dx), momentum as a generator of translation
  • From properties of translation operator to representation of momentum operator as space derivative
  • Obtaining < x | p >, overlap of position and momentum eigenstates
  • Fourier Transform as a change of basis. alpha(x) <-> alpha(p) using < x | p >
• Lecture 3 (Feb 16):
  • Time evolution of states and operators in Scrodinger and Heisenberg pictures, respectively
  • Comparing classical and quantum ways of obtaining the equations of motion for free particle, and particle in a potential V(x): classical Hamilton's equations vs. quantum evolution of operators in Heisenberg representation
  • 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 among Pi's
  • Equation of motion for Pi: Lorentz force law in QM
• Lecture 4 (Feb 19):
  • From Gauge symmetry in ED to Gauge transformation of state in QM
  • Decoherence: physical and practical / effective (due to measurement limitations)
  • Incoherent mixtures and ensemble averages
  • Density matrix: hermitian, positive semidefinite, Tr(rho)=1
  • Ensemble average [A] = Tr(rho A)
• Lecture 5 (Feb 22):
  • Pure states vs mixed states using Tr(rho^2)
  • Examples of density matrices, non-uniqueness
  • Time evolution of density matrix
  • Angular momentum operator as generator of rotation
  • Commutation relations among J from non-commuting rotations about different axes
  • Generalized notion of angular momentum as anything that satisfied the angular momentum algebra
• Lecture 6 (Feb 25):
  • 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
  • Addition of angular momentum: moving to a bigger Hilbert space
• Lecture 7 (Mar 1):
  • 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 >
• Assignment 1 : Due on Mar 15
• Lecture 8 (Mar 4):
  • Effect of rotation on states (in terms of Euler angles)
  • Wigner D-matrices and reduced Wigner functions
  • Explicit calculations of representations of angular momentum operators for any J value
  • Effect of rotation on operators
  • Vector operators: commutation relations with angular momentum
  • Cartesian vs spherical tensors
• Lecture 9 (Mar 8):
  • Spherical tensor operator: definition through rotation, using similarity with spherical harmonics
  • Spherical tensor operators: commutation relations with angular momentum operator
  • Selection rules, Wigner-Eckart theorem
• Lecture 10 (Mar 11):
  • Tutorial 1
  • Applications of Wigner-Eckart theorem
  • Angular distributions in particle decays
• Assignment 2 : Due on Mar 29
• Lecture 11 (Mar 15):
  • Variational principle for approximating ground state energy: conditions for the test function, optimizing the estimation by variation of parameters
  • Estimating how close one is to the actual ground state, estimating energy of first excited state if ground state wavefunction is known.
  • Semiclassical (WKB) approximation: conditions for validity
• Lecture 12 (Mar 18): (Tutorial by Sounak Biswas)
  • Propagator of the Schrodinger Equation, it's interpretation as the transition amplitude between Eigenkets of Heisenberg position operators
  • Composition property of the transition amplitude, hence the Path integral.
  • Explicit path integral representation of the propagator for [H=p^2/2m + V], by evaluation of matrix elements of infinitesimal evolution operator (normal ordered) .
  • The continuum limit as [\int D[x(t)] (exp(iS/h)], and the classical limit as the classical path contributing in the [h->0] limit
  • Formal expressions for expectation values of position operators in the Path integral language, outline of the proof of Eherenfest's theorem
• Lecture 13 (Mar 22):
  • Patching of WKB solution with the solution of linear potential problem, Airy functions and their asymptotic behaviour,
  • Modified quantization and tunnelling conditions with WKB
  • Estimating density of states using WKB approximation
  • Setting up of the formalism for time-dependent perturbation theory in terms of unperturbed eigenstates
• Lecture 14 (Mar 29):
  • First order correction to eigenvalues
  • Conjugate projection operator and first order correction to eigenvector, why this works only for non-degenerate cases
  • Second order corrections to eigenvalues and eigenstates
  • Normalization of perturbed eigenstates
  • Convergence of perturbative expansion
• Lecture 15 (Apr 1):
  • Degenerate time-independent perturbation theory: the right choice of basis
  • Example of Hamiltonian as a 2x2 matrix with small off-diagonal elements: non-degenerate and degenerate cases
  • Perturbation of simple harmonic oscillator with [b x^2]
  • Hydrogen-like atom in a constant electric field: quadratic Stark effect for the ground state, polarizability of the atom
  • Linear Stark effect for the excited states of electron in H, symmetry breaking giving rise to degeneracy breaking
• Lecture 16 (Apr 5):
  • Hamiltonian for electron in an Hydrogen atom: spin-orbit coupling, effect of nuclear spin
  • Fine splitting, hyperfine splitting
  • Atomic transitions: relative energies and wavelengths
• Assignment 3: Due on Apr 22
• Lecture 17 (Apr 6):
  • Hydrogen atom energy levels in the absence of external influences
  • Hamiltonian for electron in an Hydrogen atom: effect of constant electric and magnetic fields
  • Zeeman effect, Paschen-Back effect
  • Variational principle for screening in He atom
• Lecture 18 (Apr 12):
  • 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
  • Perturbative expansion in terms of coefficients [c_n], starting with [c_i=1]
• Lecture 19 (Apr 13):
  • Transition probabilities to first and second order in V(t)
  • Constant potential turned on at some time t0: Interpretation of energy-time uncertainty relation
  • Fermi's Golden rule
  • Second order transition probability, virtual transitions,
  • 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
• Lecture 20 (Apr 15):
  • Revision and Tutorial
• Midterm: Apr 17
• Lecture 21 (Apr 19):
  • Exp[eta t] potential: Second order corrections to the energy of the initial state
  • Relation of the imaginary contribution to energy with the decay rate as well as the width of the Fourier power spectrum
• Lecture 22 (Apr 22):
  • 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, transition rate between energy levels
  • Photoelectric effect: transitions to positive energy states, box normalization, transition rates
• Lecture 23 (Apr 26):
  • Validity conditions for time-dependent perturbation theory
  • Approximation techniques: separation of time / energy scales
  • Sudden approximation
  • Adiabatic approximation
• Assignment 4 : Due on May 17
• Lecture 24 (Apr 29):
  • 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)]
  • Short-range V (i.e. |x'| << r): incoming plane wave and outgoing spherical wave
  • Scattering amplitude [f(k',k)]
• Lecture 25 (May 5):
  • Interpretation of scattering amplitude, relation with differential cross section
  • First order Born approximation
  • Example of deducing lattice structure with scattering
  • Validity of Born approximation
  • 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
  • Born approximation with spherically symmetric potential
• Lecture 26 (May 6):
  • Properties of Born amplitude with spherically symmetric potential
  • 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 27 (May 9):
  • Spherically symmetric potential: total cross section in terms of phase shifts [delta_l]
  • Wavefunctions [u_l = r A_l] 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, S-wave phase shift [delta0 = -kR] and cross section [4 Pi R^2]
• Assignment 5: Due before May 31
  (Includes topics to be covered in later lectures)
• Tutorial (May 10)
• Tutorial (May 13)
• Lecture 28 (May 17):
  • Low energy scattering with a spherical finite attractive potential, resonance and perfect transmission
  • Extremely low energy limit: matching linear and long-wavelength sinusoidal wavefunctions, 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
  • High-energy scattering: cross section [2 Pi R^2], separation of "real" [f_reflection] and "imaginary" [f_shadow]
  • Resonant scattering, resonance width, total resonant cross section
• Lecture 29 (May 20):
  • The power of analyticity of scattering amplitude: connecting positive and negative energy states, bound states as poles of scattering amplitude
  • Calculating forward scattering amplitude
  • Optical theorem: relating forward scattering amplitude with total cross section
• Final Exam (May 26)
• Not covered....
  • Eikonal approximation
  • Scattering of identical particles
  • Time-dependent scattering
  • Inelastic scattering
  • Scattering on long-range potentials (Coulomb)