Topics covered: QM 2004
- Lecture 1 (Jan 27):
- 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
- Solving H = beta sigma_x (spin precession) using
U = exp(-i H t)
- Heisenberg vs Schroedinger representation: conservation of
quantities when operators commute with H
- Lecture 2 (Jan 29):
- Calculation of < sigma_z > with H = beta sigma_x using evolution
of sigma_z in Heisenberg representation
- Neutrino oscillations: starting with a flavour eigenstate,
propagating the mass eigenstates and reconstructing a flavour eigenstate at time t
- 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 >
- Particles as Gaussian wavepackets: x-modulation of the amplitude of a plane
wave corresponds to a spread in momentum
- One dimensional potential problems: Time independent Schroedinger's
equation, boundary conditions, probability density and current,
continuity relation
- Potential step: reflection and transmission coeficient
- Potential barrier: extrapolating the tunneling probability from a 'step'
barrier to one with an arbitrary shape
- Lecture 3 (Feb 3):
- Bound states of a potential well: even and odd solutions
- Scattering states (E>0) of a potential well:
Transmission coefficient T(E) as a physical quantity
- Resonances (|T(E)|^2=1) of scattering states corresponding to
bound state energies of an infinitely deep well
- Poles of T(E) corresponding to bound state energies of
a finite potential well
- Near-resonance behaviour of T(E) leading to Breit-Wigner
shape: amplitudes and phase shifts near resonances
- Analytical structure of T(E) in the complex E-plane:
branch cut, poles (bound states), resonances
- Lecture 4 (Feb 5):
- Semiclassical (WKB) approximation: tunneling probability for
a barrier, quantization condition for a well (See Schiff, pp 268--276)
- Approximating bound state spectrum using WKB
- Lecture 5 (Feb 10):
- 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 flux due to A
- Gauge transformation: A -> A + grad(Lambda) and the change of
phase of the wavefunction locally
- Propagator: as the Kernel of the integral operator,
expansion in terms of energy eigenstates,
Fourier transform of the trace of the propagator and its
poles at the energy eigenvalues
- Propagator as the transition amplitude, composition of propagators
- Discretization of Schroedinger's equation, momentum operator
mixing |x> states, Mixing of |x> states by H of electromagnetism
- Aharanov-Bohm effect using composition of discretized propagators
- Lecture 6 (Feb 12):
- Angular momentum as the generator of rotation, commutation relations
between J from non-commuting rotations about different axes
- 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 >
- Rotation of a state: Wigner functions D in terms of Euler angles,
reduced Wigner functions d and their explicit calculation for J=1/2,
Pauli matrices as representations of spin 1/2 particles
- Orbital angular momentum: L^2, L_z and L(+/-) operators,
the identity L^2 = x^2 p^2 - (x . p)^2 + i hbar x . p implying
separability of 3-dim Schroedinger's equation into r-dependent and
angle-dependent parts
- Y(l,m) as solutions to the angular part of Schroedinger's equation,
ruling out the possibility of half integer l and m
- Lecture 7 (Feb 19):
- Addition of angular momenta, Good and bad quantum numbers depending
on the interaction, alternative basis of |j1 j2 m1 m2 > and |j1 j2 j m >
- Clebsch-Gordan coefficients: selection rules, recursion relations,
ab-initio calculation using J(+/-)|j m >
- Vector operators: commutation with angular momentum, combining
two vector operators to get irreducible scalar, vector and tensor operators
- Lecture 8 (Feb 24):
- Tensor operator: definition through rotation, commutation relations
with angular momentum operator
- Selection rules, Wigner-Eckart theorem
- Lecture 9 (Feb 26):
- Wigner-Eckart theorem revisited, applications
- Relation between Y(l,m) functions and Wigner matrix elements D(l,0,m)
- Angular distributions of decays of particles through helicity amplitudes
- Lecture 10 (Mar 2):
- Pure vs. mixed states: information about the relative phases between
states
- Loss of phase information and decoherence: averaging of the phase,
fast time variation of the phase, finite time resolution or energy
resolution giving rise to ''effective'' loss of coherence. Example of
loss of interference terms in neutrino oscillations.
- Decoherence due to wavepacket separation: different speeds of mass
eigenstates
- Density matrix: pure vs. mixed ensemble, ensemble average,
definition of density matrix, identification of a pure density matrix
through the calculation of Tr(rho^2), equivalence of two ensembles
with the same density matrix
- Density matrix formalism of QM: expectation values as
< A > = Tr(rho A), transition probabilities through
|< psi | phi >|^2 = Tr(rho_psi rho_phi), time evolution of
density matrix as (i hbar d rho/dt) = - [rho, H]
- Conservation of Tr(rho^2), a pure state cannot become a mixed state
and vice versa.
- Spin 1/2 particles: Equivalence of the |+ > |-> notation
and two dimensional column vectors, all the spin operators and their
eigenkets in the two notations
- Spin precession in the |+ > |-> notation,
the change in sign of the phase when rotated through 2 pi,
neutron interferometry to detect this change in sign.
- Midterm I : Mar 4
- Lecture 12 (Mar 9):
- Symmetries in QM: symmetry operator, commutation with H,
degeneracy
- Parity: eigenvalues and eigenvectors, action on
operators like position, momentum, angular momentum. polar vs.
axial vectors, scalars vs. pseudoscalars. Action on |l m >
- If [H, Pi]=0 and |n> is a nondegerate energy eigenstate,
then |n> is also a parity eigenstate
- Symmetric double well potential, R-L oscillations
- Parity selection rules, no edm for quantum states
- Parity violating experiments: Co-60 beta decay, Pi -> mu nu
- Lecture 13 (Mar 11):
- Discrete translation T(a): a unitary but non-hermitian symmetry
operation for periodic potentials
- | theta > as the common eigenstates of H and T(a),
solving for < x | theta > to get Bloch's theorem
- Eigenvalues of T(a) in tight-binding approximation,
dispersion relation and Brillouin zones
- Time reversal / motion reversal operator Theta: need for defining
it to be antiunitary
- Algebra of antilinear operators, Theta = U K
- Action of Theta on operators like position, momentum, angular
momentum, and on wavefunctions like < x | alpha > or
< theta, phi | l m >
- If H is Theta-invariant and energy eigenvalues are nondegenerate,
then energy eigenfunctions are real
- Action of Theta^2 on spin-J systems. Half-integer spin systems have
to have degeneracy
- Lecture 14 (Mar 15):
- Lecture 16 (Mar 16):
- Time independent perturbation theory (non-degenerate):
first and second order corrections to energies and eigenkets
- Validity of expansions, convergence criteria, normalization
- Examples: SHO perturbed by kx^2/2, Quadratic Stark effect
for the ground state of electron in H atom,
upper bound on the polarizability of ground state H.
- Lecture 17 (Mar 18):
- 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 18 (Mar 23):
- 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]
- Projection theorem (a consequence of Wigner-Eckart theorem,
brought here to compute < l s j mj | Sz | l s j mj >
- Variational method: estimating the ground state energy
using trial wavefunctions, 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.
- Example: electron in a He atom by using atomic number Z as
the variational parameter
- Lecture 19 (Mar 25):
- Time dependent perturbation theory: H = H0 + V(t),
aim to calculate transition amplitudes between energy eigenkets
of H0.
- Interaction picture: states, operators and Schroedinger's
equation
- Dyson series as a solution to the evolution operator
in interaction picture
- Transition probabilities to first and second order in V(t)
- Lecture 20 (Apr 6):
- The case of constant potential turned on at some time t0
- Interpretation of energy-time uncertainty relation in
tems of transition probability
- First order transition probability, conservation of energy
and Fermi's golden rule
- Second order transition probability, virtual transitions,
energy nonconservaion
- Potential increasing as Exp[eta t], The limit eta -> 0 for
time independent perturbation theory
- Corrections to the energy of the initial state, imaginary
part giving the decay rate as well as the width of the Fourier
power spectrum
- Lecture 21 (Apr 8):
- Sudden approximation
- Harmonic perturbation: absorption and stimulated emission
- Interaction of atomic states with EM field: absorption
cross section, electric dipole approximation
- Density of final states, transition rates
- Midterm II: Apr 9
- Lecture 22 (Apr 13):
- Adiabatic approximation: rate of transition depending on
the rate at which the perturbation is varied
- Harmonic oscillator with oscillating center in two extreme
lilits: adiabatic and sudden approximation
- Lecture 23 (Apr 15):
- Scattering theory: Lippman-Schwinger equation
- Solution of the L-S equation with an incoming plane wave
- Local and small-range V giving the ''outging spherical wave''
solution
- Scattering amplitude f(k',k) and the cross section
|f(k',k)|^2
- Lecture 24 (Apr 20):
- Calculating f(k',k) in the first order Born approximation
- Diffraction pattern to find the structure of materials
- The special case of a spherically symmetric potential:
the scattering cross section real, a function only of the
scattering angle, and insensitive to the sign of the potential.
The small and large momentum limits
- Yukawa potential and the Rutherford scattering cross section as
a limiting case
- Validity domain of the Born approximation, Transition
operator T, higher order terms in Born approximation
- Lecture 25 (Apr 22):
- Optical theorem for elastic scattering
- Scattering for spherically symmetric potentials:
Transition operator T as a scalar operator (in the Wigner-Eckart
language)
- The ''partial wave'' basis |E l m> and its components
along the momentum eigenstates
- Scattering as a superposition of incoming and outgoing
spherical waves, with a phase shift between them that is caused
by the scattering potential
- Scattering amplitude and cross section in terms of the phase
shift, optical theorem in the phase shift language
- Lecture 26 (Apr 27):
- Spherically symmetric Schroedinger's equation and its solutions:
spherical Bessel (j), Neumann (n) and Hankel (h) functions, their
limits as the argument vanishes, and asymptotic limits as the
arguments increase to infinity
- Spherical Hankel functions as incoming and outgoing
spherical waves at large
distance from the potential, general advanced wave solution as a
superposition of h (or j/n) functions
- Calculating the scattering amplitude and cross section for a
hard sphere potential in the low energy and high energy limit
- Intuitive understanding of the phase shift in the presence of
a potential, attractive/repulsive potentials increasing/decreasing
the ''effective r'', resonances and no-scattering (Ramsauer-Townsend)
effect
- Lecture 27 (Apr 29):