Assignment 11

Problem 1:

Consider a composite system made of two spin ½ particles.  For t < 0 the Hamiltonian does not depend on time and can be taken to be zero.  For t > 0 the Hamiltonian is given by  H = (4∆/ħ2)S1S2, where ∆ is a constant.  Suppose that the system is in the state |+-> for t ≤ 0.  Find, as a function of time, the probability for being in each of the states |++>, |+->, |-+>, and |-->,
(a)  by solving the problem exactly, using |Ψ(t)> = U(t, t0)Ψ(t0)> and
(b)  by solving the problem assuming the validity of first-order time-dependent perturbation theory with H as a perturbation which is switched on a t = 0.
(b)  Under what conditions does the perturbation calculation disagree with the exact solution and why?

Problem 2:

In a one-dimensional structure a particle of mass m is in the ground state of a potential energy function U(x) = ½kx2.  A phase transition occurs, and the effective spring constant suddenly doubles.  What is the probability that the particle will be found in an exited state after the phase transition?  Give a numerical answer.

Problem 3:

In one dimension, consider a spinless particle trapped in a delta-function potential U(x) = -Cδ(x), C > 0.
At t = 0, a time dependent perturbation W(t) = Wcosωt is turned on.
Assume ω >> mC2/(2ħ3) so that the particle can be ejected from the trap.  Use perturbation theory to find the transition rate.
You can assume that the free particles will be in a box of size L,  L = very large.

Problem 4:

A hydrogen atom with Hamiltonian H0(r) is placed in a time-dependent electric field E = E(t) k. The perturbed Hamiltonian is H(r,t) = H0(r) + H'(r,t).
(a)  Show that H'(r,t) = qeE(t) r cos(θ).
(b)  Assuming the electron is initially in the ground state, and recalling that the first excited state of hydrogen is quadruply degenerate, to which state of the quadruply degenerate first excited states is a dipole transition from the ground state possible?  Prove this.
(c)  If the electron is in the ground state at t = 0, find  the probability (to first order in perturbation theory) that at time t the electron will have made the transition to the state determined in (b), as a function of E(t).

Problem 5:

Consider a spinless independent electron in the average spherically symmetric potential due to the nucleus and the other electrons in an atom. 
Assume the electron is interacting with a monochromatic plane wave
E(r,t) = (z/z)E0cos(ky - ωt),  B(r t) = (x/x  )B0cos(ky - ωt),  B0/E0 =  ω/k = c.
E = -∂A/∂t - Φ,  B = ×A.
Choose the Coulomb gauge, ·A = 0, and let Φ = 0,
A(r,t) = (z/z)[A0exp(i(ky - ωt)) + A0*exp(-i(ky - ωt))],
with iωA0 = E0/2 and ikA0 = B0/2.
The Hamiltonian of the electron interacting with this plane wave is
H = (1/(2m)) (p - qeA(r,t))2 + U(r) = H0 + W,
with W = -(q/m)pA + (qe2/(2m))A2.
Assume that that the intensity of the wave is low enough so that the term containing A02 can be neglected compared to terms containing A0.
(a)  Expand exp(iky) in a Taylor series expansion and evaluate W to zeroth order in ky.
(Since ky is on the order of a0/λ, it is much smaller than 1 in the visible region of the EM spectrum.)
Write the zeroth order W(0) in terms of E0.  This is the electric dipole Hamiltionian WDE.
(b)  Evaluate the commutator [z, H0], and show that the transition matrix element <Φf|pzi> can be written in terms of the matrix element <Φf|z|Φi>, thus showing that WDE is equivalent to the form we would get starting with the energy of an electric dipole in an electric field.
(c)  Show that the first order term in the expansion of W is W(1) = -(q/m)B0cos(ωt) pzy.
Using pzy = ½(ypz - zpy)  + ½(ypz + zpy) = ½Lz + ½(ypz + zpy) write W(1) = WDM + WQM, where WDM is called the magnetic dipole Hamiltonian and WQM is called the electric quadrupole Hamiltonian.
For initial and final states for which all transitions are allowed, estimate the order of magnitude of the ratio of the magnetic dipole and the electric quadrupole transition probabilities to the electric dipole transition probability.
(d)  Show that for electric dipole transitions Δl = ±1.