Do 6 problems from problems 1 – 7.
Problem 1:
The figure below shows one of the possible energy eigenfunctions ψ(x) for a particle bouncing freely back and forth between impenetrable walls located at x = -a and x = +a. The potential energy equals zero for |x| < a. If the energy of the particle is 2 eV when it is in the quantum state associated with this eigenfunction, find the energy when it is in quantum state of lowest possible energy.
Problem 2:
A particle is in the state |y> that has angular momentum j and angular momentum projection on the z-axis m such that
J2|y> = h2(j(j+1)|y>, Jz|y> = mh|y>.
Find expectation values of the angular momentum components Jx and Jy in this state.
Problem 3:
The Heisenberg Hamiltonian representing the “exchange
interaction” between two spins (S1 and S2) is given by H =
-2f(R)S1×S2,
where f(R) is the so-called exchange coupling constant and R is the spatial
separation between the two spins. Find the eigenstates and eigenvalues of the
Heisenberg Hamiltonian describing the exchange interaction between two
electrons.
HINT: The total spin operator is S =
S1
+ S2.
Problem 4:
A collection of non-interacting undistinguishable spin 1/2 particles is trapped in a 3-dimensional, isotropic harmonic-oscillator potential. How many particles can occupy the three lowest lying energy states?
Problem 5:
The La line of the characteristic X-ray spectra of heavy atoms consists of several components of different frequencies corresponding to the various allowed transitions from levels with n = 3 to levels with n = 2. Predict the number of different frequencies to be observed, on the basis of the selection rules Dl = ±1, Dj = 0, ±1.
Problem 6:
The weak interactions (for example, beta decay) are mediated by massive particles called intermediate vector bosons, which are observed in accelerator experiments to have masses in the range mc2 ~ 80-90 x 109 electron-volts. Assuming the weak interactions to occur because of the quantum-mechanical exchange of a virtual intermediate vector boson between two particles, estimate the maximum range of the weak force.
Problem 7:
Let us consider a carbon atom whose electrons are in the following configuration (1s)2 (2s)2 2p 3p. List all the expected terms 2S+1LJ on the basis of the L-S (Russell-Sanders) coupling scheme.
Do 3 problems from problems 8 – 11.
Problem 8:
Consider a two-dimensional infinite potential square well
of width L,
(V = 0 for 0 < x,y < L, V = infinite everywhere else) with an added perturbation
(a) Calculate the first order perturbation to the ground
state energy eigenvalue.
(b) Calculate the first order perturbation to the first
excited state energy eigenvalue
Problem 9:
A spin 1/2 particle is in an external magnetic field B
= B0(êx
+ êy).
Let the magnetic moment associated with the spin be
m = -gS, where
g is a constant and is known as the
gyromagnetic ratio. Use the eigenstates of Sz, |+> and |–>,
as basis kets.
(a) What is the interaction Hamiltonian of the particle
with the magnetic field? Express in matrix form.
(b) The most general spin state of the particle is |c(t)>
= a(t) |+> +
b(t) |–>.
Write down the coupled time dependent Schroedinger
equations for a(t) and
b(t).
(c) Suppose at t = 0 we have |c(0)>
= | + >. Find |c(t)>.
(d) Evaluate <Sz(t)> for the
c(t) obtained in (c).
Problem 10:
A spinless particle of mass M is scattered by a central potential of finite range such that the wave function at low energies and for large r is well described by
,
where d is a phase
shift.
(a) Find the total low-energy cross section in terms of
the quantities defined.
(b) Find the phase shift
d if the slow particle scatters off a central potential of the form
V(r) = V0, for r < a, V(r) = 0, for r > a.
Problem 11:
In the WKB approximation, find the allowed energies that a ball of mass m bouncing due to gravity on a perfectly reflecting surface, can have. You can use the fact that for this problem the WKB approximation gives
where p(q) is the momentum of the ball at the height q and the integral is over a full periodic path. You may leave your answer as an integral equation which could be solved to yield the energy levels.