2/17/12
QUANTUM MECHANICS II (524)
PROBLEM SET 5 (hand in February 24 with the possibility of an extension)
11) (30 pts) Suppose you have chosen the basis in which orbital and spin angular
momentum have been coupled to total angular momentum. For the central
Woods-Saxon eigenvalue problem that you solved numerically the ` = 1 and
` = 2 eigenvalues are degenerate when you use this basis. Calculate in
lowest-order perturbation theory the correction to the energy for the
j = ` ± 1/2 spin-orbit partners when the spin-orbit potential is considered as a
perturbation. Compared these results with the exact (numerical) eigenvalues
when the spin-orbit potential is included to all orders.
12) (10 pts) Consider the Hamiltonian for the one-dimensional harmonic oscillator
1
p2
+ mω 2 x2 .
H0 =
2m 2
Now add a perturbation of the kind
1
H1 = λ mω 2 x2 ,
2
with λ real and much smaller than 1.
a) Solve the problem exactly for the eigenvalues with inclusion of the
perturbation and then expand the eigenvalues to order λ2 .
b) Generate this result by employing stationary-state perturbation theory.
13) (30 pts) Do problem 12 in Chapter 5 of the second addition of Sakurai
“Modern Quantum Mechanics” without using the solution manual that may
be available online. The problem is given below:
A system with three states has a Hamiltonian that can be represented in that
basis by


E1 0 a
 0 E1 b  ,
a∗ b∗ E2
with E2 > E1 . Consider the constants a and b to be of the same size but small
compared to E2 − E1 . Each section is worth 10 pts.
a) Use second-order nondegenerate perturbation theory to calculate the
perturbed eigenvalues. Is this procedure correct? Comment.
b) Solve the problem exactly.
c) Now use second-order degenerate perturbation theory and compare the
three results obtained.
14) (10 pts) Consider a particle with mass m in the infinite square well
V (x) = 0
− a ≤ x ≤ a,
and ∞ elsewhere.
a) Approximate the wave function of the ground state by the simplest even
polynomial which vanishes at the boundaries:
ψ(x) = a2 − x2
− a ≤ x ≤ a,
and 0 elsewhere. Calculate the mean value of the Hamiltonian in this state
and compare with the exact result for the ground state.
b) Introduce a variational parameter α for a family of trial wave functions
given by
ψ(x) = (a2 − x2 )(a2 − αx2 )
− a ≤ x ≤ a,
and 0 elsewhere. Calculate the mean value of the Hamiltonian as a
function of α. Minimize the energy and compare to the exact result and
the result of part a).