# QUANTUM THEORY OF MATTER Homework set#13 Degenerate and non

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```					                       QUANTUM THEORY OF MATTER
Homework set #13:
Degenerate and non-degenerate stationary perturbation theory

Problem # 13.1 : Electric ﬁeld and harmonic oscillator
Consider a charged particle (of charge q) in the one-dimensional harmonic oscillator poten-
tial. Suppose we turn on a weak electric ﬁeld (E), so that the potential energy is shifted
by an amount H = −qEx.
(a) Show that there is no ﬁrst-order change in the energy levels.
(b) Calculate the second order correction to the energy levels.
o
(c) The Schr¨dinger equation for H = H0 + H can be solved directly by a change of
variables: x = x − (qE/mω 2 ). Find the exact energies, and show that they are
consistent with the perturbation theory approximation.

Problem # 13.2 : Bead on a ring
Consider a particle of mass m that is free to move on a ring of length L (e.g. a circular
wire of circumference L).
(a) Show that the stationary states can be written in the form

1
0
ψn (x) = √ e2πinx/L , −L/2 < x < L/2 ,
L

where n = 0, ±1, ±2, · · ·, and the allowed energies are
2
0     2     h
nπ¯
En =                    .
m    L

Note that, with the exception of the ground state (n = 0), these states are all doubly
degenerate.
(b) Now suppose we introduce the perturbation

2
/a2
H = −V0 e−x             ,
where a << L. This perturbation puts a little dimple in the potential at x = 0, as
though we bent the wire slightly to make a trap for the bead (particle). Find the ﬁrst
order correction to En using degenerate perturbation theory when appropriate.
Hint: To evaluate the integrals assume a << L and extend the limits of integration
from ±L/2 to ±∞, since H is essentially zero outside −a < x < a.
0      0
(c) What are the ”good” linear combinations of ψn and ψ−n for this problem? What is
the symmetry associated with these linear combinations. Which operator commutes
with the Hamiltonian?

Problem # 13.3 : Perturbed cubical well
Consider the three-dimensional inﬁnite cubical well deﬁned by the potential

0   for 0 < x < a, 0 < y < a, 0 < z < a ,
V (x, y, z) =
∞   otherwise .

Suppose we perturb this potential by placing a δ-function bump at the point
(a/4, a/2, 3a/4):
H = a3 V0 δ(x − a/4)δ(y − a/2)δ(z − 3a/4) .

(a) Find the energy, wave function(s) and degeneracy of the ground state and ﬁrst excited
state(s) for the unperturbed well.
(b) Find the ﬁrst-order corrections due to H to the energy of the ground state and the
ﬁrst excited state(s).

Problem # 13.4 : Harmonic oscillator
Consider the isotropic three-dimensional harmonic oscillator in Cartesian coordinates and
discuss the eﬀects of the perturbation

H = λx2 yz

for some constant λ.
(a) Find the energy, wave function(s) and degeneracy of the ground state and ﬁrst excited
state(s) for the unperturbed oscillator.

(b) Find the ﬁrst-order corrections due to H to the energy of the ground state and the

ﬁrst excited state(s).
Hint: The matrix elements of the displacement x for a simple harmonic oscillator are

given by
√           √
n|x|n =     ¯ /2mω
h           n δn,n −1 + nδn ,n−1   .

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