Homework3

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Physics 561 Homework Three
1. A sneaking introduction to angular momentum
Consider a two-dimensional isotropic simple harmonic oscillator with the potential energy
V (r) =
1
2
mω2r2 (1)
where r2 = x2 + y2.
(a) What is the degeneracy of each energy eigenstate?
(b) Find the position-representation wavefunctions for the ground state and each of the
excited states with energies equal to or less than 3h¯ω.
(c) Construct linear superpositions of the states within the degenerate subspace of energy
2h¯ω, which are eigenstates of the Lz, the z component of the orbital angular mo-
mentum. (Refer to the first few pages of chapter VI for the definition of the angular
momentum operators).
2. Anisotropy
Now consider a three-dimesional simple harmonic oscillator which is anisotropic with a
potential energy:
V (X,Y, Z) =
mω2
2
[(
1 +
2λ
3
)(
X2 + Y 2
)
+
(
1− 4λ
3
)
Z2
]
(2)
where ω ≥ 0 and 0 ≤ λ < 3
4
.
(a) Find the eigenstates and eigenenergies of this Hamiltonian.
(b) As a function of λ, describe the variation in the eigenenergies and degree of degeneracy
for the lowest three eigenenergies. Also explain whether these states can be written
as eigenstates of the parity operator (see FII).
(c) Can we construct simultaneous eigenstates of H and Lz, like we did in the previous
problem? If not, why not? What is the connection between this issue and whether or
not the potential is isotropic?
3. Sloshing eigenstates
Consider, in the Heisenberg picture for the simple harmonic oscillator, the time-dependent
raising and lowering operators aH(t) and a
†
H(t).
(a) Find expressions for aH(t) and a
†
H(t) in terms of the Schodinger operators a and a
†.
(b) Calculate XH and PH .
(c) Show that U †( pi
2ω
, 0)|x〉 is an eigenvector of P and find its eigenvalue. Similarly, show
that U †( pi
2ω
, 0)|p〉 is an eigenvector of X.
4. The SchroenbergHeisdinger Picture
In addition to the Schrodinger picture of time-dependent kets and time-independent oper-
ators and the Heisenberg picture of time-independent kets and time-dependent operators,
the Interaction Picture, with a mixed time dependence, is also useful, particularly for situ-
ations in which a base Hamiltonian is perturbed by an additional term, an interaction (as
may be the case in, for example, a scattering process).
Consider a quantum system described by a Hamiltonian H = H0 +W (t), which can be
written as a base Hamiltonian H0 plus a perturbation W (t). The time evolution operator
for the unperturbed system H0 is written as U0(t, t0). In the interaction picture, the state
vector |ΨI(t)〉 is defined as:
|ΨI(t)〉 = U †0(t, t0)|ΨS(t)〉 (3)
(a) Show that the time evolution of |ΨI(t)〉 is governed by
ih¯
d
dt
|ΨI(t)〉 = WI(t)|ΨI(t)〉 (4)
where WI(t) is the perturbation W (t) transformed into the interaction picture:
WI(t) = U
†
0(t, t0)W (t)U0(t, t0). (5)
Assume that the perturbation W is much smaller than H0. Which varies more rapidly
in time: |ΨI(t)〉 or |ΨS(t)〉? Why? What do we mean by the statement that one
operator is smaller than another?
(b) Sometimes it is easier to solve integral equations than differential equations. Show
that the differential equation that you derived in part (a) is equivalent to an integral
equation for |ΨI(t)〉:
|ΨI(t)〉 = |ΨI(t0)〉+ 1
ih¯
∫ t
t0
dt′WI(t′)|ΨI(t′)〉 (6)
At time t = t0, we set |ΨI(t0)〉 = |ΨS(t0)〉.
(c) One strategy for solving such an integral equation (in which |ΨI(t)〉 is written in terms
of an integral over |ΨI(t)〉 itself) is by iteration in a power series. Show that we can
write the solution for |ΨI(t)〉 in the form:
|ΨI(t)〉 =
[
1 +
1
ih¯
∫ t
t0
dt′WI(t′) +
1
(ih¯)2
∫ t
t0
dt′WI(t′)
∫ t′
t0
dt′′WI(t′′) + ...
]
|ΨI(t0)〉
(7)
If the interaction W is sufficiently weak, then this may be a useful expansion. (Even
in cases where the interaction is strong, it may be possible to sum an infinite series.
However, sometimes even the sum of the infinite series isn’t correct!)
k1 k2 k1 k1 k1k2 k2
5. Phonons
Consider a one-dimensional diatomic lattice, whose unit cell contains two atoms with the
same mass m, but with different spring constants k1, k2 describing their (harmonic) inter-
actions with nearest neighbors. The distance between the atoms is a
2
.
(a) Solve for the relationship between wavevector q and eigenfrequency ω(q) for harmonic
vibrations on this lattice. You need only consider values of the wavevector −pi
a
< q <
pi
a
, where a is the lattice constant.
(b) In the limit where k2 →∞, describe how this solution approaches the phonon disper-
sion relation derived in class, for a monoatomic lattice.
(c) In the limit where k1 = k2, describe how this solution approaches the phonon disper-
sion relation derived in class, for a monoatomic lattice.