PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 4
Sept. 22, 2004
Due: Sept. 29, 2004
Reading: Notes, Griffiths Chapter 1,2
1. Additive Hamiltonians.
1a) Show that if the potential energy V (r) in the Schrödinger equation
Hψ = Eψ
h̄2 ∇2
H = −
+V
2m
can be written as a sum of functions of a single coordinate,
V (r) = V1 (x1 ) + V2 (x2 ) + V3 (x3 ),
(1)
then the time-independent (definite energy) Schrödinger equation can be
decomposed into a set of 1D equations of the form
∂ 2 ψi (xi ) 2m
+ 2 [Ei − Vi (xi )]ψi = 0,
∂ 2 xi
h̄
i = 1, 2, 3
(2)
with ψ(r) = ψ1 (x1 )ψ2 (x2 )ψ3 (x3 ) and E = E1 + E2 + E3 .
1b) Use this principle to find the energy levels (all) for the anisotropic 3D
harmonic oscillator.
H=−
h̄2 2 m 2 2
∇ + (ω1 x + ω22 y 2 + ω32 z 2 )
2m
2
(3)
1c) Find the ground state wave function for this problem by using ladder
operators as in class.
1d) In the isotropic case ω1 = ω2 = ω3 , what is the degeneracy of the energy
levels, i.e. how many linearly independent eigensolutions correspond to
each distinct eigenvalue?
2. Classical and Quantum “Probability Densities” in SHO. The first excited wave function for the simple harmonic oscillator, corresponding to eigenvalue E1 = 3h̄ω/2, is
x − 12 ( xx )2
0
e
,
ψ1 (x) = q
x
1/2
0
2π x0
2
1
s
x0 =
h̄
mω
(4)
2a) Show this is proportional to the ladder raising operator L+ acting on the
ground state ψ0 .
2b) Write down the quantum mechanical expression for the probability density
Pq (x) for finding the particle in this state as a function of x.
2c) Find the classical expression for the same quantity, recognizing that the
classical probability Pc of finding the oscillating particle in the interval
[x, x + dx] is proportional to the amount of time it spends in this interval.
Express your result in terms of the classical amplitude of oscillation A.
2d) Identifying the classical amplitude A with quantum-mechanical quantities
by comparing the classical and quantum energies, plot using Maple Pq (x)
and Pc (x) determined above from x = −3A, 3A. Make sure both probabilities are normalized properly. If you could do one but not both of 2b)
or 2c, plot the one you were able to get.
2