PHY 4604 Fall 2008 – Homework 7
Due at the start of class on Friday, December 5. No credit will be available for
homework submitted after the start of class on Monday, December 8.
Answer all four questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
1. The isotropic harmonic oscillator. A particle of mass M moves in the potential
1
1
V (r) = M ω 2 (x2 + y 2 + z 2 ) = M ω 2 r2 .
2
2
The stationary states of this problem can be found in Cartesian coordinates by the
method of separation of variables.
(a) Based on your knowledge of the one-dimensional harmonic oscillator, write down
the general form of the normalized stationary states ψ(x, y, z) and their energies
in terms of three non-negative integer quantum numbers nx , ny , and nz .
(b) Show that the number of stationary states with energy En = (n + 3/2)~ω where
n = 0, 1, 2, . . . is (n + 1)(n + 2)/2.
Since the potential is spherically symmetric, it can also be solved by seeking a solution
ψ(r, θφ) = R(r)Ylm (θ, φ), where Ylm is a standard spherical harmonic.
(c) Recast the p
radial equation for R(r) as a differential equation for u(ρ) = rR(r),
where ρ = M ω/~ r is a dimensionless radial coordinate.
(d) Show that the radial solution should go like u(ρ) ∼ e−ρ
like u(ρ) ∼ ρl+1 for very small ρ.
2 /2
for very large ρ, and
(e) On this basis, recast the differential equation for u(ρ) as an equation for v(s) =
2
ρ−(l+1) eρ /2 u(ρ), where s = ρ2 . Your result should be of the form of the associated
Laguerre equation:
dy
d2 y
x 2 + (1 + α − x) + νy = 0.
(1)
dx
dx
Equation (1) has normalizable solutions only for non-negative integer values ν = k and
real α > −1. These solutions are the associated Laguerre polynomials, which satisfy
Lαk (x) =
Γ(k + α + 1) x −α dk −x k+α
(e x ),
e x
Γ(k + 1)
dxk
where Γ(x) is the gamma function and Γ(n) = (n−1)! for integer n. [The normalization
is chosen so that for non-negative integer values α = p, Lpk (x) is consistent with Griffiths
Eq. (4.87). Note also that, apart from a factor of two in the coefficient of ρdv/dρ,
Griffiths Eq. (4.61) for the hydrogen atom is precisely of the form of Eq. (1). This
factor of two explains why Griffiths Eq. (4.86) expresses v(ρ) as an associated Laguerre
polynomial evaluated at argument 2ρ instead of ρ.]
(f) Use the preceding information to write ψ(r, θ, φ) in terms of elementary functions,
associated Laguerre polynomials Lαk , and spherical harmonics. Show that the
energy eigenvalues can be written in the form E = (n + 3/2)~ω where n = 0, 1, 2,
. . .. Determine the allowed values of k and l for a given n, and hence verify that
the degeneracy of energy En is the same as deduced in part (b).
(g) Explicitly list all the wave functions ψ(x, y, z) for n = 2. Show how to construct
linear combinations of these states proportional to each of the wave functions
ψ(r, θ, φ) corresponding to n = 2.
2. The hydrogen atom.
(a) Show that hxi = hyi = hzi = 0 in any stationary state ψnlm of the hydrogen atom.
Hint: Write x, y, and z in spherical coordinates and focus on the angular integrals.
You may find helpful the relation Ylm (π − θ, φ) = (−1)l+m Ylm (θ, φ).
(b) Calculate hri, hr2 i, hx2 i, and hz 2 i in the ψ210 state of hydrogen.
Hint: Integrals over θ are often simplified by the identity
Z π
Z 1
f (cos θ) sin θ dθ =
f (µ) dµ
0
−1
for any function f (µ).
3. Griffiths Problem 4.49.
4. Griffiths Problem 4.55.
For parts (e) and (f), you may wish to consult the Clebsch-Gordan coefficients in
Griffiths Table 4.8.