PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 10
Nov. 17, 2004
Due: Dec. 1, 2004
Reading: Griffiths Secs. 4.1-4.3
1. Hydrogen wave functions.
(a) Plot and label the first six (i.e. n=1,2,3, all allowed values of `) radial
functions Rn` (r), as functions of r/a0 . These functions are found to be
v
uµ
u 2 ¶3 (n − ` − 1)!
Rn` (x) = t
e−x/2 x` L2`+1
n−`−1 (x)
3
na0
2n[(n + `)!]
(1)
where Ln` are associated Laguerre polynomials and x ≡ 2r/(na0 ).
• (Option 1.) You may use Maple to calculate and plot these functions. In this case note that the built-in functions L(n, a, x) Maple
calls generalized Laguerre polynomials are normalized differently from
the associated Laguerre polynomials. To generate associated Laguerre
polynomials, type:
Laguerre:=(q,xi)->expand(exp(xi)*diff(xi^ q*exp(-xi),xi$q)):
This is the def. of Laguerre polynomials, Lq (ξ) = eξ (d/dξ)q (e−ξ ξ q ).
Now type
ALaguerre:=(q,p,xi)->(-1)^ p*diff(Laguerre(q,xi),xi$p):
to define the associated Laguerre polynomials, Lpq−p (ξ) = (−1)p (d/dξ)p Lq (ξ).
To get the required L2`+1
n−`−1 occurring in Eq. (1), we need p = 2` + 1
and q = n + `. So define
Lnl:=(n,l,xi)->ALaguerre(n+l,2*l+1,xi):
Then you should get, e.g.
Lnl(3,0,xi);
18 − 18ξ + 3ξ 2
Next define a radial function Rn` using the substitute command subs:
Rnl:=(n,l,xi)->subs(xi=2*r/n,expr);
where expr is your expression for the form of the Rn` . This is necessary
because if you try to define the Laguerre polynomial Laguerre(q,2*r/n),
Maple tries to differentiate with respect to the variable 2 ∗ r/n and
can’t, whereas subs substitutes 2 ∗ r/n after differentiation.
Plot your 6 Rn` vs. r/a0 on a scale of r/a0 = 0..20 and R = −0.2..0.6.
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• (Option 2.) If you want to avoid Maple, derive the 6 Rn` from the
normalization condition and the recursion relation discussed in class.
Sketch the curves for Rn` vs. r/a0 , paying particular attention to the
values at r = 0, as well as the nodes (values of r/a0 where Rn` = 0).
N.B. Do not compare with Fig. 4.4 in Griffiths, as it contains a mistake!
2 pts. extra credit for finding it...
(b) Plot the radial probability density P (r) for finding the electron a distance
r away from the nucleus for the three cases n, `= (1,0), (2,0), and (3,0).
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Hint: the radial probability density is not Rn`
! Maple or sketch. Comment
on the number of nodes in the wavefunction and its dependence on n; also
on the most likely distance of the electron from the nucleus, and how it
depends on n.
(c) Suppose an electron is in an angular state Y1,1 + Y1,−1 . Plot the angular
probability distribution P (θ, φ) by using
with(plots):
sphereplot(P(theta,phi),phi=0..2*Pi,theta=0..Pi,grid=[51,51],
scaling=CONSTRAINED);
Sketch also ok. Specify what your plot means, but don’t worry about
normalization.
2. Harmonic Oscillator. Find the matrix elements < n|x|n0 > and < n|p|n0 >
in the basis consisting of stationary states of the simple harmonic oscillator.
Construct the corresponding (infinite) matrices X̂ and P̂ , and show that H =
P̂ 2 /2m + (1/2)mω 2 X̂ 2 is diagonal in this basis. Give the diagonal elements.
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