Homework 9
(due Wednesday, Nov. 7)
1. Dimensionalysis:
Consider the one-dimensional Schrodinger equation with a quartic potential:
−
h̄2 d2 ψ
+ αx4 ψ(x) = Eψ(x).
2m dx2
(1)
(a) What is the characteristic length scale of the ground state wave function? (Hint:
Study page 1 of the hydrogen atoms lecture notes.) The length scale should be
expressed in terms of h̄, m, and α.
(b) Based on your length scale in part (a) estimate the energy of the ground state.
2. Hydrogen atom:
Solve for the radial wave functions of the hydrogen atom for n = 2. There are two
possible l values for n = 2: l = 0 and l = 1.
(a) The first step is to solve for the coefficients of the power series using the recursion
relation
2(j + l + 1 − n)
cj+1 =
cj
(2)
(j + 1)(j + 2l + 2)
For n = 2 and l = 1 show that c1 = 0. For n = 2 and l = 0 show that c2 = 0.
(b) Using the book’s notation, the function u(ρ) is then
u(ρ) = ρl+1 e−ρ
∞
X
cj ρj .
(3)
j=0
Express u in terms of r instead of ρ, making sure to use the correct energy for
n = 2.
(c) Apply the normalization condition
Z
∞
(u(r))2dr = 1
(4)
0
to determine the coefficient c0 .
(d) Compute R(r) = u(r)/r and compare your results to those in Table 4.7 in the
book.
3. Visualization:
(a) Write down the full hydrogen atomic wave function, ψ(r, θ, φ), for n = 2, l = 1,
and m = −1, 0, 1.
(b) Eliminate θ and φ in these equations using x = r sin(θ) cos(φ), y = r sin(θ) sin(φ),
and z = r cos(θ).
(c) Find the linear combination of the n = 2 and l = 1 wave functions which is
proportional to x. Normalize this wave function.
(d) Find the linear combination of the n = 2 and l = 1 wave functions which is
proportional to y. Normalize this wave function.
(e) What is the n = 2 and l = 1 normalized wave function which is proportional to
z?
(f) Bonus: Visualize (plot in 3D) the n = 2, l = 1 wave functions from parts (a)-(e)
using Matlab or Octave.