10.57.Verify that 1 2 dx 2 1 dx0 for the 1-D
particle-in-a-box, showing that the order of the wavefunctions
inside the integral sign does not matter.
10.58.Evaluate the following integrals of the wavefunctions
of particles-in-boxes by using equation 10.28 instead of solv-
ing the integrals:
(a) 4 4 d (b) 3 4 d
(c) 4 ˆH 4 d (d) 4 Hˆ 2 d
(e) 111 111 d (f) 111 121 d
(g) 111 Hˆ 111 d (h) 223 ˆH 322 d
10.14 Time-Dependent Schrödinger Equation
10.59.Substitute (x, t) eiEt/
(x) into the time-
dependent Schrödinger equation and show that it does solve
that differential equation.
10.60.Write (x, t) eiEt/
(x) in terms of sine and co-
sine, using Euler’s theorem: eicos isin . What would
a plot of (x, t) versus time look like?
10.61.Evaluate (x, t)^2. How does it compare to (x)^2?
10.62.Construct plots of the probabilities of the first three
wavefunctions for a particle in a one-dimensional box having
length a. Identify where the nodes are.
10.63.Numerically integrate the expression for the average
value of position for 10 for a particle-in-a-box and explain the
answer.
10.64.Construct a table of energies of a particle in a 3-D
box versus the quantum numbers nx, ny, and nz, where the
quantum numbers range from 1 to 10. Express the energies
in h^2 /8ma^2 units. Identify all examples of accidental degen-
eracies.
10.65.Numerically integrate the 1-D particle-in-a-box wave-
function product 3 * 4 over all space and show that the two
functions are orthogonal.314 Exercises for Chapter 10
Symbolic Math Exercises