Chemistry 3730 Fall 2000 Quiz 1

Name:

For each question, write out the expressions which you need to evaluate in standard mathematical notation. Note that information required for the solution of these problems is given on the reverse of this page.

1.

Calculate the probability that a particle in a box is to be found in the leftmost quarter ($x\le L/4$) of the box for n=1, 2 and 3. Give your answers to 4 significant figures. Compare the relative values of your answers and relate them to the shapes of the wavefunctions. [10 marks]

Maple hint: $\pi$ is typed `Pi' in Maple.

2.

Calculate the average potential energy for a harmonic oscillator for n=0, 1 and 2. Give your answers in terms of $\hbar$ and $\omega_0$. Can you discern a pattern in your answers? [10 marks]

Maple hints: To use the Hermite polynomials, type

with(orthopoly,H);

In Maple, H_n(u) is typed H(n,u) and e^u is exp(u). It is best to do all the calculations in terms of $\beta$ and to simplify by hand at the end. It will be necessary to inform Maple that $\beta$ is a positive parameter.

Useful information

The particle-in-a-box wavefunctions are of the form

$$\psi_n(x) = \sqrt{2/L}\sin\left(\frac{n\pi x}{L}\right).$$

The harmonic oscillator potential energy is

$$V(x) = \frac{1}{2}kx^2$$

and the wavefunctions are of the form

$$\psi_n(x) = A_nH_n(\beta x)e^{-\beta^2x^2/2}$$

where H_n(u) is a Hermite polynomial and

$$A_n = \sqrt{\frac{\beta}{2^nn!\sqrt{\pi}}}.$$

Additionally,

$$\beta=(km)^{1/4}/\hbar^{1/2}$$

and

$$\omega_0 = (k/m)^{1/2}.$$