Chemistry 3730 Fall 2000 Quiz 2

Name:

In questions requiring the use of Maple, outline your calculations in standard mathematical notation. Note that information required for the solution of these problems is given on the last page of this paper.

1.

(a)

Calculate approximate energies for the ground and first excited state of a particle experiencing the potential energy

$$V(x) = \frac{1}{2}kx^2 + V_0e^{-\alpha^2x^2}$$

in one dimension assuming that V₀ is small. [10 marks]  

Maple hints: e^u is exp(u), $\sqrt{u}$ is sqrt(u) and $\pi$ is Pi in Maple. To use the Hermite polynomials, enter the following into your Maple session:

with(orthopoly,H);

The Hermite polynomial H_n(u) is then accessed by H(n,u). Maple will need to know that $\alpha$ and $\beta$ are positive parameters. Do not define $\beta$ in your Maple session.

(b)

Sketch (qualitatively) the probability densities for the ground and first excited states of the system described in question 1a along with the corresponding densities for the harmonic oscillator. [5 marks]

2.

Is x compatible with $p_x^2-\hbar p_x$? [10 marks]

Note: I know of no physical interpretation to this question.

Useful information

The energy of a system perturbed by an additive Hamiltonian term $\hat{H}^{(1)}$ can be expressed in the form E = E⁽⁰⁾ + E⁽¹⁾, where

$$E^{(1)} = \langle\psi\vert\hat{H}^{(1)}\vert\psi\rangle.$$

The harmonic oscillator potential energy is $V(x) = \frac{1}{2}kx^2$and the wavefunctions are of the form

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

for $n=0,1,2,\ldots$; H_n(u) is a Hermite polynomial,

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

and $\beta=(km)^{1/4}/\hbar^{1/2}$. The total energy of the oscillator is $E = \hbar\omega_0\left(n+\frac{1}{2}\right)$where $\omega_0 = (k/m)^{1/2}.$

If $\hat{A}$, $\hat{B}$ and $\hat{C}$ are arbitrary operators and k is a constant,

$$\begin{eqnarray*}[\hat{A},\hat{A}]& = & 0\\ \mbox{}[\hat{A},\hat{B}] & = & -[\h... ...t{C}] & = & \hat{A}[\hat{B},\hat{C}] + [\hat{A},\hat{C}]\hat{B} \end{eqnarray*}$$

Additionally, $[\hat{x},\hat{p}_x] = i\hbar$.