Chemistry 3730 Fall 2000 Assignment 3

Due: Friday, Oct. 6, 10:00a.m.

1.

Consider an electron in a box of length 1nm in which a small potential of the form

$$V(x) = \left\{\begin{array}{l@{\quad\textrm{if}\quad}l} 2\ti... ...10^{-20}\,\mathrm{J} & x<L/3,\\ 0 & x>L/3 \end{array}\right.$$

acts on the particle.

(a)

Use first-order perturbation theory to calculate the energies for this problem for n=1, 2 and 3. Sketch a diagram (by hand) showing the original particle-in-a-box energy levels along with the energy levels for this ``bumpy box'' problem. [10 marks]

Hints: Work with explicit (numerical) values of the parameters right from the start. To make your answers as similar to mine as possible, use $h = 6.6261\times 10^{-34}\,\mathrm{J/Hz}$ and $m_e = 9.1094\times 10^{-31}\,\mathrm{kg}$. It will be most convenient to make your zero-order wavefunctions a function of n and x only. Similarly, if you want to turn E⁽⁰⁾_n into a function, make n the only formal argument of your function. Don't try to turn V(x) into a function. Since it is zero for x>L/3, any integral involving V reduces to an integral from x=0 to L/3.

(b)

Does the wavelength of light emitted during a transition from n=2 to n=1 increase or decrease in this problem relative to the original particle in a box? [2 marks]

(c)

Calculate the normalized ground-state wavefunction for the particle in a bumpy box using perturbation theory. Keep only the first few (largest) terms of the series. Wherever you decide to cut off your series, make sure that the last term you keep is at least five times larger than any of the discarded terms.¹ [10 marks]

Hints: The coefficients fall off quite slowly and they do not fall off monotonically. Therefore, start by computing c_1,k for many different values of k, looking for sensible places to cut off the infinite series.

(d)

Plot your wavefunction along with $\psi_1^{(0)}(x)$. Does the result make sense physically? [4 marks]

(e)

Compute the average kinetic and potential energies for your bumpy-box ground-state wavefunction. Compare the sum of values to the total energy. [10 marks]

Hints: You will first have to defined $\hbar$.

Bonus:

Can you explain why $\langle K\rangle + \langle V\rangle$ is slightly different than E₁?

2.

Are r² = x²+y²+z² and the total energy E compatible observables for systems in which the potential energy depends only on the position? [10 marks]

Note: You don't have to right down every step if you feel that some simplifications are obvious.

3.

Show that the spherical harmonics $Y_\ell^{m_z}$ are eigenfunctions of the operator $\hat{L}_x^2+\hat{L}_y^2$. [5 marks]

Hint: $\hat{L}_x^2+\hat{L}_y^2 = \hat{L}^2 - \hat{L}_z^2$ and use the fact that the spherical harmonics are eigenfunctions of $\hat{L}^2$ and $\hat{L}_z$.

Footnotes

... terms.¹

This rule of thumb ensures that the terms you are keeping really are significantly larger than any of the remaining terms.