Physics 251 Spring 1999
Modern Physics
Homework Assignment # 7, Due (in class) Wednesday March 17

Reading Assignment: Krane, Chapter 7 omitting section 7.8
Reading Assignment: Start Krane, Chapter 8, for lecture 3/15 Problem Assignment:

1.

Krane, Chapter 7, p.232, #1.

2.

Krane, Chapter 7, p.232, #7.

3.

Consider the n=2, $\ell$=1, m=0 level of the H atom. The wavefunction is given by eq. 7.5 and table 7.1. At a fixed distance r from the nucleus, the wavefunction has angular dependence $\cos\theta$and the probability density has angular dependence $\cos^2 \theta$.Plot this probability density versus $\theta$ from 0 to 180$^{\circ}$.How much uncertainty is there in the angular position ($\theta,\phi$)of the electron? It is said that this state has an angular momentum which makes an angle of 90$^{\circ}$ with the z axis. It might be more appropriate just to say that this is a state with no evidence for any angular momentum.

4.

Only by mixing states with different angular momenta does one begin to get some definition of the angular position of an electron. For example, using the wavefunctions in table 7.1, consider the state which is a 50/50 admixture of $\Psi_{200}$ and $\Psi_{210}$The n=2 state has a maximum probability to be at a distance r=4a₀ from the nucleus. Show that at this distance, the wavefunction has an angular dependence of the form $2\cos\theta-1$. Plot the corresponding probability density versus $\theta$ from 0 to 180$^{\circ}$.There is now a large uncertainty in the value of $\vert\vec{L}\vert$ and a reduced uncertainty in the angular position. Approximately how much uncertainty is there in the angular position?

5.

Krane, Chapter 7, p.233, #15.

6.

Krane, Chapter 7, p.233, #20.

7.

Krane, Chapter 7, p.233, #21.

8.

Krane, Chapter 7, p.233, #23.