January 29, 1998

PHY 742 - Problem Set # 6

In the following problems, express your results in terms of the ${\cal{F}}$ and ${\cal{G}}$ integrals where

$${\cal{F}}^L_{n_1l_1n_2l_2} \equiv \int_0^{\infty} dr_1 \: r_... ...\frac{r_<^L}{r_\gt^{L+1}} R^2_{n_1l_1}(r_1) R^2_{n_1l_1}(r_2)$$ (1)

and

$${\cal{G}}^L_{n_1l_1n_2l_2} \equiv \int_0^{\infty} dr_1 \: r_... ...{n_1l_1}(r_1) R_{n_2l_2}(r_1) R_{n_1l_1}(r_2)R_{n_2l_2}(r_2).$$ (2)

You may which to use the computer program ``gaunt" for some of these.

Evaluate the following expection values of the electron-electron interaction $v_{12} \equiv \frac{e^2}{\vert{\bf{r_1 - r_2}}\vert}$:

1.

<³S(1s2s)|v₁₂|³S(1s2s)>.

2.

<¹S(1s2s)|v₁₂|¹S(1s2s)>.

3.

<³P(1s2p)|v₁₂|³P(1s2p)>.

4.

<¹P(1s2p)|v₁₂|¹P(1s2p)>.

5.

<¹S(2p²)|v₁₂|¹S(2p²)>.

6.

<³P(2p²)|v₁₂|³P(2p²)>.

7.

<¹D(2p²)|v₁₂|¹D(2p²)>.