September 1, 1998

PHY 742 - Problem Set # 4

1.

Use the variational principle to find the value of $\beta$ which minimizes the ground state energy of a He-like ion of nuclear charge Z, assuming the two-electron wavefunction to have the form:

$$\Psi(r_1,r_2) = \frac{\beta^3}{ a_0^3 \pi} e^{-\beta(r_1 + r_2)/a_0},$$ (1)

where the Hamiltonian takes the form:

$${\cal{H}} = -\frac{\hbar^2}{2m} \nabla_1^2 -\frac{\hbar^2}{2... ...} -\frac{Z e^2}{r_2} + \frac{e^2}{\vert{\bf{r_1 - r_2}}\vert}.$$ (2)

Hint: Show that

$$<\Psi\vert{\cal{H}}\vert\Psi\gt = 2 \beta \frac{\hbar^2}{2m} -2 Z \beta \frac{e^2}{a_0} + \frac{5}{8} \beta \frac{e^2}{a_0}.$$ (3)

Use your result to estimate the ground state energy of the He-like ion.