Sample Test Problems

1. We wish to get a good upper limit on the Helium ground state energy. Use as a trial wave function the 1s hydrogen state with the parameter a screened nuclear charge $Z^*$ to get this limit. Determine the value of $Z^*$ which gives the best limit. The integral $\langle (1s)^2\vert{e^2\over\vert\vec{r}_1-\vec{r}_2\vert}\vert(1s)^2\rangle={5\over 8}Z^*\alpha^2mc^2$ for a nucleus of charge $Z^*e$.
2. A Helium atom has two electrons bound to a $Z=2$ nucleus. We have to add the coulomb repulsion term (between the two electrons) to the zeroth order Hamiltonian.
   
   $$H={p_1^2\over 2m}-{Ze^2\over r_1}+{p_2^2\over 2m}-{Ze^2\over r_2}+{e^2\over \vert\vec{r}_1-\vec{r}_2\vert}=H_1+H_2+V$$
   
   
   The first excited state of Helium has one electron in the 1S state and the other in the 2S state. Calculate the energy of this state to zeroth order in the perturbation V. Give the answer in eV. The spins of the two electrons can be added to give states of total spin $\vec{S}$. The possible total spin states are $s=0$ and $s=1$. Write out the full first excited Helium state which has $s=1$ and $m_s=-1$. Include the spatial wave function and don't forget the Pauli principle. Use bra-ket notation to calculate the energy shift to this state in first order perturbation theory. Don't do any integrals.

3. The lowest energy excited states of Helium have the electrons in a (1s)(2s) configuration in space and can have total spin 0 or 1. Write out the full (space and spin) states for a) $s=1$ and $m_s=0$ and for b) $s=0$ and $m_s=0$. Make sure to take account of the Pauli principle in your states. Which state will have the lower energy?