Homework Problems

1. Calculate the lowest order energy shift for Helium $\Delta E^{(t)}_{2,l}$ where $\ell=0,1$.

2. Consider the lowest state of ortho-helium. What is the magnetic moment? That is what is the interaction with an external magnetic field?
3. A proton and neutron are bound together into a deuteron, the nucleus of an isotope of hydrogen. The binding energy is found to be -2.23 MeV for the nuclear ground state, an $\ell=0$ state. Assuming a potential of the form $V(r)=V_0{e^{-r/r_0}\over r/r_0}$, with $r_0=2.8$ Fermis, use the variational principle to estimate the strength of the potential.

4. Use the variational principle with a gaussian trial wave function to prove that a one dimensional attractive potential will always have a bound state.

5. Use the variational principle to estimate the ground state energy of the anharmonic oscillator, $H={p^2\over 2m}+\lambda x^4$.