[QUE/QM-23018]

1. For a relativistic harmonic oscillator, show that the Hamiltonian can be approximated by \[ H = \frac{p^2}{2m} +\frac{1}{2}m \omega^2 x^2 -\frac{p^4}{8c^2m^3}\]
2. Show  that \[\langle{n}\vert{p^4}\vert{0}\rangle = \frac{m\hbar\omega^2}{2}\Big(3\delta_{n,0} -6\surd 2 \delta_{n,2} - 2\surd6 \delta_{m,4}\Big)\]
3. Calculate the leading non-vanishing energy shift of the ground state due to this relativistic perturbation.
4. Calculate the leading corrections to the ground state eigenvector \(\vert{0}\rangle\).

{Daniel F Styer*}