Problem $1$

Consider a three$-$dimensional isotropic harmonic oscillator described by Hamiltonian

hat$(H)=\frac{\text{h}\text{a}\text{t}(p{)}_x^2+\text{h}\text{a}\text{t}(p{)}_y^2+\text{h}\text{a}\text{t}(p{)}_z^2}{2m_e}+\frac{1}{2}{m}_e{}^2(x^2+y^2+z^2)$

The oscillator is perturbed by a perturbation potential of the form

$V=\frac{V_0{m}_e}{}(x+y+z).$

Using the degenerate perturbation theory find first order corrections to the energy of the first excited state and the corresponding "correct" zero order eigenvectors.

$2.$ You want to observe dipole allowed transitions between the ground state and the formerly degenerate states found in part $1$ of the problem in a an absorption spectrum of your system. Determine polarizations of the incident electromagnetic waves needed to observe each of possible absorption lines and compute the corresponding transition probabilities assuming that incident electromagnetic wave is a Gaussian pulse with time dependence given by $E_0$exp$[-\frac{t^2}{{}^2}],$ where $$ is a constant characterizing the duration of the pulse, $E_0$ is the maximum magnitude of the field, but its direction $($polarization$)$ must be determined by you. Assume that the perturbation starts acting at $t=-.$