a. Calculate the first-order shift in the energies of a harmonic oscillator due to the addition of an anharmonic term Cx4 to the potential. b. A free particle moves in 1-D between impenetrable walls...

a. Calculate the first-order shift in the energies of a harmonic oscillator due to the addition of an anharmonic term Cx⁴
to the potential.

b. A free particle moves in 1-D between impenetrable walls at x = 0 and x = L. The particle is moving so fast that we cannot entirely neglect the effects of relativity. This has the approximate effect of adding a term to the kinetic energy:

where c is the speed of light. (Consult a textbook on special relativity to find the origin of this term.) Find the first-order changes to the energy levels and the stationary state wave functions due to this perturbation.