Discuss the below:
First-order energy correction in case of 1-D delta-function
Recall that a 1-D delta-function potential well of the form V (x) = -B delta(x) had exactly one bound state, with a double-tailed exponential wave function.
(a) Apply a harmonic oscillator perturbation of the form V ′(x) = (m omega^2 x^2)/2. Calculate the ground-state energy for this perturbed system to first order.
(b) Imagine instead that the harmonic oscillator is the original unperturbed system, and the delta-function potential is the perturbation. Calculate the ground state energy for this system to first order.
(c) Do your results agree with each other? Discuss why or why not.