Perturbation theory problem

3. H.O with anharmonic perturbation ( ).
We add an anharmonic perturbation to the Harmonic Oscillator problem.
Since this is a symmetric perturbation we expect that it will give a nonzero result in first order perturbation theory.
First, write x in terms of and and compute the expectation value as we have done before.

4. Lowering and Ladder operator acting on state

5. A perturbed particle in a box
Estimate the energy of the ground-state and first excited-state wavefunction within first-order perturbation
theory of a system with the following potential energy
Solution
The first step in any perturbation problem is to write the Hamiltonian in terms of a unperturbed component
that the solutions (both eigenstates and energy) are known and a perturbation component
For this system, the unperturbed Hamiltonian and solutions is the particle in an infinitely high box and the
perturbation is a shift of the potential within the box by Vo.
H¹=V°

6. We can calculate expectations energy by its integral formula or better yet,
instead of evaluating this integrals we can simplify the expression that will give us the E¹n
While this is the first order perturbation to the energy, it is also the exact value.