EDMAR G. PANTOHAN
This report is based on the very interesting researches of L. de Broglie on what he called “phase waves”. The advantages of the wave theory,

The laws of motion and quantum condition can be derived from Hamiltonian principle.
The discrepancy between the frequency of motion and frequency of emission disappears when the latter frequencies coincide [...]

Karl Patrick S. Casas
Consider a three-dimensional infinite cubical well

The stationary states are

and the allowed ground state energy is given by

.
The first excited state is triply degenerate,

and we denote each degenerate state as

Now, we introduce the perturbation   (shown in the figure above),

We can get the first-order correction to the ground state energy by

Using the integral formula,
.
And [...]

Rommel J. Jagus
Find the 2nd-order correction to the energies for the potential
Solution:

This is zero unless both m and n are odd in which case it is

But

So,

Since
For n=1

…

For n=3

…

Therefore
if n is even
if n is odd
Source: Problem 6.4 A in “Introduction to Quantum Mechanics” by David J. Griffiths

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