Question

Take an eigenfunction for the harmonic oscillator and the corresponding energy eigenvalue, and substitute them into the Schrodinger equation. Then prove that the equation is satisfied. Do this for n =4, show how n is substituted.

Answer 1

The harmonic potential in one dimension is given by

.

If be an energy eigenfunction with energy for this system, then it must satisfy the time independent equation as follows:

.

The energy eigenfunctions for a 1D harmonic oscillator are given by

with energy level

,

where

Now, for n=4, the energy eigenfunction is

with energy . Now, let us check if satisfies the aforesaid time independent equation. For this, we see that

.

Hence satisfies the time independent equation.