# First order perturbation in an infinite spherical well

Please see attached question.

Update: The correction is due to relativistic effects.

-(1/2mc^2)(E^2-2E[V]+[V^2])

Where [V] is the expectation value of the potential

I updated the question. Please let me know if you need more.

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#### Solution Preview

The standard radial equation for the radial wave function is:

(1.1)

Then for l=0 and for the region where V=0 we get:

(1.2)

Using the substitution we obtain:

(1.3)

After simplifying it simply becomes:

(1.4)

We define

(1.5)

Then the solution of the harmonic equation is:

(1.6)

If we do a little coordination transformation we see that

(1.7)

Only now, instead of having (due to the infinite potential ...

#### Solution Summary

The solution shows how to obtain the wave functions of an infinite spherical well for l=0 modes. It then goes on calculating the energies and the first order corrections due to perturbation.