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    First order perturbation in an infinite spherical well

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    Please see attached question.

    Update: The correction is due to relativistic effects.

    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:

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

    Using the substitution we obtain:

    After simplifying it simply becomes:


    We define
    Then the solution of the harmonic equation is:
    If we do a little coordination transformation we see that


    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.