A Modern Approach to Quantum Mechanics · Problem 11.7

Identical Particles — John S. Townsend

Physics — Quantum Mechanics · finite-proton-size-hydrogen

Problem

Assume that the proton is a uniformly charged sphere of radius

R

. (a) Show for the hydrogen atom that the potential energy of the electron in the field of the proton is given by

V(r) = \begin{cases} -\dfrac{3e^2}{2R^3}\left(R^2 - \dfrac{1}{3}r^2\right) & r < R \\ -\dfrac{e^2}{r} & r > R \end{cases}

*Hint:* Use Gauss's law and remember that the potential energy

V(r)

must be continuous. (b) Calculate the energy shift for the

1s

and

2p

states of hydrogen if the potential energy in (a) is used. What effect does this shift have upon the Lyman

\alpha

wavelength? *Suggestion:* You can use the fact that

R \ll a_0

to simplify the integrand before evaluating the integrals.