For , taking the reduced mass equal to the electron mass as an approximation, the…

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · One-electron atoms and the origins of atomic structure

Reading the Spectrum, Ch The Coulomb spectrum from wave mechanics: quantum numbers and degeneracy

Problem

For

l=0

, taking the reduced mass equal to the electron mass

m_e

as an approximation, the radial equation in terms of

u(r)=rR(r)

-\frac{\hbar^2}{2m_e}\frac{d^2u}{dr^2} - \frac{e^2}{4\pi\epsilon_0 r}u = Eu.

The Bohr radius is defined by

a_0 = 4\pi\epsilon_0\hbar^2/m_e e^2

. ((a)) Substitute the trial solution

u(r) = r\,e^{-r/a_0}

into the equation above and show that it is a solution only for one particular value of

E

. ((b)) Give that value of

E

in eV and compare it with

E_1 = -hcR_{\rm H} = -13.598\,\mathrm{eV}

, commenting on the size of the difference.

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