4B (Quantum Mechanics)

The first term is radial kinetic energy from the Laplacian, the second is the attractive Coulomb potential, and the third is the centrifugal energy from the angular-momentum eigenvalue

L^{2} = l (l + 1) ℏ^{2}

. Single-valued spherical harmonics require

l = 0, 1, 2, \dots .

b

Substitute

g = r^{a} e^{- b r}

into the radial Schrodinger equation and equate the coefficients of

r^{- 2}

r^{- 1}

, and the constant term. Regularity selects

a = l, b = \frac{m e ^{2}}{4 π ε _{0} ℏ ^{2} ( l + 1 )} .

The corresponding bound-state energy is

E_{l} = - \frac{m e ^{4}}{2 ( 4 π ε _{0} ) ^{2} ℏ ^{2} ( l + 1 ) ^{2}} .

l + 2

and the lower one

l + 1

. Energy conservation for the emitted photon gives

ν = \frac{m e ^{4}}{2 h ( 4 π ε _{0} ) ^{2} ℏ ^{2}} (\frac{1}{( l + 1 ) ^{2}} - \frac{1}{( l + 2 ) ^{2}}) .

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