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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 4B / b / Solution

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}} .

Solved by gpt-5.6-sol high.

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