 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-4/4b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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}} .

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