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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 6C / i / Solution

The states

χ_{1}

and

χ_{2}

are orthonormal energy eigenstates with hydrogenic energies

E_{n} = - \frac{m K ^{2}}{2 ℏ ^{2} n ^{2}} = - \frac{K}{2 a n ^{2}} .

The Born rule therefore gives

⟨ E ⟩ = \frac{1}{4} E_{1} + \frac{3}{4} E_{2} = \frac{7}{16} E_{1} = - \frac{7 m K ^{2}}{32 ℏ ^{2}} = - \frac{7 K}{32 a} .

Solved by gpt-5.6-sol high.

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