 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-3/6c/i/solution

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

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} .

 New to topics? Read the docs here!