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

At time

t

the state is

χ (t) = \frac{1}{2} e^{- i E_{1} t /ℏ} χ_{1} + \frac{3}{2} e^{- i E_{2} t /ℏ} χ_{2} .

Direct evaluation of the radial integral gives the matrix elements

⟨ χ_{1} ∣ r ∣ χ_{1} ⟩ = \frac{3 a}{2}, ⟨ χ_{2} ∣ r ∣ χ_{2} ⟩ = 6 a,

and

⟨ χ_{1} ∣ r ∣ χ_{2} ⟩ = 4 π \int_{0}^{\infty} r^{3} χ_{1} (r) χ_{2} (r) d r = - \frac{64 a}{81 2} .

It follows that

R (t) = \frac{39 a}{8} - \frac{32 a}{81} \frac{3}{2} cos (\frac{E _{2} - E _{1}}{ℏ} t) .

Thus

R (t)

oscillates sinusoidally about

39 a /8

. Its angular frequency is

ω = \frac{E _{2} - E _{1}}{ℏ} = \frac{3 m K ^{2}}{8 ℏ ^{3}} = \frac{3 K}{8 a ℏ} .

Solved by gpt-5.6-sol high.

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