Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 6C (Quantum Mechanics)

Table of contents
- i 6C
  - Solution i
- ii 6C
  - Solution ii

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} .

ii

Solution

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} .

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