Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 14C (Quantum Mechanics)

Table of contents
- i 14C
  - Solution i
- ii 14C
  - Solution ii

i

Solution

For a normalized wavefunction obeying the infinite-wall Dirichlet boundary condition, integration by parts gives the energy expectation

⟨ H ⟩ = \int_{0}^{a} (\frac{ℏ ^{2}}{2 m} ∣ ψ^{'} (x) ∣^{2} + U (x) ∣ ψ (x) ∣^{2}) d x \geq 0.

Thus every energy eigenvalue is nonnegative.

For

0 < E < U_{0}

, the Time-independent Schrodinger equation and the wall conditions give

ψ (x) = {A sin (k x), B sinh (l (a - x)), 0 \leq x \leq a /2, a /2 \leq x \leq a,

where

k = 2 m E /ℏ

and

l = 2 m (U_{0} - E) /ℏ

. Continuity of

ψ

and

ψ^{'}

at the finite potential step gives

A sin (ka /2) = B sinh (l a /2),

A k cos (ka /2) = - Bl cosh (l a /2) .

Dividing and rearranging yields the bound-state quantization condition

\frac{1}{k} tan \frac{ka}{2} = - \frac{1}{l} tanh \frac{l a}{2} .

ii

Solution

Before the change, the normalized ground state of the infinite square well is

ψ_{0} (x) = \frac{2}{a} sin \frac{π x}{a} .

For an allowed post-quench energy

E \in (0, U_{0})

satisfying part (i), define the unnormalized eigenfunction

χ_{E} (x) = ⎩ ⎨ ⎧ sin (k x), \frac{sin ( ka /2 )}{sinh ( l a /2 )} sinh (l (a - x)), 0 \leq x \leq a /2, a /2 \leq x \leq a .

Its normalization factor is

N_{E}^{- 2} = \int_{0}^{a /2} sin^{2} (k x) d x + \frac{s i n ^{2} ( ka /2 )}{s i n h ^{2} ( l a /2 )} \int_{a /2}^{a} sinh^{2} (l (a - x)) d x .

The Born rule therefore gives

prob (E) = \frac{2 N _{E}^{2}}{a} \int_{0}^{a /2} sin \frac{π x}{a} sin (k x) d x + \frac{sin ( ka /2 )}{sinh ( l a /2 )} \int_{a /2}^{a} sin \frac{π x}{a} sinh (l (a - x)) d x^{2} .

For a value of

E

that is not an eigenvalue, this probability is zero. The sudden change leaves the wavefunction fixed, while the energy eigenbasis changes.

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