Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section A / A1

Table of contents
- a A1
  - Solution a
- b A1
  - Solution b

a

Solution

The probability density is proportional to

∣ a + i x ∣^{2} = a^{2} + x^{2}

, not to the complex amplitude itself. For equal small intervals approached from inside the support,

P_{0} : P_{a} = a^{2} : 2 a^{2} = 1 : 2.

b

Solution

Wavefunction normalization cancels in the ratio of integrals:

P (a /2 < x < a) = \frac{\int _{a /2}^{a} ( a ^{2} + x ^{2} ) d x}{\int _{0}^{a} ( a ^{2} + x ^{2} ) d x} = \frac{19 a ^{3} /24}{4 a ^{3} /3} = \frac{19}{32} .

A normalized version of the specified wavefunction is

3 (a + i x) / (2 a^{3/2})

on its support.

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