 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2015/paper-1/section-a/1/solution

The Born rule uses the squared modulus, so the imaginary term contributes positively:

∣ a + i x ∣^{2} = a^{2} + x^{2}

. For

a > 0

, wavefunction normalization would multiply the supplied expression by

N = 3/ (4 a^{3})

. Equivalently, take the ratio of the integrals directly:

P = \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} .

Using

ψ^{2}

instead of

∣ ψ ∣^{2}

would fail to produce a real, nonnegative probability density.

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