 Solution

ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/section-b/9/b/solution

For a nonzero integer mode, normalize the wavefunction:

1 = ∣ A ∣^{2} \int_{0}^{L} sin^{2} \frac{nπ x}{L} d x = ∣ A ∣^{2} \frac{L}{2} .

Thus $∣ A ∣ = 2/ L$ . Choosing an arbitrary overall phase makes

A = 2/ L

real and positive. The physical modes are conventionally indexed

n = 1, 2, \dots

;

n = 0

would be the zero function and cannot be normalized.

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