 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/4/c/solution

By the two Fourier transform rules in part (b),

F [f^{''}] = - k^{2} f, F [x^{2} f] = i^{2} f^{''} = - f^{''} .

Consequently transforming the differential equation gives

- k^{2} f + f^{''} = μ f,

f^{''} (k) - k^{2} f (k) = μ f (k) .

The inverse Fourier transform gives the reverse implication under the same regularity hypotheses. This expresses the Fourier symmetry of the quantum harmonic oscillator operator.

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