 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b/section-b/33/c/i/solution

Write

g = e^{- q^{2} /2}

and

f = 2 q^{2} - a

\hat{H} (f g) = [- \frac{1}{2} f^{''} + q f^{'} + \frac{1}{2} f] g = (5 q^{2} - 2 - \frac{a}{2}) g .

For an eigenfunction, this must equal

E (2 q^{2} - a) g

at every

q

. Matching the

q^{2}

E = 5/2

, and matching constants then gives

5 a /2 = 2 + a /2

. Hence $a = 1$ and $E = 5/2$ . This is the even

n = 2

oscillator state. Its normalized wavefunction is

ϕ_{2} = (2 q^{2} - 1) e^{- q^{2} /2} / (2 π^{1/4})

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