 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2014/2/a/solution

The dimensionless coordinate for the quantum harmonic oscillator gives

d / d x = mω /ℏ d / d ξ

- ψ_{ξξ} + ξ^{2} ψ = \frac{2 E}{ℏ ω} ψ .

For

ψ = e^{- ξ^{2} /2} f

, direct differentiation yields

ψ_{ξξ} = e^{- ξ^{2} /2} [f^{''} - 2 ξ f^{'} + (ξ^{2} - 1) f] .

Cancelling the nonzero exponential reduces the equation to the Hermite differential equation

f^{''} - 2 ξ f^{'} + (ϵ - 1) f = 0, ϵ = \frac{2 E}{ℏ ω} .

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