 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2016/s3/2/b/solution

For the Gaussian,

ϕ_{0}^{''} = (x^{2} / σ^{4} - 1/ σ^{2}) ϕ_{0}

. Using

ℏ/ (mω) = σ^{2}

, the

x^{2}

terms cancel and leave

E_{0} = ℏ ω /2

. For

ϕ_{1} \propto x ϕ_{0}

ϕ_{1}^{''} = (x^{2} / σ^{4} - 3/ σ^{2}) ϕ_{1}

, and the same cancellation gives

E_{1} = 3ℏ ω /2

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