 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/section-b/b7/b/solution

⟨ \overset{a}{^}_{+}^{2} ⟩ = ⟨ \overset{a}{^}_{-}^{2} ⟩ = 0

, while the ladder coefficients give

⟨ \overset{a}{^}_{+} \overset{a}{^}_{-} ⟩ = n

and

⟨ \overset{a}{^}_{-} \overset{a}{^}_{+} ⟩ = n + 1

. Squaring the expressions for

\overset{x}{^}

and

\overset{p}{^}

therefore gives

⟨ x^{2} ⟩ = \frac{ℏ}{mω} (n + \frac{1}{2}), ⟨ p^{2} ⟩ = m ℏ ω (n + \frac{1}{2}) .

Thus

⟨ V ⟩ = \frac{1}{2} m ω^{2} ⟨ x^{2} ⟩ = \frac{1}{2} ℏ ω (n + \frac{1}{2}), ⟨ K ⟩ = \frac{⟨ p ^{2} ⟩}{2 m} = \frac{1}{2} ℏ ω (n + \frac{1}{2}) .

E_{n} = ℏ ω (n + 1/2)

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