 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/3/b/solution

The normalized initial Gaussian wave packet is real and even, so the integrands for

⟨ x^{'} ⟩_{0}

and

⟨ p^{'} ⟩_{0}

⟨ x^{'2} ⟩_{0} = 1/ (2 α)

. Since

\partial_{x} Ψ = - αx Ψ

and

\partial_{x}^{2} Ψ = (α^{2} x^{2} - α) Ψ

⟨ p^{'2} ⟩_{0} = ℏ^{2} (α - α^{2} ⟨ x^{'2} ⟩_{0}) = \frac{ℏ ^{2} α}{2} .

Moreover

⟨ x^{'} p^{'} ⟩_{0} = i ℏ α ⟨ x^{'2} ⟩_{0} = i ℏ/2

and

⟨ p^{'} x^{'} ⟩_{0} = - i ℏ/2

. Thus

⟨ x^{'} ⟩_{0} = ⟨ p^{'} ⟩_{0} = ⟨ x^{'} p^{'} + p^{'} x^{'} ⟩_{0} = 0, ⟨ x^{'2} ⟩_{0} = \frac{1}{2 α}, ⟨ p^{'2} ⟩_{0} = \frac{ℏ ^{2} α}{2} .

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