 Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2018/paper-2/section-a-ii/14/ii/b/solution

Orthonormal energy eigenstates contribute incoherently to the expectation of energy:

⟨ E ⟩ = \frac{ϵ}{S} [1 (0.2)^{2} + 4 (0.3)^{2} + 9 (0.4)^{2} + 16 (0.843)^{2}] = \frac{13.210384}{1.000649} ϵ .

Therefore $⟨ E ⟩ ≃ 13.2018 ϵ$ for the normalized literal decimals. Ignoring the small rounding-induced norm error gives

13.2104 ϵ

, conventionally reported at this precision as about $13.2 ϵ$ . The mean need not equal an individual energy eigenvalue.

 New to topics? Read the docs here!