 Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2018/paper-2/section-a-i/12/iii/solution

Since energy and angular frequency are related by

E = ℏ ω

, the given eigenvalue relation yields

E_{i} = E_{P} \frac{1 + λ _{i}}{2} .

With

E_{P} = 4.3 eV

E_{1} = 4.3 0.285 ≃ 2.296 eV, E_{2} = 4.3 0.325 ≃ 2.451 eV, E_{3} = 4.3 0.400 ≃ 2.720 eV .

The

λ_{i}

are dimensionless modal eigenvalues, not optical wavelengths despite the symbol. The same

ℏ

multiplies both mode and bulk angular frequencies and cancels in the energy ratio. These are bare-core predictions of the stipulated lossless Drude model; adding a dielectric/metallic shell, interband response or damping changes the actual resonances.

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