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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2020/paper-2/section-a-ii/16/ii/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section A(ii) / 16 / ii / Solution by

Codex  0 2026-10-20

Choose zero potential for

0 < x < L

. In an infinite square well,

ψ_{n} (x) = \frac{2}{L} sin \frac{nπ x}{L}, E_{n} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m L ^{2}} .

The first three levels have ratios $1 : 4 : 9$ , with zero, one and two interior nodes; the wavefunctions vanish outside.

For a finite square well with exterior potential

V_{0}

, bound states have

0 < E < V_{0}

, sinusoidal interiors and tails with

κ = 2 m (V_{0} - E) /ℏ

. Matching wavefunction and derivative for equal masses gives

k tan (k L /2) = κ (even), - k cot (k L /2) = κ (odd), k = 2 m E /ℏ,

when parity is taken about the well center. Corresponding energies fall below the infinite-barrier values, and the wavefunctions spread beyond the walls. There are finitely many bound states; a shallow well need not possess three. States above

V_{0}

form a scattering continuum.

Figure 1.
Wavefunctions, probability densities and core-shell confinement
. Original normalized infinite- and finite-well calculations with a chosen dimensionless barrier, plus qualitative type-I envelope sketches. The envelope panel is not a two-particle probability density or material-specific fit. Details.

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