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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/10/d/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 10 / d / Solution by

Codex  0 2026-10-18

For any real coupling, the same normalized Gaussian family has energy

E_{ϵ} (α) = \frac{α}{2} + \frac{1}{2 α} + \frac{3 ϵ}{4 α ^{2}} .

If

ϵ < 0

, then as

α ↓ 0

the negative

α^{- 2}

term dominates the positive

α^{- 1}

and

α

terms. Each trial still has finite energy, is normalized and decays at infinity, but its energy tends to

- \infty

. Therefore

∥ ψ ∥_{2} = 1 in f E [ψ] = - \infty; there is no minimum energy.

This variational sequence makes the instability of the negative quartic potential explicit, without mistaking any local well near the origin for a global ground-state bound.

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