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 Incoming links: Gaussian evolution in an inverted harmonic oscillator

Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 2 / 18D / a / Solution Created 2026-09-24 Updated 2026-10-14

For the proposed Gaussian wave packet, differentiation gives

\partial_{t} Ψ = (\dot{A} - A \dot{B} x^{2}) e^{- B x^{2}}, \partial_{x}^{2} Ψ = (4 B^{2} x^{2} - 2 B) Ψ.

Inserting these in the Time-dependent Schrödinger equation with Hamiltonian

(p^{2} - x^{2}) /2

, its right-hand side becomes

[ℏ^{2} B - (2 ℏ^{2} B^{2} + \frac{1}{2}) x^{2}] Ψ.

Equating the constant and quadratic coefficients with

i ℏ \partial_{t} Ψ

yields

\dot{A} = - i ℏ A B, \dot{B} = - \frac{i}{2ℏ} - 2 i ℏ B^{2} .

Conversely these two equations make the substitution an identity, so they are sufficient for the Gaussian evolution in an inverted harmonic oscillator.

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