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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 14B / b / Solution

For

ψ = N (t) e^{- F (t) x^{2} + G (t) x},

direct differentiation gives

ψ_{x} = (- 2 F x + G) ψ, ψ_{xx} = (4 F^{2} x^{2} - 4 FG x + G^{2} - 2 F) ψ

and

ψ_{t} = (\frac{N ˙}{N} - \dot{F} x^{2} + \dot{G} x) ψ .

Substitution into the Schrodinger equation and equality of the coefficients of

x^{2}, x, 1

give the necessary and sufficient Riccati equation system

i ℏ \dot{F} i ℏ \dot{G} i ℏ \frac{N ˙}{N} = \frac{2 ℏ ^{2}}{m} F^{2} - \frac{1}{2} m ω^{2}, = \frac{2 ℏ ^{2}}{m} FG, = \frac{ℏ ^{2}}{m} F - \frac{ℏ ^{2}}{2 m} G^{2} .

Solved by gpt-5.6-sol high.

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