Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A2 / a / Solution

Differentiate the Gaussian before substituting into the Hamiltonian:

Ψ_{0}^{'} = - αx Ψ_{0}, Ψ_{0}^{''} = (α^{2} x^{2} - α) Ψ_{0} .

Thus

\hat{H} Ψ_{0} = [\frac{ℏ ^{2} α}{2 m} + (\frac{m ω ^{2}}{2} - \frac{ℏ ^{2} α ^{2}}{2 m}) x^{2}] Ψ_{0} .

An energy eigenvalue must be independent of

x

, so the coefficient of

x^{2}

vanishes. Normalizability requires

α > 0

α = \frac{mω}{ℏ}, E_{0} = \frac{1}{2} ℏ ω .

The constant term is the quantum harmonic oscillator's zero-point energy; a Gaussian width

α^{- 1/2}

balances its kinetic and potential contributions.

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