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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 35B / a / Solution

Let

H

be the one-dimensional Hamiltonian operator and let

E_{0} \leq E_{1} \leq E_{2} \leq \dots

be its discrete energy eigenvalues. Expanding a normalized trial wavefunction in an orthonormal energy basis,

∣ ψ ⟩ = \sum_{n} c_{n} ∣ n ⟩,

gives

⟨ ψ ∣ H ∣ ψ ⟩ = \sum_{n} ∣ c_{n} ∣^{2} E_{n} \geq E_{0} \sum_{n} ∣ c_{n} ∣^{2} = E_{0} .

Thus the Rayleigh-Ritz variational principle gives

E_{0} \leq \frac{\int _{R} ψ ^{*} ( x ) H ψ ( x ) d x}{\int _{R} ∣ ψ ( x ) ∣ ^{2} d x}

for every nonzero admissible trial wavefunction. One chooses a parameterized family and minimizes the quotient.

When

V (x) = V (- x)

, parity commutes with

H

. In one dimension the nondegenerate ground state is even and the first excited state is odd. Restricting the trial family to odd wavefunctions makes every trial state orthogonal to the ground state, so

E_{1} \leq \frac{⟨ ψ _{odd} ∣ H ∣ ψ _{odd} ⟩}{⟨ ψ _{odd} ∣ ψ _{odd} ⟩} .

This is the odd-state variational principle for an even potential.

Solved by gpt-5.6-sol high.

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