Introduction to Quantum Mechanics · Problem 8.4

The Variational Principle — David J. Griffiths

Physics — Quantum Mechanics · variational_excited_states

Problem

(a) Prove the following corollary to the variational principle: If ⟨ψ∣ψgs⟩=0\langle\psi|\psi_{\text{gs}}\rangle = 0, then ⟨H⟩≥Efe\langle H\rangle \geq E_{\text{fe}}, where EfeE_{\text{fe}} is the energy of the first excited state. *Comment:* If we can find a trial function that is orthogonal to the exact ground state, we can get an upper bound on the *first excited state*. In general, it's difficult to be sure that ψ\psi is orthogonal to ψgs\psi_{\text{gs}}, since (presumably) we don't *know* the latter. However, if the potential V(x)V(x) is an *even* function of xx, then the ground state is likewise even, and hence any *odd* trial function will automatically meet the condition for the corollary. (b) Find the best bound on the first excited state of the one-dimensional harmonic oscillator using the trial function ψ(x)=Axe−bx2.\psi(x) = Axe^{-bx^2}.

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