= Solution
For a self-adjoint <Hamiltonian (quantum mechanics)> bounded below, every nonzero admissible trial <wavefunction> satisfies the <quantum variational principle>
$$
\boxed{\mathcal E[\psi]=\frac{\langle\psi|\hat H|\psi\rangle}{\langle\psi|\psi\rangle}\ge E_0.}
$$
Here $E_0$ is the lowest exact <energy>, and the trial function must obey the physical <boundary conditions> and have finite relevant integrals. Equality holds for a <wavefunction> in the ground-state eigenspace. The quotient is unchanged by normalization or any nonzero overall scalar.
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