 Solution

Perturbation theory expands known eigenvalues and eigenfunctions about a solvable Hamiltonian. Small corrections require appropriate energy-gap and matrix-element scales; degeneracies must first be diagonalized within their subspace. The variational method instead minimizes a normalized trial state's energy expectation. It gives an upper bound on the ground energy without requiring a small perturbation; excited-state bounds need orthogonality constraints or an appropriate symmetry subspace.

Normalize the unperturbed oscillator states:For the Gaussian perturbation of a quantum harmonic oscillator, the first-order shifts are its expectation values:For $\beta \gg \alpha$, they scale as $V_0\alpha /\beta$ and $V_0(\alpha /\beta {)}^3$ respectively: a narrow central perturbation couples weakly to the odd state's central node. For $\beta \ll \alpha$, both approach $V_0$ because the perturbation is nearly constant over the wavefunctions. The leading corrections are $V_0[1-{\beta }^2/(2{\alpha }^2)]$ and $V_0[1-3{\beta }^2/(2{\alpha }^2)]$.

For an improved variational estimate, replace the fixed width $\alpha$ by an adjustable positive width $s$ in each even/odd Gaussian. Their energy functionals areMinimize each separately. At $s=\alpha$ they recover the unperturbed energy plus the first-order shift, so optimization cannot raise those trial estimates. For this even one-dimensional potential the lowest odd state is the first excited state, making its odd-sector minimum an upper bound on $E_1$. Enlarging each parity-preserving trial basis can improve the estimates further.