Factorized-Hamiltonian raising trial state ($H{\psi }_1=-i{\hslash }^2{f}^{\prime \prime }{\psi }_0/m+\hslash f^{\prime }{\psi }_1/m$)

For $Q=p-if(x)$, $H=Q^{†}Q/(2m)$ and $Q{\psi }_0=0$, the trial state ${\psi }_1=Q^{†}{\psi }_0$ obeys the displayed identity. It follows from $[Q,Q^{†}]=2\hslash f^{\prime }$ and $[p,f^{\prime }]=-i\hslash f^{\prime \prime }$. The coefficients are generally position-dependent multiplication functions, not constants. For linear positive $f$, the first term vanishes and the second becomes the harmonic-oscillator level spacing.