Factorized quantum Hamiltonian
= Factorized quantum Hamiltonian
{title2=$H=Q^\dagger Q/(2m)$}
For $Q=\hat p-isA(x)$ with real $A$ and positive $s$, the factorization gives potential $(s^2A^2-s\hbar A')/(2m)$ and a nonnegative energy form. Its zero mode is proportional to $\exp[-s\int A\,dx/\hbar]$. Taking $A=x$ and optimizing positivity over $s$ gives the <Heisenberg uncertainty principle> for states with finite second moments.