= Quadratic form of a positive quantum Hamiltonian
{title2=$q_H(\psi)=\|H^{1/2}\psi\|^2$}
For a positive self-adjoint <Hamiltonian operator>, finite energy means that a normalized state lies in $D(H^{1/2})$, its form domain, and has <expectation value> $q_H(\psi)$. This domain can be larger than $D(H)$. With a complete orthonormal basis of <energy eigenstates>, $\psi=\sum_nc_ne_n$ and $He_n=E_ne_n$ give $q_H(\psi)=\sum_nE_n|c_n|^2$ by applying $H^{1/2}$ and taking its squared norm. Membership in $D(H)$ instead requires $\sum_nE_n^2|c_n|^2<\infty$. For a Dirichlet free-particle <Hamiltonian operator> on an interval, the energy form is $\hbar^2\int|\psi'|^2/(2m)$ on functions with square-integrable weak derivative and zero endpoint values.
Back to article page
