Quadratic form of a positive quantum Hamiltonian ($q_H(\psi )=\Vert H^{1/2}\psi {\Vert }^2$)

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, $\psi ={\sum }_n{c}_n{e}_n$ and $H{e}_n=E_n{e}_n$ give $q_H(\psi )={\sum }_n{E}_n|c_n{|}^2$ by applying $H^{1/2}$ and taking its squared norm. Membership in $D(H)$ instead requires ${\sum }_n{E}_n^2|c_n{|}^2<\infty$. For a Dirichlet free-particle Hamiltonian operator on an interval, the energy form is ${\hslash }^2\int |{\psi }^{\prime }{|}^2/(2m)$ on functions with square-integrable weak derivative and zero endpoint values.