Let $Q$ be a dense linear subspace of the Hilbert space, $\mathcal{H}$. Then there is a mapping $q: Q \times Q \to \mathbb{C}$, known as a quadratic form. For two states $\psi, \phi \in Q$,and a self-adjoint operator $A$ on $\mathcal{H}$, then
\begin{equation} q(\psi,A \phi) = \int \overline{\psi(x)} A \phi(x) dx. \end{equation}
But what is the physical meaning of such a quadratic form between two different states?
More specifically, if $A = T^{00}$ the energy density operator, does this then measure the change of energy required to change the state $\phi \to \psi$?
