Inner product of wavefunctions

Question

im trying to understand the meaning of this inner product:
$⟨psi_a|H|psi_b⟩$.
$H$ can be a time-independent hamiltonian.
I know that $⟨psi_a|H|psi_a⟩$ is the expectation value, but I don't know the meaning of the first inner product.
Can we say that $|langlepsi_a|H|psi_brangle|^2$ is equal to the probability that the measurement of H over the state $psi_b$ gives the state $psi_a$?

Nelson Vanegas A. · Accepted Answer

As in any inner product you can take it as a projection of the state vector $|psi_arangle$ in the direction of $H|psi_brangle.$ The latter is just another state vector, for example $H|psi_b rangle = |psi_bprimerangle,$ then $langle psi_a |psi_bprimerangle$ is just a projection of one in the direction of the other. You can also take it as the Matrix element of the Hermitian operator $H_{ab},$ if you allow for $|psi_irangle$ to be a basis of the State Space. In a more physical interpretation you could argue that you could obtain the probability of a state |$psi_brangle$ transitioning to $|psi_arangle$ in time if $$|psi_b (x,t)rangle = e^{tH/hbar} |psi_b(x,0)rangle,$$ then the term $langle psi_a |psi_bprimerangle$ would determine the probability amplitude of the transition from $b$ to $a.$ (Expand $|psi_b(x,0)rangle$ in a basis of eigenvectors of $H$ and the same for $|psi_a(x,0)rangle$ for this to work as a possible interpretation.

Correct answer by Nelson Vanegas A. on August 2, 2021

hulsey · Answer

You should think of $langle psi_a |H|psi_brangle$ as a "matrix element" of H (i.e. $langle psi_a |H|psi_brangle = (H)_{ab}$), especially in the case that the indices $a,b$ run over an orthonormal basis of states.

While $langle psi_a |H|psi_arangle$ is real, if $a neq b$ the same cannot be said for $langlepsi_a |H|psi_brangle^* = langle psi_b |H|psi_arangle$. However, in any calculation of a measurable quantity that involves these states, you'll end up taking the real or imaginary part of this mixed quantity.

Asking about the "meaning" is subjective; it depends on what the states $|psi_a rangle$ are meant to represent. If they are energy eigenstates, then $langle psi_a |H|psi_brangle propto delta_{ab}$. Otherwise, it is difficult to say.

Inner product of wavefunctions

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