# What is the physical meaning of a general 'sandwich' in quantum mechanics?

Considering a sandwich of the form $$\langle{\psi}|{\hat{O}}|{\phi}\rangle$$, where $$\hat{O}$$ is an operator and $$\psi, \phi$$ are state vectors, I know that if $$\psi=\phi$$ then the sandwich represents the expectation value of that operator $$\langle \hat{O} \rangle$$ i.e. the average value of all the possible outcomes of a measurement under identical conditions. But if $$\psi \ne \phi$$ does the sandwich have a physical meaning?

Moreover if the operator $$\hat{O}$$ is not hermitian, then the expectation value does not represent an obserable, so does $$\langle{\psi}|{\hat{O}}|{\psi}\rangle$$ still have a physical interpretation?

• You are asking about the "physical meaning" of an arbitrary matrix element of an arbitrary matrix? Dec 7, 2022 at 15:23
• Do people really call such matrix elements, such inner products, not only overlaps but also sandwiches? As for interpreting them, they're fixed by more easily-understood diagonal ($\psi=\phi$) elements if $\hat{O}$ is positive-definite; indeed, $\hat{O}$ is the difference of two such operators.
– J.G.
Dec 7, 2022 at 16:26
• @J.G. I've heard people use the word sandwich for them a lot in the US. Also as a verb e.g. "to sandwich the operator between $\langle \phi |$ and $|\psi \rangle$ Dec 7, 2022 at 17:42
• @doublefelix Thanks for identifying a relevant nation. My experience with "overlap" is UK-based.
– J.G.
Dec 7, 2022 at 19:15

In general there is no physical meaning I am aware of even if $$O$$ is hermitian. Even when $$O$$ is the momentum operator, for example, taking $$\langle \psi | \hat{p} | \phi \rangle$$ has no specific interpretation for arbitrary $$\psi, \phi$$.
• The probability of transitioning from an initial state $$|\psi \rangle$$ at $$t=0$$ to an eigenstate $$|\lambda \rangle$$ of some operator $$O$$ such that $$O |\lambda \rangle = \lambda |\lambda \rangle$$, is (if $$\lambda$$ has only 1 orthonormal eigenvector) $$|\langle \lambda | e^{-itH/\hbar }|\psi \rangle|^2$$
• Letting $$P_\lambda$$ be a projector to the eigenspace corresponding to eigenvalue $$\lambda$$, the probability of getting outcome $$\lambda$$ when measuring a state $$|\psi \rangle$$ without evolution in between is $$\langle \psi | P_\lambda | \psi \rangle$$
• The expectation value of an operator $$O$$ when measuring a particle in state $$|\psi \rangle$$ is $$\langle \psi | O | \psi \rangle$$