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If the current state of some quantum system is $| \psi \rangle$, what is the physical interpretation of $$ \langle \phi | A | \psi \rangle $$ where $|\phi\rangle$ is some other -maybe the same- quantum state, and $A$ is a generic self-adjoint operator?

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The matrix element $\langle \phi | A | \psi \rangle$ can have a lot of meanings depending on context.

  • If $A$ is self-adjoint and $|\phi \rangle = |\psi \rangle$, then it is the expectation value of $A$.
  • If $A$ is unitary, it can be a time evolution operator, so $A|\psi\rangle$ is what $|\psi \rangle$ evolves into. Then the matrix element tells us how much the final state is like $|\phi\rangle$.
  • If $A$ is self-adjoint and the vectors are general, it doesn't really have a general meaning, except in some special cases. For example, suppose the Hamiltonian has a term proportional to $A$. Then there will be a piece of the time evolution that looks like $e^{-iAt}$, in which case the matrix element tells us about the transition rate from $|\psi\rangle$ to $|\phi\rangle$ (see Fermi's golden rule).
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Given a system in a quantum state $|\psi\rangle$, we act on it with the operator $A$ (this could mean many different things depending on what the operator $A$ is) which leaves the system in the state $A | \psi \rangle$. The quantity $| \langle \phi| A | \psi \rangle |^2$ then gives the probability that the system is in the state $|\phi\rangle$ after the action of $A$.

For example, if $A = e^{- i H t}$ where $H$ is the Hamiltonian, then acting on a system with $A$ simply means allowing it to evolve by a time $t$.

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