How to understand the off-diagonal elements

[phdphysics]

In quantum mechanics, observable variables are represented by operators, and thus can be replaced by matrix in a certain basis.
If we have H|n>=E(n)|n>, where |n> are eigenfunctions of Hamilton matrix. Here is the problem: what's the physical meaning of <m|P|n>, namely off-diagonal elements of another observable operator, such as momentum?
I have known that sometimes this has something to do with quantum transition. But I want to know is there any universal understanding of off-diagonal elements?
Thanks!

 

[naima]

same question forthe off diagonal elements in the density matrix. Those which disappear during decoherence.

 

[The_Duck]

what's the physical meaning of <m|P|n>, namely off-diagonal elements of another observable operator, such as momentum?
I have known that sometimes this has something to do with quantum transition. But I want to know is there any universal understanding of off-diagonal elements?
Thanks!

Many operators can be interpreted as generators of infinitesimal transformations on states. For example consider the momentum operator $P$. If I have a state $| \psi \rangle$ and $\epsilon$ is an infinitesimal distance, then $(1 + i \frac{\epsilon}{\hbar} P ) | \psi \rangle$ is a version of the state $| \psi \rangle$ that has been translated to the left by a distance $\epsilon$.

Then $\langle m | P | n \rangle$ can be interpreted as something like "the change in the overlap of $| m \rangle$ and $| n \rangle$ when one of them is translated by a small distance."