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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!

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!

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