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How to understand the off-diagonal elements

  1. Oct 18, 2013 #1
    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!
     
    Last edited: Oct 18, 2013
  2. jcsd
  3. Oct 18, 2013 #2

    naima

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    Gold Member

    same question forthe off diagonal elements in the density matrix. Those which disappear during decoherence.
     
  4. Oct 18, 2013 #3
    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."
     
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