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Commutator of Position and Energy

  1. Jul 21, 2009 #1
    This is a question about simple non-relativistic quantum mechanics in one dimension.

    If the energy operator is [tex]\imath \frac{h}{2\pi}\frac{\partial}{\partial t}[/tex], then it would appear to commute with the position operator [tex]x[/tex]. Then, if the energy and position operators commute, I ought to be able to find simultaneous eigenstates of them.

    However, it is clear that in general the Hamiltonian does not commute with [tex]x[/tex], and in general these two operators do not have any simultaneous eigenstates.

    What is wrong with my thinking? Does it make sense to think of [tex]\imath \frac{h}{2\pi} \frac{\partial}{\partial t}[/tex] as the energy operator, and is that supposed to be the same as the Hamiltonian? Am I running into a problem because I am thinking on the one hand of a time-independent problem and on the other of a time-dependent one? Alternatively, is it incorrect to state that any two operators which commute must have simultaneous eigenstates?

    Thank you,
  2. jcsd
  3. Jul 22, 2009 #2
    Time derivative is not a valid operator in the Hilbert space, and its commutator with x does not make sense. Normally, vectors in the Hilbert space do not depend on time. The time dependence arises only when you look at the state from different (time displaced) reference frames.

    Schroedinger equation does not say that "the time derivative and the Hamiltonian are equivalent operators" It says: "the time derivative of the state vector is equal to the action of the Hamiltonian on the state vector".
  4. Jul 23, 2009 #3


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    [H,X] is usually taken to be a velocity operator.
    I.e., energy and position don't commute in general.
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