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Energy vs. position eigenstates

  1. Dec 24, 2012 #1
    This might be a silly question, but I'm not sure about the answer...

    Commuting operators have identical eigenstates. For example: energy and position operators seem to commute:
    [E,x] = Ex - xE = 0

    Does it mean that position and energy operators share identical eigenstates? Because eigenstates of energy are stationary, are position eigenstates stationary as well?
     
  2. jcsd
  3. Dec 24, 2012 #2
    1) The hamiltonian (energy) and position do not commute in general.

    2) If two operators, say A and B, commute they share a mutual complete set of eigenkets. This is not to say all eigenkets of A are necessarily eigenkets of B.
     
  4. Dec 25, 2012 #3

    dextercioby

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    Defining commutative operators in the general case is impossible, one needs certain simplifying assumptions, such as (essential) self-adjointness. Two (essentially) self-adjoint operators commute iff the (generalized) projectors from their spectral decomposition commute. This mathematical reasoning applies thoroughly to all operators in quantum mechanics which describe observables.

    In 1D, the free-particle Hamiltonian commutes with the momentum operator, but not with the coordinate operator, due to the kinetic term (proportional to p^2) and the fundamental commutation relations of Born and Jordan (1925).
     
  5. Dec 25, 2012 #4
    Thank you for the answers.
     
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