Energy vs. position eigenstates

[mpv_plate]

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?

 

[Jorriss]

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?

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.

 

[dextercioby]

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).

 

[mpv_plate]

Thank you for the answers.