# Energy vs. position eigenstates

1. Dec 24, 2012

### 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?

2. Dec 24, 2012

### Jorriss

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.

3. Dec 25, 2012

### 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).

4. Dec 25, 2012

### mpv_plate

Thank you for the answers.

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