Energy vs. position eigenstates

In summary, the conversation discusses the concept of commuting operators and their identical eigenstates in quantum mechanics. It is mentioned that the hamiltonian and position operators do not commute in general, but if two operators commute, they share a mutual set of eigenkets. It is also noted that the free-particle Hamiltonian commutes with the momentum operator but not with the coordinate operator due to the kinetic term and fundamental commutation relations.
  • #1
mpv_plate
87
4
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?
 
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  • #2
mpv_plate said:
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.
 
  • #3
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
Thank you for the answers.
 

1. What are energy and position eigenstates?

Energy and position eigenstates are quantum mechanical states that describe the energy and position of a particle in a system, respectively. They are represented by mathematical functions called wavefunctions, which contain information about the probability of finding the particle at a specific energy or position.

2. How are energy and position eigenstates related?

Energy and position eigenstates are related through the uncertainty principle, which states that the more precisely the position of a particle is known, the less precisely its energy can be known, and vice versa. This means that a particle in an energy eigenstate will have a range of possible positions, and a particle in a position eigenstate will have a range of possible energies.

3. What is the significance of energy and position eigenstates?

Energy and position eigenstates are important because they allow us to make predictions about the behavior of quantum systems. By representing the energy and position of a particle as eigenstates, we can calculate the probabilities of different outcomes and make predictions about the behavior of the particle.

4. How are energy and position eigenstates measured?

Energy and position eigenstates cannot be measured directly, as they are not physical quantities. However, the probabilities associated with these states can be measured experimentally through techniques such as spectroscopy for energy measurements and position detectors for position measurements.

5. Can a particle be in both an energy and position eigenstate at the same time?

No, a particle cannot be in both an energy and position eigenstate simultaneously. This is because energy and position are complementary variables, meaning that they cannot both be known with precision at the same time. A particle can exist in a superposition of different energy or position states, but it cannot be in two specific states at once.

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