Momentum Eigenstate with a non-zero potential

In summary, the conversation discusses the proof that a particle starting in a momentum eigenstate will remain in an eigenstate when under the influence of a potential c*y. The momentum operator and eigenstate are defined, and it is noted that a momentum eigenstate can only exist for systems with zero potential. The question of whether a momentum eigenstate can have a time-dependent eigenvalue is also posed.
  • #1
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Homework Statement


Prove that if a particle starts in a momentum eigenstate it will remain forever in a eigenstate given the potential c*y where c is a constant and y is a spatial variable.

Homework Equations



(h/i)d/dx is the momentum operator and a momentum eigenstate when put in the momentum operator gives an eigenvalue times the momentum eigenstate.

The Attempt at a Solution



If p commutes with H then a eigenstate of H is an eigenstate of p always. My problem is that p does not commute with H and I always thought that you can only have momentum eigenstates for systems with with zero potential. So I'm at a loss where to begin because if I workout the Heisenberg equations I get dp/dt where p is the operator in the Heisenberg picture to be none zero. Any help will be much appreciated.[/B]
 
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  • #2
Can an momentum eigenstate have a eigenvalue which is time dependent.
 

FAQ: Momentum Eigenstate with a non-zero potential

What is a momentum eigenstate with a non-zero potential?

A momentum eigenstate with a non-zero potential is a state in quantum mechanics where the particle's momentum is well-defined and the potential energy is not zero. This means that the particle has a definite momentum and is subject to a force due to the non-zero potential.

How is the momentum eigenstate with a non-zero potential different from the momentum eigenstate with zero potential?

In a momentum eigenstate with a non-zero potential, the particle has a well-defined momentum and is also subject to a force due to the non-zero potential. In contrast, in a momentum eigenstate with zero potential, the particle has a well-defined momentum but is not subject to any force.

What is the significance of the potential in a momentum eigenstate?

The potential in a momentum eigenstate determines the force acting on the particle. In a non-zero potential, the particle experiences a force and its momentum is affected. This allows for the study of the dynamics of a particle in a non-zero potential.

How is the momentum eigenstate with a non-zero potential used in experiments?

The momentum eigenstate with a non-zero potential is used in experiments to study the dynamics of particles in varying potentials. By controlling the potential, scientists can manipulate the force acting on the particle and observe its effect on the particle's momentum.

Can a particle have a momentum eigenstate with a non-zero potential and still have a well-defined position?

Yes, it is possible for a particle to have a momentum eigenstate with a non-zero potential and also have a well-defined position. This is known as a stationary state and is characterized by a wavefunction that does not change with time. In this state, the particle's momentum is well-defined and its position is also well-defined at a specific point in space.

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