Symmetric Potential: Reasons for Eigenstate Solutions

[jostpuur]

I never learned this in the lectures (maybe I was sleeping), but now I think I finally realized what is the reason that eigenstate solutions of SE with a symmetric potential are either symmetric or antisymmetric. Is the argument this:

"The Hamiltonian and the space reflection operator commute, therefore they have common eigenstates" ?

If it is this, can somebody explain me why does a constant potential (which is also symmetric) have plane wave solutions, that are not symmetric or antisymmetric.

 

[Nesk]

Any quantum particle in a constant potential is in an un-bound state, i.e. a plain plane wave.

 

[denian]

Yes, your understanding is correct. The reason for the eigenstate solutions of the Schrödinger equation (SE) with a symmetric potential being either symmetric or antisymmetric is because the Hamiltonian and the space reflection operator commute, meaning they have common eigenstates. This is known as the symmetry principle in quantum mechanics.

To explain this further, let's consider a particle in a symmetric potential. The potential is symmetric if it is the same on both sides of a central point, such as a potential well or barrier. In this case, the Hamiltonian and the space reflection operator (which flips the position of the particle) commute, meaning they share the same eigenstates. This means that the eigenstates of the Hamiltonian must also be eigenstates of the space reflection operator, and vice versa.

Now, the space reflection operator has two possible eigenvalues: +1 for symmetric states and -1 for antisymmetric states. This means that the eigenstates of the Hamiltonian must also have these two possible eigenvalues, leading to the eigenstate solutions being either symmetric or antisymmetric.

As for the case of a constant potential, which is also symmetric, it may seem contradictory that the eigenstate solutions are not always symmetric or antisymmetric. However, this is because the potential itself does not determine the symmetry of the eigenstates. It is the combined effect of both the potential and the Hamiltonian that determines the symmetry of the eigenstates. In the case of a constant potential, the Hamiltonian is simply the kinetic energy operator, which has no effect on the symmetry of the eigenstates. Therefore, the eigenstates can take on any symmetry, including plane wave solutions which are neither symmetric nor antisymmetric.

I hope this explanation helps clarify the concept of eigenstate solutions in symmetric potentials.