Symmetric Potentials - Eigenstates & Ground States

In summary, the conversation discusses the relationship between symmetry and eigenstates of a potential. The participants agree that a symmetric potential does not necessarily mean that the eigenstates will also be symmetric. However, they mention that the parity operator can be used to choose a basis consisting of symmetric and antisymmetric eigenstates. They also mention that there are no degenerate eigenvalues in one dimension, but this statement may not hold in all cases. The conversation ends with a discussion about the effects of infinitely high walls on this relationship.
  • #1
MJC3Jh
2
0
Hi,

Can anyone help me to understand the following please? If a potential is symmetric does this mean that the eigenstates are either symmetric or antisymmetric? Is the ground state always symmetric and the first excited state always antisymmetric?

Thanks!
 
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  • #2
Yes and yes.
 
  • #3
Why is the second bit true?
 
  • #4
The ground state wave function in one dimension has no zeroes. You can probably google for a proof.
 
  • #5
MJC3Jh said:
If a potential is symmetric does this mean that the eigenstates are either symmetric or antisymmetric?
Not necessarily, IMO. You can always choose the basis consisting of symmetric and antisymmetric eigenstates of the Hamiltonian though, as the parity operator commutes with the Hamiltonian. But this is not the same as what you ask, as a symmetric and an asymmetric eigenstates can have the same eigenvalue, so their linear combination will also be an eigenstate.
 
  • #6
There are no degenerate eigenvalues in one dimension.
 
  • #7
Avodyne said:
There are no degenerate eigenvalues in one dimension.
I guess this statement should be qualified somehow, because it is clearly doubtful in the case of a constant potential. My guess is the statement does not hold water in a more general case either, at least for the continuous spectrum. If you have infinitely high walls, I don't know, maybe you're right.
 

1. What is a symmetric potential?

A symmetric potential is a mathematical representation of a physical system that exhibits symmetry, meaning that certain properties of the system remain unchanged under certain transformations. In the context of quantum mechanics, a symmetric potential is a potential energy function that is symmetrical about a certain axis or point.

2. What are eigenstates in the context of symmetric potentials?

Eigenstates, also known as stationary states, are quantum states that represent the possible energy levels of a system. In the context of symmetric potentials, eigenstates are solutions to the Schrödinger equation that describe the system's energy levels and corresponding wavefunctions.

3. How do eigenstates relate to ground states in symmetric potentials?

The ground state of a system is the lowest energy state that the system can occupy. In symmetric potentials, the ground state is typically the state with the lowest energy eigenvalue. This means that the ground state is the most stable state of the system and is often used as a reference point for understanding the system's other energy levels.

4. What is the significance of finding the ground state in symmetric potentials?

Finding the ground state in symmetric potentials is important because it allows us to understand the behavior of the system at its lowest energy state. This can provide insight into the system's stability, as well as its potential for energy transitions and other physical processes.

5. How do scientists use eigenstates and ground states in the study of symmetric potentials?

Scientists use eigenstates and ground states as tools for understanding the behavior of physical systems described by symmetric potentials. By analyzing the energy levels and corresponding wavefunctions, scientists can make predictions about the behavior of the system and test these predictions through experiments and observations.

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