Two fold degeneracy perturbation question

In summary, a two fold degeneracy exists in the unperturbed states, Hamiltonian, and eigenvalue. The linear combination of the two states also has an eigenvalue of E. When a perturbation is introduced, the energy level splits into two values and two states. Removing the perturbation will result in two unique linear combinations of Psi_a and Psi_b, which are now degenerate eigenstates with the same eigenvalue. This allows for a basis to be chosen for the previously degenerate eigenspace.
  • #1
mmwave
647
2
consider a two fold degeneracy such that
H Psi_a = E Psi_a and H Psi_b = E Psi_b and <Psi_a | Psi_b> = 0

All of the above are the unperturbed states, Hamiltonian and eigenvalue. Notice the two states share the eigen value E.

Form the linear combination of the two states

Psi = a * Psi_a + b * Psi_b

Clearly H Psi = E Psi and the eigenvalue of the linear combination is still the value E.

Now a perturbation to the hamiltonian H will split the energy level into two values E+ and E- and two states Psi+ and Psi-.

Finally the question:

I am told that if the perturbation is "dialed down to zero" then Psi+ and Psi- reduce to two different linear combinations of Psi_a and Psi_b. What does this mean?

Shouldn't the states be simply Psi_a and Psi_b? Why would there be two unique linear combinations instead of just the two eigenstates themselves?
 
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  • #2
Originally posted by mmwave
Shouldn't the states be simply Psi_a and Psi_b? Why would there be two unique linear combinations instead of just the two eigenstates themselves?

why would you think it would be Psi_a and Psi_b? you know that if the perturbation is nonzero, than the eigenstates are not necessarily Psi_a, Psi_b.

any linear combination is an eigenstate of the unperturbed hamiltonian. the perturbation just chooses a particular linear combination that diagonalize the perturbation as well.
 
  • #3


Originally posted by lethe
why would you think it would be Psi_a and Psi_b? you know that if the perturbation is nonzero, than the eigenstates are not necessarily Psi_a, Psi_b.


Because the question comes when the perturbation is removed. It seems to me that the system should go back to where it was with eigenstates Psi_a and Psi_b. I think there may be an ustated assumption I am ignorant of here.
 
  • #4


Originally posted by mmwave
Because the question comes when the perturbation is removed. It seems to me that the system should go back to where it was with eigenstates Psi_a and Psi_b. I think there may be an ustated assumption I am ignorant of here.

when the perturbation is very very small, you get two orthogonal eigenstates that have very very close to the same eigenvalue.

when the perturbation is zero, the two orthogonal eigenstates have exactly the same eigenvalue. they are now degenerate.

all you have done is choose a basis for the degenerate eigenspace, when there was no good choice before you turned on the perturbation.
 

1. What is two fold degeneracy perturbation question?

The two fold degeneracy perturbation question is a problem in quantum mechanics that involves calculating the energy levels of a system that has two states with the same energy. It is a common concept used in perturbation theory, which is a method for approximating the behavior of a system when it is slightly perturbed from its original state.

2. How do you solve a two fold degeneracy perturbation question?

To solve a two fold degeneracy perturbation question, you first need to determine the unperturbed energy levels of the system. Then, you use perturbation theory to calculate the first-order corrections to the energy levels caused by the perturbation. Finally, you use these corrections to determine the new energy levels of the system in the perturbed state.

3. What is the significance of two fold degeneracy in quantum mechanics?

In quantum mechanics, two fold degeneracy refers to the situation where two different quantum states have the same energy. This can occur in certain systems, such as atoms and molecules, and has important implications for their behavior and properties. For example, it can affect the selection rules for transitions between energy levels and the stability of the system.

4. Can two fold degeneracy occur in classical systems?

No, two fold degeneracy is a concept that is specific to quantum mechanics. In classical systems, all states have distinct energies and there is no overlap or degeneracy between energy levels. However, in some cases, classical systems can exhibit behavior that is similar to two fold degeneracy, but it is not the same phenomenon as in quantum mechanics.

5. What are some real-life applications of two fold degeneracy perturbation question?

Two fold degeneracy perturbation questions have many applications in quantum mechanics, which is the basis for many modern technologies. For example, they can be used to study the behavior of atoms and molecules, which is important in fields such as chemistry and materials science. They are also used in the design of electronic devices, such as transistors, and in the development of quantum computing and other quantum technologies.

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