Is it possible to show that the 6 states generated are not all unique?

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In summary, Kramers theorem states that for a time-reversal invariant Hamiltonian, the energy levels must have an even degree of degeneracy. However, it can be shown that threefold degeneracy is not possible, as it would result in a contradiction with the pairwise exchange operators for three particles.
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Homework Statement


Kramers theorem states that if the Hamiltonian of a system is invariant under time reversal, and if [tex]T^2|\psi>=-|\psi>[/tex] where T is the time reversal operator, then the energy levels must be at least doubly degenerate. In fact the degree of degenercay must be even. Show explicitly that threefold degeneracy is not possible.


Homework Equations





The Attempt at a Solution


So I have assumed that there are three eigenstates with the same eigenvalue, let's write them as |A1>,|A2>,|A3> and the eigenvalue is A, so if H is the Hamiltonian then:
H|A1>=A|A1>
H|A2>=A|A2>
H|A3>=A|A3>
A is a real valued number.
Now, HT|A1>=AT|A1>
so T|A1> is one of the three above eigenstates, either A1,A2 or A3.
Now I need to show that either way we get a contradiction.
Now if T|A1>=|A1>, then -|A1>=T^2|A1>=T|A1>=|A1> which is impossible.
Now i am left with two options, either T|A1>=|A2> or T|A1>=|A3>, here is where I am stuck I didn't succeed in showing a contradiction.

Any hints?
 
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  • #2
I have this same problem. I think you need to use the fact that Kramer's Theorem only applies to odd numbers of particles. If you use 3 electrons (a, b, c) then you can create 6 states from the combinations. You can also have 3 pairwise exchange operators P12, P13, P23.

Then you can show that for any T |abc> choice, you have to get T and T^2 equivalent to one of the pairwise operators, which is false since P^2=1.
 

1. What is Kramers Theorem?

Kramers Theorem, also known as the Kramers-Kronig relations, is a mathematical theorem that relates the real and imaginary parts of a complex function. It is used to analyze the properties of systems described by complex functions, such as light propagation in optical materials.

2. Who discovered Kramers Theorem?

Hendrik Anthony Kramers, a Dutch physicist, discovered Kramers Theorem in 1927. He developed it as a generalization of the Kramers dispersion formula, which was first proposed by his mentor, Hendrik Lorentz.

3. What is the significance of Kramers Theorem?

Kramers Theorem has significant applications in various fields of science, including optics, electromagnetism, and quantum mechanics. It is used to determine the optical properties of materials, such as refractive index and absorption coefficient, and to analyze the behavior of physical systems described by complex functions.

4. How is Kramers Theorem applied in optics?

In optics, Kramers Theorem is used to determine the refractive index and extinction coefficient of materials by measuring their optical properties, such as reflectance, transmittance, and absorption. It is also used to analyze the dispersion of light in materials, which is essential for designing optical devices.

5. Is Kramers Theorem applicable to all complex functions?

Kramers Theorem is applicable to any complex function that satisfies the requirements of causality and analyticity. However, it is most commonly used in cases where the function describes the properties of physical systems, such as electromagnetic fields or material properties.

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