Degenerate time indep. perturbation theory

In summary, in time independent degenerate perturbation theory, we diagonalize the matrix of the perturbation part of the Hamiltonian instead of the original Hamiltonian because the unperturbed part is proportional to the unit matrix due to all states in the degenerate subspace having the same energy. This means that any linear combination of states within the subspace will also be an eigenstate of the unperturbed part with the same energy. Therefore, the focus is on diagonalizing the perturbation part, which can be done by choosing a basis within the degenerate subspace that diagonalizes it, allowing for regular (non-degenerate) perturbation to be performed.
  • #1
ice109
1,714
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why in time independent degenerate perturbation we diagonalize the matrix of the perturbation part of the hamilitonian and not the original hamiltonian?
 
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  • #2
Because the unperturbed part is just proportional to the unit matrix since all states in the degenerate subspace have the same energy.
 
  • #3
jensa said:
Because the unperturbed part is just proportional to the unit matrix since all states in the degenerate subspace have the same energy.
what? the bold part is true and i agree with.

the italicized part isn't true, firstly since i just did a problem where the matrix representation of the unperturbed part had off diagonals, secondly because an unperturbed part which was proportional to the identity would imply that it's already diagonalized and hence no degeneracy and hence no need for degenerate theory.

i'm thinking it's because the the operator/matrix that diagonalizes the perturbed part also diagonalizes the unperturbed part. but I'm probably wrong.
 
  • #4
Of course it depends on what basis you are using if the matrix representation is diagonal or not. I assumed that you are working in an eigen-basis of the unperturbed Hamiltonian. In this basis the matrix representation of the unperturbed Hamiltonian is diagonal with entries corresponding to the eigen-energies, right? As far as the degenerate subspace is concerned the hamiltonian is diagonal with the same eigen-energies, hence it is proportional to the unit matrix.

The point of degenerate perturbation theory is that can choose a basis of the degenerate subspace arbitrarily and obviously it is most convenient to choose this basis such that it diagonalizes the perturbation. Then once we have made this choice of basis we can perform regular (non-degenerate) perturbation.

The reason why we do not care about diagonalizing the unperturbed part is because any linear combination of states within the degenerate subspace will also be an eigenstate of the unperturbed part with the degenerate energy. This is just another way of saying that the unperturbed Hamiltonian in this subspace is represented by a matrix proportional to the unit matrix.
 

Related to Degenerate time indep. perturbation theory

What is degenerate time-independent perturbation theory?

Degenerate time-independent perturbation theory is a mathematical method used in quantum mechanics to calculate the energy levels and wavefunctions of a system when it is subject to a weak perturbation. It is specifically used for systems where multiple states have the same energy, known as degenerate states.

When is degenerate time-independent perturbation theory used?

Degenerate time-independent perturbation theory is used when a system has degenerate states and the perturbation is weak enough that it does not significantly change the energy levels or wavefunctions of the system.

How does degenerate time-independent perturbation theory work?

Degenerate time-independent perturbation theory works by first calculating the unperturbed energy levels and wavefunctions of the system. Then, the perturbation is introduced and the first-order correction to the energy levels and wavefunctions is calculated. This correction is added to the unperturbed values to obtain the perturbed values.

What are the limitations of degenerate time-independent perturbation theory?

One limitation of degenerate time-independent perturbation theory is that it only works for weak perturbations. If the perturbation is strong, higher order corrections may need to be considered. Additionally, this method can only be applied to systems with degenerate states, so it is not applicable to all quantum mechanical systems.

What are some real-world applications of degenerate time-independent perturbation theory?

Degenerate time-independent perturbation theory has many applications in physics, chemistry, and engineering. It is used in the study of atomic and molecular spectra, in the design of electronic devices, and in the analysis of quantum systems in condensed matter physics. It is also used in the development of quantum algorithms for computing and data encryption.

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