Coupled Two-Level Eigensystem Problem
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Summary: An in-depth analysis of a simple coupled,two-level system that shows how the strength of the coupling perturbation affects the resulting repulsion of the eiqenvalues and the mixing of the original eigenstates.
A simple, two-level system is probably the simplest system that can be analyzed to see the effects of coupling between eigenstates. It is surprisingly rich problem with very interesting results that have connections to both classical and quantum physics applications.
Consider a system described by a 2x2 Hermitian matrix operator, M0. Such a system had two real eigenvalues, and when represented in the basis given by its eigenvectors, M0 is diagonal:
Now consider the addition of a perturbing operator, M', that consists purely of terms that couple the original eigenstates. That is, the perturbing operator does not add any constant offsets to the original eigenvalues because it has only off-diagonal elements. A new, perturbed operator, M, is the sum of M0, and M'.
The question is, in terms of the original eigenvalues and eigenvectors, what are the new eigenvalues and eigenvectors for the new perturbed operator, M?
The secular is equation is used to determine the eigenvalues:
This analysis clearly shows a number of key features:
To solve for the eigenvectors, we simply plug the eigenvalues back into the eigenvalue equation. Below, we denote the coefficients associated with the positive and negative root solutions by "+" and "-" subscripts, respectively.
We can see several important properties of the wavevectors from the above analysis:
In summary, the new eigenvalues and eigenstates are given by
An in-depth analysis of the time-dependent eigenvectors will show that in the presence of a coupling field, a system initially prepared in a state corresponding to an original unperturbed eigenstate, will appear to oscillate back and forth between the two original eigenstates. This corresponds to the absorption and emission of a photon by a two-level atomic system. See the three spring problem module for the temporal analysis of a degenerate system.
Originally published in Connexions (CNX): https://web.archive.org/web/20131013080112/https://cnx.org/content/m27882/latest/
© 2023 by Stephen Wong