In quantum mechanics, operators correspond to observable variables, eigenvectors are also called eigenstates, and the eigenvalues of an operator represent those values of the corresponding variable that have non-zero probability of occurring.
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An example of this phenomenon may be found in conventional superconductivity, in which the phonon-mediated attraction between conduction electrons leads to the formation of correlated electron pairs known as Cooper pairs.
Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller as we go to higher order.
Imagine that we have two or more perturbed eigenstates with different energies, which are continuously generated from an equal number of unperturbed eigenstates that are degenerate.
Note that the eigenstates m> would be independent of time if the parameters of the Hamiltonian did not change; their only time dependence is through the change of the parameters.
The two eigenstates of the Hamiltonian represent the particle's spin aligned parallel or antiparallel to the magnetic field.
Similarly, in the three-dimensional problem, you can define eigenstates that are continuous and differentiable everywhere in the parameter space except the origin and the negative z-axis (or, alternately, the positive z-axis).
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