Degenerate perturbation theory is the quantum method for handling states that share the same energy when a small perturbation is added. In Physical Chemistry II, you use it to split degenerate levels and find the corrected eigenstates and energies.
Degenerate perturbation theory is the version of perturbation theory you use when two or more quantum states have the same unperturbed energy. In Physical Chemistry II, that usually means the normal first-order correction formula is not enough, because the degenerate states mix with each other as soon as you add the perturbing Hamiltonian.
The basic move is to stop treating each state separately and work inside the degenerate subspace. You build a matrix from the perturbation operator using the degenerate basis states, with elements like <i|H'|j>. That matrix tells you how the states couple to one another under the small change in the Hamiltonian.
Then you diagonalize that matrix. The eigenvalues of the perturbation matrix give the first-order energy shifts, and the eigenvectors give the new linear combinations of the original degenerate states. So instead of one shared energy level, the degeneracy is usually split into distinct perturbed energy levels.
This is why degenerate perturbation theory feels a little different from the non-degenerate case. In the non-degenerate case, a state mostly shifts by itself. In the degenerate case, the perturbation can force a mixing of basis functions, so the right starting basis is often not the obvious one. The physically meaningful states are the combinations that the perturbation itself selects.
A classic chemistry example is an atom or molecule placed in an external electric or magnetic field. A previously degenerate set of levels can split through the Stark or Zeeman effect, and the symmetry of the perturbation often tells you which matrix elements are zero. That means the math is not just calculation for its own sake, it is a way to see how symmetry and small external influences reshape the energy landscape.
Degenerate perturbation theory shows up whenever symmetry creates matching energy levels and then something breaks that symmetry. In Physical Chemistry II, that can mean an external field, a weak interaction, or a molecular distortion that changes how quantum states relate to each other.
This matters because many real systems are not perfectly isolated. If you want to explain spectral line splitting, predict which states separate under a field, or justify why two basis functions mix, this is the tool that turns the abstract Hamiltonian into a usable answer. It connects the math of eigenvalues and eigenvectors to observable changes in spectroscopy and molecular energy levels.
It also sharpens your reading of symmetry. If a perturbation has certain symmetry properties, some matrix elements vanish, and that can simplify the whole problem. So this topic is not just about getting numbers, it is about learning how to spot the structure of the problem before you start calculating.
Keep studying Physical Chemistry II Unit 4
Visual cheatsheet
view galleryPerturbation Theory
Degenerate perturbation theory is a special case of perturbation theory. The general method still starts with a solvable Hamiltonian plus a small perturbing Hamiltonian, but degeneracy changes the workflow because states with the same unperturbed energy can mix. If you know the general idea first, the degenerate case makes more sense as the point where the usual shortcut breaks down.
Hamiltonian
The Hamiltonian is the operator you split into the unperturbed part and the perturbation part. In this topic, the perturbing Hamiltonian is turned into a matrix inside the degenerate subspace, and that matrix is what you diagonalize. So the structure of the Hamiltonian determines both the splitting pattern and the new eigenstates.
Eigenstate
The original degenerate eigenstates are usually not the final physical states after the perturbation is applied. Degenerate perturbation theory finds new eigenstates that are linear combinations of the original ones. That shift matters because measurements and selection rules are tied to the perturbed eigenstates, not just the old basis you started with.
perturbed energy levels
This topic is one of the main ways you calculate perturbed energy levels when degeneracy is present. Instead of one shared energy, the levels split according to the eigenvalues of the perturbation matrix. In spectroscopy problems, that splitting is often what you compare to observed line patterns or field-induced shifts.
A problem set or quiz question will usually give you a degenerate set of states and a small perturbing Hamiltonian, then ask you to find the first-order energy shifts. The move is to write the perturbation matrix in the degenerate basis, diagonalize it, and read off the corrected energies and eigenstates. If the matrix has off-diagonal terms, that is your signal that the original states mix.
You may also be asked to explain why a degeneracy splits in an electric or magnetic field, or to use symmetry arguments to show that certain matrix elements are zero. In spectroscopy questions, the result often appears as a splitting of lines or a change in allowed transitions.
These two methods look similar at first, but they handle different starting points. Non-degenerate perturbation theory assumes each unperturbed energy level belongs to one state, so you can correct that state directly. Degenerate perturbation theory is needed when two or more states share the same energy, because the perturbation can mix them and you must diagonalize within that subspace first.
Degenerate perturbation theory is the quantum method for handling states that have the same unperturbed energy.
The main step is to build and diagonalize the perturbation matrix inside the degenerate subspace.
The eigenvalues of that matrix give the first-order energy shifts, and the eigenvectors give the new physical states.
Degeneracy often splits when symmetry is broken by an external field or another weak interaction.
If the perturbation couples the states, you cannot treat each degenerate state as if it were separate.
It is the quantum method used when two or more states share the same energy and a small perturbation is added. You form a matrix from the perturbing Hamiltonian inside that degenerate set, then diagonalize it to find the shifted energies and the new states.
Non-degenerate perturbation theory assumes each energy level is isolated, so the correction is found state by state. Degenerate perturbation theory is needed when levels are equal in energy, because the perturbation can mix them and the correct basis is found by diagonalizing the perturbation matrix.
A field changes the Hamiltonian, and that new term often has different effects on states that used to be equal in energy. The perturbation picks out certain linear combinations of the original states, so the shared level separates into distinct perturbed energy levels, as in Stark or Zeeman splitting.
Not necessarily. The method is required when the unperturbed states are exactly degenerate, or when the perturbation is large enough that the mixing matters. If the states are only close but not exactly equal, you may still need to check whether the near-degeneracy makes the non-degenerate formula unreliable.