Diagonalize inside a degenerate subspace first
Construct the correct zeroth-order combinations and connect near degeneracy with avoided crossings.
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The question for today
What you’ll understand
- Project a perturbation into a degenerate eigenspace.
- Compute first-order split energies and the correct basis.
- Identify when coupling opens an avoided crossing.
At exact degeneracy, an unperturbed energy labels a whole subspace rather than one preferred state. An arbitrarily weak perturbation can choose entirely different combinations within that subspace. Dividing by the zero internal energy differences is therefore the wrong operation. We will instead diagonalize a small projected matrix, then compare the result with the exact spectrum of a nearly degenerate pair to understand level repulsion and its symmetry exceptions.
The unperturbed basis has genuine freedom
Suppose several orthonormal states have the same isolated unperturbed energy E₀. Any normalized superposition in their span is also an eigenstate of H₀. There is no physically distinguished choice among those combinations until additional information, such as a symmetry label or the perturbation, selects one. Attempting the nondegenerate state-correction formula in an arbitrary basis gives zero denominators, because it tries to treat an unrestricted internal rotation as a small correction to a fixed state.
Let P project onto this degenerate subspace. Projecting the first-order eigenvalue equation gives an ordinary eigenproblem for W=PVP restricted to the subspace. Its eigenvectors determine the combinations that can be continued as perturbed eigenstates, and its eigenvalues are their first-order energy coefficients. This diagonalization must retain the perturbation matrix elements within the whole relevant degenerate block. Discarding an off-diagonal element just because its original basis vectors looked different can erase the actual splitting.
Separate internal mixing from external corrections
After diagonalizing the internal block, coupling to states outside it produces higher-order corrections with nonzero external denominators. The perturbation must still be small compared with those separations if that expansion is to be useful. If W remains degenerate, first-order theory has not selected every state; a higher-order effective operator or further symmetry analysis is required inside the unresolved block. Diagonalizing one matrix does not automatically guarantee that the entire original degeneracy has disappeared.
Near degeneracy, it is often better to diagonalize the close pair together rather than expand each level separately. A general Hermitian two-level block has a mean energy, a diagonal detuning δ, and an off-diagonal coupling g. Its exact eigenvalues are separated by the square root of δ²+4|g|². At zero detuning the minimum separation is 2|g|. This finite gap replaces the artificial divergence of a nondegenerate series whose denominator was allowed to approach zero.
Repulsion depends on an allowed coupling
Levels do not universally repel merely because two plotted curves come close. If an exact symmetry forces their coupling to vanish, states in different symmetry sectors may cross without mixing. An avoided crossing occurs when the relevant states can couple and the effective off-diagonal element is nonzero. The eigenvectors exchange their original basis character while the energy branches stay separated. Following a static eigenvector through this exchange is different from solving a finite-speed time-dependent sweep.
Inspect two possible combinations in the quantum lab. Reset and set both controls to mixing angle 90 degrees and relative phase zero; predict equal basis probabilities and Bloch x equal to plus one. Reset again and set mixing angle 90 degrees and phase 180 degrees; predict the same probabilities and Bloch x equal to minus one. These are symmetric and antisymmetric combinations. The lab has no perturbation matrix or energy-gap readout, and its phase precession does not implement a detuning sweep.
Work it through
Split an exactly degenerate doublet
Two unperturbed states both have energy 5 in one energy unit. Their projected perturbation is W=[[2,1],[1,2]] in that unit, multiplied by λ=0.1. Find its first-order energies and the selected normalized combinations.
Acting with W on the symmetric vector (1,1) gives three times that vector, while acting on (1,−1) gives one times that vector. The first-order coefficients are therefore 3 and 1.
Normalize each vector by dividing by √2. The symmetric and antisymmetric states are orthogonal, so they provide the correct basis inside the originally degenerate space.
Multiply the coefficients by λ and add E₀. The symmetric state has energy 5.3 and the antisymmetric state 5.1 to first order. Couplings to external levels would contribute at higher orders.
The doublet splits into 5.1 for (|a⟩−|b⟩)/√2 and 5.3 for (|a⟩+|b⟩)/√2, to first order.
The living notebook
Explore the model
Interactive lab · quantum
Equal populations can label different eigencombinations
Reset and set BOTH controls to θ=90° and φ=0°, predicting Bloch x=1. Reset and set θ=90° and φ=180°, predicting Bloch x=−1 with the same basis probabilities. No energy spectrum is calculated.
- Probability |0⟩
- 71.1%
- Probability |1⟩
- 28.9%
- Bloch (x, y, z)
- 0.78, 0.45, 0.42
A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forRelative phase distinguishes symmetric and antisymmetric states even when their basis populations match.
Pause & reason
Check your understanding
Your turn
Try it yourself
A two-level block has mean energy 3, detuning δ=0.6, and real coupling g=0.4 in common energy units. Find both energies and their separation. What would its minimum separation be if detuning could be varied while g stayed fixed?
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
Evaluate the sum of the squared half-detuning and squared coupling before taking the square root.
Compare with the solution
The square root is √(0.3²+0.4²)=0.5. The energies are 2.5 and 3.5, separated by 1.0. The smallest separation occurs at δ=0 and equals 2g=0.8. A true crossing would require the effective coupling to vanish as well.
Keep the idea
Take it with you
- Degenerate perturbation theory first resolves a projected finite-dimensional eigenproblem.
- External energy gaps still control the later corrections.
- An avoided crossing requires an allowed nonzero coupling.
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