Problem and prediction
Where should a perturbative calculation be trusted?
A common failure mode of perturbation theory arises when two unperturbed energy levels approach each other. The off-diagonal mixing can become larger than their separation, even when the perturbation appears small on an absolute energy scale. This laboratory gives exact energies for direct comparisons and displays the expansion parameter itself.
Exact diagnostics
Spectrum and level repulsion
Exact energies and uncoupled levels
Blue/orange solid: exact eigenvalues. Gray dashed: ±|x|, the energy-ordered uncoupled levels. Red points indicate your current detuning. Level repulsion appears for g≠0, with minimum gap 2|g|.
Perturbative accuracy
Where does second-order theory fail?
Absolute difference between exact E+ and E+⁽²⁾=|x|+g²/(2|x|). This nondegenerate expansion is undefined at x=0 and becomes inaccurate when |g/x| is large. The central breakdown region is omitted from the curve; its physical reason and exact spectrum are described directly below the plot.
Eigenvectors matter
What is the character of the upper state?
P(↑z|E+)=½[1+x/√(x²+g²)]. At finite g it changes smoothly from nearly zero to nearly one; for g=0 it jumps at the degeneracy where the eigenvectors are not unique.
Mathematical method
Degenerate and nondegenerate perturbation
In a basis of two unperturbed levels |↑z〉,|↓z〉, consider the real symmetric Hamiltonian H(x,g)=xσz+gσx. For g=0 its energies cross at x=0. For g≠0, the off-diagonal perturbation mixes them, and the 2×2 characteristic equation is E²=x²+g².
For |g/x|≪1, expand the positive root in g at fixed nonzero x:
The correction diverges as x approaches zero because the expansion parameter g/x becomes large. At x=0, diagonalize the perturbation matrix gσx inside the degenerate subspace: the correct splitting is ±|g| already at first order in its magnitude.
Derive the exact eigenstate mixing probability
H points along the effective Bloch vector (g,0,x). The upper eigenstate aligns with the normalized vector (g,0,x)/√(x²+g²). Since σz has eigenvalues ±1, its upper-state probability is [1+〈σz〉]/2=[1+x/√(x²+g²)]/2. At finite coupling its spin character rotates smoothly over a detuning scale |x|~|g|.
What does a real avoided crossing prove?
For a one-parameter family of 2×2 Hermitian matrices with a fixed nonzero off-diagonal coupling, two levels repel. Additional symmetries, more parameters or coupling g=0 can permit exact crossings. The present experiment isolates the simplest algebraic origin of level repulsion without invoking a time-dependent sweep or Landau–Zener transition probabilities.
Three investigations
Learn to diagnose an approximation
- Weak-coupling limit. With g=.2, place x near ±3 and compare exact and second-order values. Now decrease |x| toward zero. Observe the error and the growing ratio |g/x|.
- Basis reconstruction. Fix g=.4 and sweep x through zero. Predict the ↑z population of the upper eigenstate on each side of the crossing.
- Exact degeneracy. Set g=0 and x=0. Discuss the missing unique upper eigenvector and why the response should report this ambiguity explicitly.
Self-check: At x=0 and g=.4, what is the exact energy gap?
Interpretation and source notes
Exact 2×2 diagonalization supplies an independent benchmark for truncated nondegenerate perturbation theory. At g=x=0 the two-dimensional eigenspace is degenerate and an individual eigenstate has no unique Bloch orientation. This is a static spectrum problem; transition dynamics under an externally swept detuning are studied separately.
Read the manuscript: Degenerate Perturbation Theory. Connections: eigenproblems (Chapter 1), coherent two-level dynamics (Chapter 2) and adiabatic gaps (Chapter 8).