Quantum mechanics

Quantum perturbation theory

See how a perturbation λV splits degenerate energy levels and creates an «avoided crossing». Verify the first-order shift $E_n\approx E_n^{(0)}+\lambda\langle n|V|n\rangle$ and the minimum splitting $2|g|$ at the crossing point.

Advanced

Equipment

Energy-spectrum plot versus level offset (panel 1)
Coupling g, level-offset δ, and strength λ sliders
Panel 2 comparing perturbation orders

Procedure

Observe the avoided crossing
Set g = 0.5, λ = 0.7 and sweep «Level offset δ» from −3 to +3. The two energy branches approach near δ = 0 but never cross — an «avoided crossing». Read the minimum separation; it approximates $2|g|$ , the gap opened by the coupling.
Vary the coupling g
Lower g toward 0: the two levels cross cleanly at δ = 0 (true degeneracy). Raise g to 1.5: the gap widens. Record the splitting at δ = 0 versus g and check the linear relation $\Delta E_{min}=2|g|$ .
Verify the perturbative orders
In panel 2, compare the first-order and higher-order curves with the «exact» one. For small λ, first order matches well far from the degeneracy: $E\approx E^{(0)}+\lambda\langle V\rangle$ ; near δ = 0 and large λ it misses — second order or exact diagonalization is needed.
Limits of validity
Find the λ where first-order error exceeds 10% relative to the exact curve at δ = 0, then explain: when $\lambda|V_{12}|\sim|E_1^{(0)}-E_2^{(0)}|$ the unperturbed/perturbed split breaks down — degenerate cases must be diagonalized inside the degenerate subspace first.

Simulation

Experiment history

Stationary perturbation theory descends from celestial mechanics — Euler, Lagrange, and Laplace computed planetary orbit perturbations from the eighteenth century. In quantum mechanics, Schrödinger systematized the method in 1926 (the Rayleigh–Schrödinger series) to solve the hydrogen Stark effect: levels shift linearly with the electric field — the young theory's first quantitative test. For degenerate problems, Dirac developed time-dependent perturbation theory (1926–27) — ancestor of the Dyson series in QED, whose terms became Feynman diagrams. The deep lesson: perturbation series are usually asymptotic rather than convergent — the point Dyson famously argued in 1952 for the QED expansion.

Related physicists

Paul Dirac Werner Heisenberg Erwin Schrödinger Richard Feynman

Equipment

Procedure

Simulation

Experiment history

Related physicists

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