• Landau–Zener white tee front view with raven-masked pixel character, avoided-crossing curves and transition probability.

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Landau–Zener

Avoided Crossing Physics

\[P_{\mathrm{D}}=\exp\!\left(-\frac{2\pi |V|^2}{\hbar |\alpha|}\right)\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Landau–Zener stands at an avoided crossing: a raven-masked figure between two energy levels that approach but never meet. The front carries the character and transition probability; the back maps the crossing and its minimum energy gap. Landau–Zener on a pixel-art unisex tee. The physics panel below has the equation, every symbol and the sources.

Related reading: Zeno – Quantum Zeno Effect, Decoherex – Quantum Decoherence, Interfera – Quantum Interference, and Zeno lore. Browse Quantum and Deep Cuts, or read What is quantum measurement?

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

For scale: size M is 20 in (50.80 cm) wide, about 3.143 × 1034 Planck lengths (lP = 1.616255 × 10−35 m). A tape measure is quicker.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

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The physics

In the two-state Landau–Zener model, a constant coupling opens an avoided crossing between levels whose uncoupled energies sweep past each other linearly in time. The probability of staying on the same diabatic state after the crossing is P_D = exp[−2π|V|²/(ℏ|α|)]: slow sweeps follow an adiabatic branch, fast sweeps stay diabatic. The formula assumes an isolated linear sweep with constant coupling and asymptotically distant start and end points.

Two curves lean toward each other and never touch. Landau–Zener stands in the gap.

In the ideal two-state model, the energy difference between the uncoupled, or diabatic, states changes linearly with time: E₁ − E₂ = αt. A constant coupling V opens an avoided crossing between the coupled, or adiabatic, levels. Their minimum separation is 2|V|.

Start in one diabatic state far before the crossing. Far after it, the probability of still being in that same diabatic state is P_D = exp[−2π|V|²/(ℏ|α|)]. Here V is the coupling energy, α is the sweep rate of the diabatic energy difference, and ℏ is the reduced Planck constant.

With fixed coupling, a sufficiently slow sweep makes P_D approach zero: the system follows an adiabatic energy branch and changes its diabatic character. A sufficiently fast sweep makes P_D approach one: the system keeps its diabatic character instead of following that adiabatic branch. The adiabatic-following probability is 1 − P_D.

The same transition was found independently in 1932 by Landau, Zener, Stueckelberg and Majorana. The formula on this page is exact for an isolated linear sweep with constant coupling and asymptotically distant endpoints; it is not a general rule for every driven quantum system.

Equations

\[P_{\mathrm{D}}=\exp\!\left(-\frac{2\pi |V|^2}{\hbar |\alpha|}\right)\]
SymbolMeaningUnit
\(P_{\mathrm{D}}\) probability of remaining in the initial diabatic state after the crossing (jump probability) \(\mathrm{1}\)
\(V\) off-diagonal coupling between the diabatic states (half the minimum adiabatic gap) \(\mathrm{J}\)
\(2\lvert V\rvert\) minimum energy gap of the avoided crossing \(\mathrm{J}\)
\(\alpha\) sweep rate of the diabatic energy difference, d(E₁−E₂)/dt \(\mathrm{J·s⁻¹}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(E₁, E₂\) diabatic (uncoupled) energy levels \(\mathrm{J}\)
\(t\) time \(\mathrm{s}\)

Sources

  1. Zener, C. (1932) Non-adiabatic crossing of energy levels, Proc. R. Soc. Lond. A 137, 696–702
  2. Landau, L. (1932) Zur Theorie der Energieübertragung. II, Phys. Z. Sowjetunion 2, 46–51
  3. Majorana, E. (1932) Atomi orientati in campo magnetico variabile, Nuovo Cimento 9, 43–50
  4. Stueckelberg, E. C. G. (1932) Theorie der unelastischen Stösse zwischen Atomen, Helv. Phys. Acta 5, 369–422
  5. Shevchenko, S. N.; Ashhab, S.; Nori, F. (2010) Landau–Zener–Stückelberg interferometry, Phys. Rep. 492, 1–30

Source Zener, C. (1932). Non-adiabatic crossing of energy levels. Proceedings of the Royal Society of London A 137, 696–702.doi.org/10.1098/rspa.1932.0165

Field notes

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