Landau–Zener Transition Probability Calculator
Introduction to Landau–Zener transition probabilities
This Landau–Zener calculator is built for the classic avoided-crossing problem in a driven two-level quantum system. When two diabatic levels approach one another and a finite coupling keeps them from crossing exactly, the system can either follow the adiabatic eigenstate or remain on its original diabatic branch. The balance between those outcomes is what the standard Landau–Zener formula captures, and that is the quantity this page evaluates.
Use the tool when you want a quick estimate of how often a passage through the crossing will leave population in the starting state. The calculator takes the coupling energy Δ, which sets how strongly the two states mix, and the sweep rate α, which sets how quickly the level spacing changes with time. From those inputs it produces the diabatic survival probability and the complementary transition probability, then gives a simple regime label so you can see whether the result is closer to adiabatic, intermediate, or diabatic behavior.
The model shows up whenever a linear sweep carries a quantum system through a near crossing. Molecular collisions, driven qubits, spin dynamics, cold-atom ramps, and other two-state settings often reduce to the same basic question: is the passage slow enough for the state to adapt, or fast enough that it effectively freezes in place? The Landau–Zener expression is valuable because it turns that physical intuition into a compact exponential law that is easy to scan across different parameter choices.
How an Avoided Crossing Sets the Landau–Zener Result
The Landau–Zener picture treats two diabatic states whose energy difference changes linearly in time, E1 − E2 = αt, while the coupling Δ stays constant. Without coupling, the levels would cross exactly; with coupling, the crossing becomes avoided and the system has a finite chance to change branch near the anticrossing. The larger the coupling and the slower the sweep, the more time the system has to respond to the changing eigenstates.
In the adiabatic limit, the state follows the instantaneous eigenstate through the crossing region. In the diabatic limit, it largely preserves its original diabatic identity and emerges on the same branch it started on. The Landau–Zener probability formula is what connects those two limits, and its exponential dependence is why apparently small changes in Δ or α can move the outcome from one regime to the other.
The probability of remaining in the initial diabatic state after the passage is
The complementary probability of making the transition to the other diabatic branch is PT = 1 − PD. A small value of PD indicates that the passage was mostly adiabatic, while a value near 1 means the system behaved mostly diabatically. Because Δ appears squared in the exponent, the result is especially sensitive to the coupling energy.
How to Use the Landau–Zener calculator
Using this Landau–Zener calculator takes only a moment. Enter the coupling energy Δ in electronvolts and the sweep rate α in electronvolts per second, making sure both values are positive. Once you submit the form, the page converts the values to SI units internally, evaluates the Landau–Zener expression, and reports the two probabilities.
It helps to choose numbers with the physics of the crossing in mind. Δ is the off-diagonal coupling that mixes the two states, so larger values widen the avoided crossing and favor adiabatic following. α is the rate at which the diabatic energy difference changes, so larger values mean the system moves through the crossing more quickly and has less time to adjust. If your starting data are in joules or joules per second, convert them before entering them here so the calculator receives the units it expects.
After you calculate, read the output in three pieces. PD is the chance of staying on the original diabatic branch. PT is the chance of ending on the other branch. The regime label gives a quick summary of the result: adiabatic means PD is very small, diabatic means PD is very close to 1, and intermediate sits between those extremes. That label is only a shorthand, but it is helpful when comparing several candidate sweeps.
Formula for the Landau–Zener transition probability
A convenient way to express the Landau–Zener calculation is through the adiabaticity parameter δ, defined by
With that definition, the result becomes PD = e−2πδ. This form makes the trend easy to read. When δ is much larger than 1, the exponent is strongly negative and PD becomes tiny, so the evolution is adiabatic. When δ is much smaller than 1, the exponent is close to zero and PD stays near 1, so the evolution is diabatic.
The calculator uses the same expression numerically. It converts Δ from eV to joules and α from eV/s to J/s, then evaluates γ = Δ2/(ℏα). In the script, that quantity is named gamma, but it plays the same role as the adiabaticity parameter in the exponential. The final probabilities are then computed as PD = exp(−2πγ) and PT = 1 − PD. Because the formula depends on the absolute value of α, only the sweep-rate magnitude matters in this idealized model.
The underlying two-state Schrödinger equation is
That equation describes a pair of amplitudes whose level spacing changes linearly in time while the coupling remains constant. Solving it exactly leads to the exponential probability law above. The derivation is mathematically involved, but the final result is compact enough to be used as a practical estimate in quantum control and spectroscopy.
Example: sweeping a two-level avoided crossing
This Landau–Zener example starts with a modest coupling, Δ = 0.01 eV, and a sweep rate of α = 1 eV/s. For that kind of input pair, the result is a useful middle case because neither the weak-coupling nor the fully adiabatic limit dominates. That makes it a good starting point for seeing how sensitively the passage responds when you adjust either parameter.
Now increase the coupling to Δ = 0.1 eV while keeping α = 1 eV/s. Because the coupling enters quadratically, the exponential suppression becomes much stronger and the diabatic survival probability drops sharply. In practical terms, a stronger gap gives the system more opportunity to follow the adiabatic path through the avoided crossing.
You can see the opposite trend by keeping Δ fixed and increasing α. A faster sweep shortens the time spent near the anticrossing, so the system has less opportunity to adjust and PD rises toward 1. That is why this calculator is handy for quick parameter scans: it shows immediately whether widening the gap or accelerating the ramp moves the passage into a different dynamical regime.
Interpreting Landau–Zener probabilities
The Landau–Zener output should be read as the result of a single idealized passage through one avoided crossing.
It does not report the phase accumulated during the sweep, and it does not include interference from repeated crossings. If the system is driven through the same anticrossing more than once, the probabilities from different passages can combine with phase information to produce Landau–Zener–Stückelberg interference. In that situation, this calculator still gives the single-pass building block, but it is not the full multi-pass solution.
It is also important to keep the basis straight. PD refers to staying in the same diabatic state, not necessarily staying in the same adiabatic eigenstate. That distinction matters because many papers and textbooks use the word “transition” without making the basis explicit. This calculator reports the standard diabatic survival probability and its complement, so if your application is written in adiabatic-state language, interpret the numbers accordingly.
Limitations and Assumptions of the Landau–Zener model
This Landau–Zener calculator assumes the textbook two-level model: an isolated pair of states, constant coupling, and a diabatic energy difference that varies linearly with time. Real systems often deviate from one or more of those assumptions. If the sweep is nonlinear, if the coupling changes during the passage, if more than two states participate, or if the system interacts strongly with an environment, the true probability can differ from the simple expression used here.
Decoherence and noise matter in experiments. Environmental coupling can wash out coherent dynamics, alter effective transition probabilities, and suppress interference effects. Likewise, multiple nearby avoided crossings can turn the motion into a sequence of transitions rather than a single isolated event. In those cases, a multistate model or direct numerical integration of the time-dependent Schrödinger equation may be more appropriate than a single Landau–Zener estimate.
Another practical limitation is unit convention. This calculator expects Δ as an energy and α as an energy sweep rate. Some references place factors of 2 differently in the Hamiltonian, so published formulas can look slightly different even when they describe the same physical situation. When you compare this result with a paper or textbook, make sure the author’s notation for the coupling and slope matches the convention used here.
Even with those caveats, the calculator remains useful because it captures the dominant scaling cleanly. It shows how strongly the outcome depends on the square of the coupling and inversely on the sweep rate. That insight is often enough to guide experiment design, estimate whether an adiabatic protocol is realistic, or build intuition before moving on to a more detailed simulation.
Where the Landau–Zener approximation is useful
The Landau–Zener approximation is a workhorse whenever a driven system crosses a narrow gap. In superconducting circuits, it helps estimate whether a qubit driven through an avoided crossing will remain in the intended adiabatic state. In atomic and molecular physics, it provides a first estimate for transitions during collisions or field ramps. In condensed-matter settings, it appears in tunneling problems, Bloch oscillations, and driven band crossings. The same mathematical structure also shows up in broader semiclassical analyses, where exponentially small probabilities emerge from nonperturbative dynamics.
For students, this calculator is a bridge between abstract theory and numerical intuition. By changing Δ and α and watching the probabilities respond, you can see why “slow compared with the gap” is the core idea behind adiabaticity. For researchers, it provides a fast check before running a full simulation. In both cases, the main lesson is the same: avoided crossings are governed not just by how close the levels come, but by how strongly they couple and how quickly the system is driven through the crossing.