Landau–Zener Transition Probability Calculator
Introduction
The Landau–Zener problem is one of the classic models of quantum dynamics. It describes what happens when a two-level quantum system is driven through an avoided crossing: two energy levels approach each other, interact through a finite coupling, and then separate again. If the sweep is slow enough, the system has time to adjust and follow the adiabatic eigenstate. If the sweep is too fast, the system tends to remain on its original diabatic branch instead. This calculator estimates those probabilities from the standard Landau–Zener formula.
In practical terms, the tool takes two inputs. The first is the coupling energy, usually written as Δ, which sets the size of the interaction between the two states. The second is the sweep rate, written as α, which tells you how quickly the diabatic energy difference changes with time. From those values, the calculator computes the probability of staying in the initial diabatic state after the passage and the complementary probability of ending on the other diabatic branch. It also labels the result as adiabatic, intermediate, or diabatic so the output is easier to interpret at a glance.
This model appears in many areas of physics. It is used for molecular collisions, superconducting qubits, spin systems, cold-atom ramps, and any setting where a pair of levels is driven through a near crossing. The reason it remains so important is that it captures a surprisingly rich physical process with a compact exact result. Even when a real experiment is more complicated than the idealized two-level picture, the Landau–Zener formula is often the first estimate researchers use to understand whether a sweep will be gentle enough for adiabatic following or abrupt enough to produce nonadiabatic transitions.
Dynamics of an Avoided Level Crossing
Consider two diabatic states whose energy difference varies linearly with time, E1 − E2 = αt. If there were no coupling, the levels would cross exactly. A nonzero coupling Δ opens a gap and turns the crossing into an avoided crossing. Near that region, the system can exchange population between the two states. The competition is simple in concept: larger coupling gives the system more opportunity to mix, while a faster sweep gives it less time to respond.
When the sweep is very slow or the coupling is large, the evolution is adiabatic. In that regime, the system follows the instantaneous eigenstate smoothly through the anticrossing. When the sweep is very fast or the coupling is tiny, the evolution is diabatic. In that case, the state effectively keeps its original diabatic identity. The Landau–Zener formula quantifies this balance and shows that the dependence is exponential, which is why small parameter changes can produce dramatic shifts in the outcome.
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 means the passage is strongly adiabatic, while a value close to 1 means the passage is strongly diabatic. Because Δ appears squared in the exponent, the result is especially sensitive to the coupling energy.
How to Use
Using the calculator is straightforward. Enter the coupling energy Δ in electronvolts and the sweep rate α in electronvolts per second. Both values must be positive. After you submit the form, the page converts the inputs into SI units internally, evaluates the Landau–Zener expression, and reports both probabilities.
It helps to think about the inputs physically before typing numbers. The coupling energy Δ is the off-diagonal interaction that mixes the two states. A larger Δ means a wider avoided crossing and stronger mixing. The sweep rate α is the rate at which the diabatic energy difference changes with time. A larger α means the system passes through the crossing more quickly. If you are working from a Hamiltonian written in joules rather than electronvolts, convert to electronvolts before entering the values. If your sweep is quoted in another unit system, convert it to eV/s first so the result remains consistent.
After calculation, read the output in three parts. First, PD is the probability of staying on the original diabatic branch. Second, PT is the probability of ending on the other diabatic branch. Third, the regime label gives a quick interpretation: adiabatic means PD is very small, diabatic means PD is very close to 1, and intermediate means neither limit dominates. This label is only a convenience, but it is useful when scanning parameter ranges.
Formula
A useful way to organize the calculation is through the adiabaticity parameter δ, defined by
With that definition, the Landau–Zener result becomes PD = e−2πδ. This form makes the physics easier to read. If δ is much larger than 1, the exponent is strongly negative and PD becomes tiny, so the evolution is adiabatic. If 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 formula numerically. It converts Δ from eV to joules and α from eV/s to J/s, then evaluates γ = Δ2/(ℏα). In the script, this 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 expression depends on the absolute value of α, only the magnitude of the sweep rate matters for the probability in this ideal 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 sophisticated, but the final result is compact enough to be used as a practical engineering and physics estimate.
Example
Suppose the coupling energy is Δ = 0.01 eV and the sweep rate is α = 1 eV/s. The adiabaticity parameter is then roughly δ = Δ2/(ℏα), after converting the units consistently. Because ℏ is very small in SI units, even modest laboratory values can produce a substantial exponent. For these sample numbers, the calculator returns a PD value around the middle range rather than extremely close to 0 or 1, so the passage is neither fully adiabatic nor fully diabatic. That means the system has a meaningful chance both to remain on its original diabatic branch and to transfer to the other one.
Now increase the coupling to Δ = 0.1 eV while keeping α = 1 eV/s. Since the coupling enters quadratically, the exponent grows by a factor of 100. The probability of staying on the original diabatic branch then becomes extremely small, and the transition probability approaches 1. In plain language, the system now has enough interaction strength and enough time near the crossing to follow the adiabatic path almost perfectly.
You can also see the opposite trend by keeping Δ fixed and increasing α. A faster sweep reduces the time spent near the avoided crossing, so the system has less opportunity to adjust. As α grows, PD rises toward 1 and the behavior becomes more diabatic. This is why the calculator is useful for quick parameter scans: it reveals immediately whether changing the gap or the ramp speed has pushed the system into a different dynamical regime.
Interpreting the Result
The output should be read as an idealized single-passage probability. It does not tell you the phase accumulated during the sweep, and it does not include interference from repeated crossings. If you sweep through the same avoided crossing multiple times, the probabilities from each passage can combine with phase information to produce Landau–Zener–Stückelberg interference. In that situation, the single-pass result from this calculator is still useful, but it is only one ingredient in the full description.
It is also important to keep track of the basis. PD refers to staying in the same diabatic state, not necessarily staying in the same adiabatic eigenstate. In many discussions, people casually say “transition probability” without specifying the basis, which can be confusing. Here the calculator reports the standard Landau–Zener diabatic survival probability and its complement. If your application is phrased in adiabatic-state language, interpret the result accordingly.
Limitations and Assumptions
The Landau–Zener formula is exact for a very specific model: an isolated two-level system with a constant coupling and a diabatic energy difference that varies linearly with time. Real systems often depart from one or more of these 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 are especially important in experiments. Environmental coupling can wash out coherent dynamics, alter effective transition probabilities, and suppress interference effects. Likewise, if there are multiple nearby avoided crossings, the system may undergo 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 interpretation. This calculator assumes Δ is entered as an energy and α as an energy sweep rate. Some texts define related parameters with factors of 2 placed differently in the Hamiltonian, so published formulas can look slightly different even though they describe the same physics after consistent definitions are applied. When comparing 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 valuable 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 alone 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.
Further Context
The Landau–Zener model has become a cornerstone of quantum control because it links a time-dependent process to a compact closed-form probability. In superconducting circuits, it helps estimate whether a qubit driven through an avoided crossing will remain in its intended adiabatic state. In atomic and molecular physics, it provides a first approximation for transitions during collisions or field ramps. In condensed matter, it appears in tunneling problems, Bloch oscillations, and driven band crossings. The same mathematical structure also appears in broader semiclassical settings, where exponentially small probabilities emerge from nonperturbative dynamics.
For students, this calculator is a useful bridge between abstract theory and numerical intuition. By changing Δ and α and watching the probabilities respond, you can see immediately why “slow compared with the gap” is the essence of adiabaticity. For researchers, it serves as a quick check before running a full simulation. In both cases, the main lesson is the same: avoided crossings are not governed only by whether levels come close, but by how strongly they couple and how quickly the system is forced through the crossing.