Quantum Tunneling Probability Calculator (WKB)
Find the probability a particle tunnels through a barrier using the WKB approximation T ≈ e^(−2κL).
🚧 What is the Quantum Tunneling Probability Calculator?
The quantum tunneling calculator finds the probability that a particle passes through an energy barrier it classically shouldn't be able to cross. Enter the barrier height, the particle's energy, its width, and a particle mass, and it returns the tunneling probability, the decay constant inside the barrier, and the decay length.
Quantum tunneling is one of the most striking predictions of quantum mechanics: a particle with energy E less than a barrier's height V0 still has a nonzero chance of appearing on the far side. Classically this is forbidden, but because a quantum particle is described by a wavefunction that only decays (rather than vanishing instantly) inside the barrier, some of that wave leaks through, appearing on the other side as a transmitted particle.
This calculator uses the standard thick-barrier approximation, T ≈ e^(−2κL), where κ = √(2m(V0−E))/ħ measures how quickly the wavefunction decays inside the barrier. The key relationship is exponential: probability falls off rapidly with barrier width L and with the square root of the particle's mass, which is why light particles like electrons tunnel readily across nanometre-scale barriers while heavier particles need much thinner or lower barriers to do the same.
This calculator is useful for physics students and anyone exploring the physical basis of scanning tunneling microscopy, tunnel diodes, or nuclear alpha decay, since it handles the constants and unit conversions automatically and shows the working.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Electron through a 1 nm barrier
Example 2 - Same electron, half the barrier width
Example 3 - Low, thick barrier (STM-like scenario)
❓ Frequently Asked Questions
🔗 Related Calculators
What is quantum tunneling?
Quantum tunneling is the phenomenon where a particle passes through an energy barrier even though it does not have enough energy to classically surmount it. A classical ball rolling toward a hill higher than its kinetic energy would always bounce back, but a quantum particle has a nonzero probability of appearing on the far side, a direct consequence of its wave-like nature.
What is the formula for tunneling probability?
For a rectangular barrier of height V0 and width L, the approximate transmission probability is T ≈ e^(−2κL), where κ = √(2m(V0 − E)) / ħ, m is the particle's mass, E is its energy, and ħ is the reduced Planck constant. This is the standard thick-barrier approximation used in introductory quantum mechanics.
Why does tunneling probability fall off so fast with barrier width?
Because T depends on e raised to a negative multiple of L, tunneling probability decreases exponentially with barrier width. Doubling the barrier width squares the already-small probability, which is why tunneling is significant only across barriers a few nanometres wide or thinner, such as in atomic-scale devices.
Why do electrons tunnel more easily than protons?
The decay constant κ is proportional to the square root of the particle's mass, so heavier particles decay far more sharply inside the barrier for the same energy deficit. An electron, being roughly 1800 times lighter than a proton, tunnels through the same barrier with a dramatically higher probability.
What is a real-world application of quantum tunneling?
The scanning tunneling microscope (STM) images individual atoms by measuring the tunneling current between a sharp metal tip and a surface, a current that depends exponentially on the tip-surface gap. Tunneling is also responsible for nuclear alpha decay, and it underlies flash memory, tunnel diodes, and some models of the early universe's quantum fluctuations.
What is the decay length in the tunneling formula?
The decay length is 1/κ, the distance over which the wavefunction's amplitude inside the barrier falls by a factor of 1/e (about 37%). A shorter decay length means the wavefunction dies off faster inside the barrier, which corresponds to a lower overall tunneling probability for a given barrier width.
Is the WKB rectangular-barrier formula exact?
No, it is a widely-used approximation valid when the barrier is 'thick' compared to the decay length (κL >> 1). The exact quantum-mechanical transmission coefficient for a rectangular barrier is a more complex formula involving hyperbolic sine functions, but T ≈ e^(−2κL) captures the dominant exponential behaviour and is accurate to within a numerical prefactor in the thick-barrier regime.
Can tunneling probability ever be zero or one?
In this approximation T can get arbitrarily close to zero for a very wide or very high barrier, but it is never exactly zero for a finite barrier, there is always some nonzero chance of tunneling. As the barrier width approaches zero, the approximation breaks down and the particle simply passes over or through with probability approaching one, better described by the exact transmission formula.
What happens if the particle's energy exceeds the barrier height?
If E is greater than V0, the particle is not tunneling at all, classically and quantum mechanically it simply passes over the barrier, though quantum mechanics still predicts some probability of reflection even in this case. This calculator only covers the E < V0 tunneling regime, since that is where the exponential formula T ≈ e^(−2κL) applies.
How does temperature affect quantum tunneling?
The WKB tunneling formula itself does not depend on temperature, it describes a single particle-barrier interaction. However, temperature affects how many particles have enough thermal energy to approach the barrier in the first place, and at very low temperatures, tunneling can become the dominant (or only) mechanism for a process to occur, since thermal activation over the barrier becomes negligible.