WKB Approximation Simulator — Quantum Tunneling Probability
Experience how a particle quantum-mechanically tunnels through a potential barrier it could never cross classically. Adjust the particle energy, barrier height and width, and mass to see the WKB transmission probability T and the decay constant κ inside the barrier update in real time.
Parameters
Particle energy E
eV
Kinetic energy of the particle incident on the barrier
Barrier height V0
eV
Potential energy of the rectangular barrier
Barrier width L
nm
Thickness of the barrier the particle tunnels through
Particle mass m
mₑ
In units of the electron mass. Heavier particles tunnel less readily
Results
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Tunneling probability T
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Decay constant κ (nm⁻¹)
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Penetration depth 1/κ (nm)
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E/V0 ratio
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Barrier gap V0−E (eV)
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Classical verdict
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Potential barrier and wavefunction ψ(x)
The blue line is the energy E and the grey rectangle is the barrier V0. The wavefunction ψ(x) oscillates before the barrier, decays exponentially inside it, and oscillates again with a smaller amplitude after it (the transmitted wave).
The general WKB transmission probability. κ is the decay rate of the wavefunction inside the barrier, and x₁, x₂ are the classical turning points (where E = V). The larger the integral in the exponent, the smaller the transmission.
$$T = e^{-2\kappa L}$$
For a rectangular barrier of height V0 and width L the constant κ makes the integral simply κL, so the transmission is this clean exponential. The penetration depth is 1/κ, the distance over which the wavefunction decays by a factor of 1/e.
The WKB approximation is valid when the potential varies slowly compared with the wavelength. A rectangular barrier is strictly an abrupt potential, but it is widely used as a teaching example to capture the essence of the decay.
What is the WKB Approximation?
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"Quantum tunneling" means a particle passes straight through a wall, right? Does something that science-fiction really happen?
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It does, and it is a real, everyday phenomenon. Roughly speaking: throw a ball at a wall and in classical mechanics it will never reach the far side — if it lacks the energy, it bounces back 100% of the time. But a microscopic particle like an electron has wave nature, and inside the wall its wavefunction does not just vanish; it decays exponentially while penetrating. If the wall is thin, a little amplitude leaks out the other side before it fully decays. That is tunneling.
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I see. But can you actually calculate "how much leaks out"? The Schrödinger equation looks intimidating.
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That is exactly where the WKB approximation comes in — named after Wentzel, Kramers and Brillouin. Instead of solving the Schrödinger equation exactly, the idea is that where the potential varies slowly compared with the wavelength, the wavefunction can be approximated as an exponential, exp(±∫k dx). With that, the tunneling probability falls out of a single integral, T ≈ exp(−2∫κ dx). It is also called the semiclassical approximation.
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When I increase the barrier width L on the left, the tunneling probability T drops incredibly fast. It falls in a straight line even though the y-axis is logarithmic.
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Good catch. For a rectangular barrier T = exp(−2κL). The width L sits in the exponent, so increasing L makes T fall exponentially — which is exactly why it is a straight line on a log axis. At the default values T ≈ 2×10⁻⁸: roughly "2 escapes per 100 million attempts". It is tiny, but it is not zero — that is the wonder of quantum mechanics. Halve the barrier width and T jumps by about a factor of ten thousand.
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Making the mass m heavier also reduces tunneling. Why is that?
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Because the decay constant κ is proportional to √(m(V0−E)). A heavier particle gives a larger κ and decays faster inside the barrier. So electrons tunnel fairly readily, but a proton — about 1800 times heavier — does so far less. That is also why alpha decay, where a whole nuclear fragment tunnels, can have half-lives on geological timescales. Bring E closer to V0 and (V0−E) shrinks, κ drops, and T rises sharply. Try it on the energy chart below.
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When I push E above V0, the verdict changes to "classically transmitted". Isn't that tunneling anymore?
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Right — once E ≥ V0 the particle simply rides over the top of the barrier. It would pass even in classical mechanics, so it is no longer tunneling. In that case (V0−E) goes negative, the square root no longer holds, so we set κ = 0 and the tunneling formula T = exp(−2κL) no longer applies. The tool flags this case explicitly in the verdict. Tunneling only has meaning in the E < V0 regime — the situation a particle could never escape classically.
Frequently Asked Questions
The WKB approximation (Wentzel–Kramers–Brillouin) is a semiclassical method for solving the Schrödinger equation. Where the potential varies slowly compared with the de Broglie wavelength, the wavefunction can be approximated as an exponential of the form exp(±∫k dx). This makes problems that are hard to solve exactly — such as tunneling — tractable: the transmission probability follows from a single integral, T ≈ exp(−2∫κ dx). This tool applies that WKB formula to a rectangular barrier.
In classical mechanics a particle with energy E below a barrier height V0 can never cross the barrier and is reflected 100% of the time. In quantum mechanics the particle behaves as a wave, and inside the barrier the wavefunction does not vanish — it decays exponentially while penetrating. If the barrier has finite width, some amplitude survives before fully decaying and leaks out the other side. That leakage is tunneling, and the transmission probability is set by how much the wavefunction decays inside the barrier.
For a rectangular barrier the WKB transmission probability is T = exp(−2κL), where κ is the decay constant inside the barrier and L is the barrier width. Since κ = √(2m(V0−E))/ℏ, T decreases exponentially with the barrier width L and also with the square root of the particle mass m and the gap (V0−E). Halving the barrier width, or lowering it, raises T dramatically; heavier particles or thicker, taller barriers make T fall off sharply.
The WKB approximation works when the potential varies slowly compared with the particle wavelength. It becomes less accurate where the potential changes abruptly — for instance at the sharp corners of a rectangular barrier, or near classical turning points. The rectangular barrier in this tool is strictly an abrupt potential, but the WKB formula T = exp(−2κL) is used here because it captures the essence of exponential decay so clearly. For smooth barriers such as alpha decay or scanning tunneling microscopy, the WKB approximation is highly accurate.
Real-World Applications
Scanning tunneling microscope (STM): An STM detects the current of electrons tunneling across the tiny vacuum gap between a sharp tip and a sample surface, imaging individual atoms. The tunneling current follows T = exp(−2κL), so it depends exponentially on the gap width L — a change of just 0.1 nm shifts the current by roughly a factor of ten. That extreme sensitivity is the source of the STM's atomic resolution. Move the width L slightly in this tool and you can see T change by orders of magnitude.
Alpha decay and nuclear physics: Alpha particles emitted from a radioactive nucleus escape the confining Coulomb barrier by quantum tunneling. In 1928 Gamow used precisely the WKB approximation to explain alpha-decay half-lives. Because the tunneling probability is exponential in the barrier, half-lives span more than 20 orders of magnitude, from 10⁻⁷ seconds to 10¹⁷ years. This is the physical basis of the Geiger–Nuttall law.
Semiconductor devices: Tunnel diodes, the program and erase steps of flash memory, and leakage through ultra-thin gate oxides — modern electronics is inseparable from tunneling. As devices shrink, gate dielectrics thin to a few nanometres and the unwanted tunneling leakage current can no longer be ignored. The WKB approximation is still used as a first estimate of this leakage current.
Nuclear fusion and stellar energy: Even at the centre of the Sun the Coulomb barrier between protons is too high for fusion to occur classically. Protons tunnel through the barrier, fusion proceeds, and stars shine. Because the tunneling probability is acutely sensitive to temperature, the fusion rate inside a star changes greatly with a small difference in core temperature. The WKB approximation is used to estimate this reaction rate (the Gamow factor).
Common Misconceptions and Pitfalls
The biggest misconception is the idea that a tunneling particle "borrows energy" inside the barrier. In tunneling the particle's energy is conserved — inside the barrier it still has the same E it had on entry. The picture of "borrowing energy briefly via the uncertainty principle to get over the wall" is intuitive but not strictly correct. What actually happens is that inside the barrier the wavefunction becomes an exponentially decaying evanescent wave rather than an oscillation, and that penetration simply reaches the far side; no energy is borrowed. The wavefunction diagram in this tool shows the wave purely decaying, not oscillating, inside the barrier.
Next, the assumption that the WKB approximation is exact for a rectangular barrier. WKB assumes the potential varies slowly compared with the wavelength, but a rectangular barrier jumps discontinuously at its corners and does not really satisfy that condition. The clean form T = exp(−2κL) makes it popular for teaching, but the exact transmission of a rectangular barrier carries a prefactor, and especially when E is close to V0 or the barrier is thin, the WKB exponent alone can be off by a factor of several. WKB shows its true accuracy on smooth barriers such as alpha decay.
Finally, the complacency that "the tunneling probability is small, so we can ignore it". Even at the tool's defaults T ≈ 2×10⁻⁸ looks negligible, but tunneling matters wherever the number of trials per particle is enormous. In a gate oxide, for example, there are a huge number of collisions per second, so even 10⁻⁸ adds up to a non-negligible leakage current across a whole chip. Alpha decay is observable for the same reason — "the probability per atom is minuscule, but the number of atoms is vast". Judge by the expected value — probability multiplied by the number of trials — not by the bare probability.
How to Use
Enter particle energy (eNum, eRange in eV) — typically 0.1–10 eV for electron tunneling through semiconductor junctions
Set potential barrier height (vNum, vRange in eV) — use 5–15 eV for common oxide layers or 0.5–2 eV for metal-semiconductor interfaces
Define barrier width (wNum, wRange in nm) — semiconductor gate oxides range 1–3 nm, tunnel junctions 0.5–1.5 nm
Specify particle mass (mNum, mRange in electron masses m₀) — use m₀ for free electrons, 0.26m₀ for GaAs conduction band, 0.42m₀ for silicon holes
Execute simulation to obtain tunneling probability T, decay constant κ, and penetration depth
Worked Example
Consider an electron (m = m₀ = 9.109×10⁻³¹ kg) approaching a 2 nm SiO₂ barrier with V₀ = 8 eV and E = 6 eV. The WKB transmission coefficient yields T ≈ 0.031 (3.1%), decay constant κ ≈ 1.24 nm⁻¹, and penetration depth 1/κ ≈ 0.81 nm. Lowering E to 4 eV increases the barrier gap to 4 eV, reducing T to ~4×10⁻⁴ (0.04%), demonstrating exponential sensitivity to energy offset. Classical mechanics predicts zero transmission; quantum tunneling enables non-zero probability.
Practical Notes
Tunneling probability drops exponentially with barrier width — doubling wNum from 1 nm to 2 nm typically reduces T by factor of 100–1000 depending on V₀−E
Effective mass matters critically; light carriers (small m) tunnel more readily — electrons in GaAs tunnel ~3× more efficiently than holes in silicon for equivalent barriers
Use E/V₀ ratio output to identify classically forbidden regime (E/V₀ < 1); regions with E/V₀ > 0.9 show suppressed tunneling due to minimal barrier penetration