10,000,000,000
m⁻¹
0.7242
0.234957
0.606534
0.393466
60.6534
%
2.17
dB
1.380880e-10
m
10,000,000,000
m⁻¹
0.7242
0.234957
0.606534
0.393466
60.6534
%
2.17
dB
1.380880e-10
m
The Quantum Tunneling Calculator computes the probability of a quantum particle penetrating through a potential energy barrier that it classically could not surmount. Quantum tunneling is one of the most striking and counterintuitive predictions of quantum mechanics: a particle with energy E less than the barrier height V₀ has a nonzero probability of appearing on the other side.
Classically, a ball rolling toward a hill taller than its kinetic energy allows will always bounce back. In quantum mechanics, the particle's wavefunction does not abruptly vanish at the barrier edge. Instead, it decays exponentially inside the barrier as $$\psi \propto e^{-\kappa x}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$ If the barrier is thin enough, the wavefunction emerges on the other side with reduced but nonzero amplitude. The transmission coefficient in the WKB approximation is: $$T \approx e^{-2\kappa a}$$ where a is the barrier width.
The exact transmission coefficient for a rectangular barrier is: $$T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)}\sinh^2(\kappa a)}$$ which reduces to the exponential approximation when κa >> 1 (thick or tall barrier).
Quantum tunneling is not merely an academic curiosity — it underlies numerous physical phenomena and technologies. Alpha decay occurs when an alpha particle tunnels out of the nuclear potential well. Scanning tunneling microscopes (STM) exploit the exponential sensitivity of tunneling current to probe surfaces with atomic resolution. Tunnel diodes and Josephson junctions (the basis of superconducting quantum computers) rely on electron tunneling through thin barriers. Nuclear fusion in stars occurs because protons tunnel through the Coulomb barrier at temperatures far below what classical physics would require.
The tunneling probability depends exponentially on barrier width, barrier height above the particle energy, and particle mass. This extreme sensitivity means that even small changes in these parameters can change the tunneling rate by orders of magnitude. Heavier particles (protons vs. electrons) tunnel far less readily, and wider or taller barriers suppress tunneling exponentially.
This calculator provides both the approximate (exponential) and exact (sinh-based) transmission coefficients, along with the decay constant, penetration depth, reflection coefficient, and attenuation in decibels. It is essential for understanding radioactive decay, scanning probe microscopy, and quantum device physics.
The calculator solves the quantum tunneling problem for a rectangular potential barrier:
Decay Constant:
$$\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$
This determines how rapidly the wavefunction decays inside the barrier.
Approximate Transmission (WKB):
$$T \approx e^{-2\kappa a}$$
Valid when κa >> 1 (thick barrier limit).
Exact Transmission:
$$T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)}\sinh^2(\kappa a)}$$
This is the full quantum mechanical result for a rectangular barrier of height V₀ and width a.
Reflection Coefficient:
$$R = 1 - T$$
Penetration Depth:
$$\delta = \frac{1}{\kappa}$$
The distance over which the wavefunction amplitude drops by a factor of e inside the barrier.
The transmission coefficient T gives the probability that the particle tunnels through the barrier (0 to 1). The approximate and exact values agree well for thick barriers (κa > 3) but can differ significantly for thin barriers. The decay constant κ determines how quickly the wavefunction attenuates inside the barrier — larger κ means faster decay and lower tunneling probability. The penetration depth 1/κ is the characteristic length scale of the evanescent wave inside the barrier. The attenuation in dB provides a logarithmic measure useful for comparing tunneling rates across different systems.
Inputs
Results
An electron with 3 eV hitting a 5 eV barrier that is 1 Å wide has about 52.5% transmission probability. The barrier is relatively thin (κa = 0.72), so the exact formula gives much higher transmission than the exponential approximation.
Inputs
Results
Increasing the barrier width to 5 Å reduces transmission to about 0.1%. The exponential sensitivity to barrier width is clear: 5× wider barrier reduces tunneling by ~500×.
Quantum tunneling is the phenomenon where a particle penetrates through a potential energy barrier that it classically lacks the energy to surmount. It arises because quantum particles are described by wavefunctions that decay exponentially inside barriers rather than vanishing abruptly. If the barrier is thin enough, the wavefunction has nonzero amplitude on the far side, giving a finite probability of the particle appearing beyond the barrier.
Inside the barrier, the wavefunction decays as e^{–κx}. After traversing a barrier of width a, the amplitude is reduced by e^{–κa}, and the probability (amplitude squared) by e^{–2κa}. This exponential dependence means doubling the barrier width squares the attenuation factor, making tunneling extremely sensitive to barrier dimensions.
The penetration depth δ = 1/κ is the distance inside the barrier over which the wavefunction amplitude falls to 1/e (about 37%) of its value at the barrier edge. For an electron with V₀ – E = 1 eV, δ ≈ 0.2 nm. For a proton with the same energy difference, δ ≈ 0.005 nm — heavier particles penetrate much less due to their larger κ.
Tunneling is ubiquitous: alpha radioactive decay (alpha particle tunnels out of the nucleus), nuclear fusion in stars (protons tunnel through Coulomb barriers), enzyme catalysis (proton/hydrogen tunneling), scanning tunneling microscopy, tunnel diodes, flash memory (electron tunneling through oxide layers), and Josephson junctions in superconducting circuits used for quantum computing.
The decay constant κ scales as √m, so heavier objects have exponentially smaller tunneling probabilities. For a 1 kg ball facing a 1 m barrier, κa would be approximately 10³⁴ — the tunneling probability T ≈ e^{–2×10³⁴} is indistinguishable from zero. Tunneling is only significant for particles of atomic or subatomic mass encountering barriers of nanometer-scale width.
The approximate formula T ≈ e^{–2κa} (WKB approximation) is simple but ignores wave reflections at the barrier edges. The exact formula includes interference effects through the sinh²(κa) term and a prefactor involving V₀²/[4E(V₀–E)]. For thick barriers (κa > 3), both agree well. For thin barriers (κa < 1), the exact formula can give significantly higher transmission due to resonance-like effects.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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