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Calcimator

Quantum Tunneling Calculator

Calculate quantum tunneling probability, transmission coefficient, and penetration depth through a potential barrier.

Inputs

eV
eV
m
kg

Results

Tunneling Probability

35.9%

Transmission Coefficient

0.36

Reflection Coefficient

0.641

Decay Constant (κ)5,123,000,000 m⁻¹
Penetration Depth0 m
Barrier Opacity (κL)0.512
How to Use This Calculator
  1. Enter Particle Energy (eV) — the kinetic energy of the particle approaching the barrier.
  2. Enter Barrier Height (eV) — the barrier must be higher than particle energy for quantum tunneling to apply.
  3. Set Barrier Width (m): atomic-scale widths (1e-10 m = 1 Å) produce significant tunneling; wider barriers exponentially suppress it.
  4. Set Particle Mass (kg): the default 9.109e-31 kg is an electron; protons (1.67e-27 kg) tunnel far less.
  5. Read Tunneling Probability (%) and Transmission Coefficient — even small probabilities matter in STM, nuclear fusion, and semiconductor devices.

How the result changes with Barrier Height

Barrier HeightTunneling ProbabilityTransmission CoefficientReflection Coefficient
104.63%0.050.9537
350.25%00.9975
650.03%00.9997
900.01%00.9999

What each input means

Particle Energy
Kinetic energy of the particle
Barrier Height
Height of the potential barrier
Barrier Width
Width of the potential barrier
Particle Mass
Mass of the particle (default: electron)

How this is calculated

Formula

T ≈ e^(-2κL), κ = √(2m(V-E))/ℏ

Worked example, using the default values

  1. Identify Input Parameters
    4 parameters
    Particle Energy = 1, Barrier Height = 2, Barrier Width = 1e-10, Particle Mass = 9.109e-31 = 4 input(s) provided
  2. Calculate Tunneling Probability
    Tunneling Probability = Number(tunnelingProbability.toExponential(4))
    35.897 = 35.897
  3. Calculate Transmission Coefficient
    Transmission Coefficient = Number(transmissionCoefficient.toExponential(4))
    0.35897 = 0.35897
  4. Calculate Reflection Coefficient
    Reflection Coefficient
    0.641 = 0.641
  5. Calculate Decay Constant
    Decay Constant = Number(decayConstant.toExponential(3))
    5123000000 = 5123000000
  6. Calculate Penetration Depth
    Penetration Depth = Number(penetrationDepth.toExponential(3))
    1.952e-10 = 1.952e-10

Engine last updated . Checked against 4 independently-derived tests — how we verify calculators.

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