Quantum Tunneling Probability Calculator Formula

Understand the math behind the quantum tunneling probability calculator. Each variable explained with a worked example.

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Formulas Used

Transmission Coefficient

transmission = exp(-2 * barrier_width * sqrt(2 * particle_mass * (barrier_height - particle_energy) * 1.602e-19) / 1.0546e-34)

Variables

VariableDescriptionDefault
barrier_heightBarrier Height (V)(eV)5
particle_energyParticle Energy (E)(eV)3
barrier_widthBarrier Width (d)(m)1e-10
particle_massParticle Mass(kg)9.109e-31

How It Works

Quantum Tunneling

In quantum mechanics, a particle can pass through a potential barrier even if its energy is below the barrier height.

Approximate Transmission

T = exp(-2d sqrt(2m(V-E)) / hbar)

  • *V* = barrier height (eV, converted to J)
  • *E* = particle kinetic energy
  • *d* = barrier width
  • *m* = particle mass
  • *hbar* = reduced Planck constant

    • The probability drops exponentially with barrier width and the square root of the energy deficit.

    Worked Example

    Electron (m = 9.109e-31 kg) with E = 3 eV hitting a 5 eV barrier, 1 angstrom wide.

    barrier_height = 5particle_energy = 3barrier_width = 1e-10particle_mass = 9.109e-31
    1. 01V - E = 2 eV = 2 * 1.602e-19 = 3.204e-19 J
    2. 02sqrt(2m(V-E)) = sqrt(2 * 9.109e-31 * 3.204e-19) = sqrt(5.837e-49) = 7.640e-25
    3. 03Exponent = -2 * 1e-10 * 7.640e-25 / 1.0546e-34 = -1.449
    4. 04T = exp(-1.449) = 0.235

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