Transmission Coefficient Calculator

What Is a Transmission Coefficient Calculator?

The transmission coefficient T gives the probability that a quantum particle passes through a potential barrier instead of reflecting off it. Our Transmission Coefficient Calculator models a rectangular barrier of height V0 and width L, covering both the tunneling case (E below V0) and the oscillatory case (E above V0).

How the Calculation Works

When the particle energy E is below the barrier height, the wavefunction decays inside the barrier and T is:

$$T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0 - E)}\right]^{-1}, \qquad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$

When E is above the barrier, the interior wave propagates and the expression becomes oscillatory:

$$T = \left[1 + \frac{V_0^2 \sin^2(k L)}{4E(E - V_0)}\right]^{-1}, \qquad k = \frac{\sqrt{2m(E - V_0)}}{\hbar}$$

Where:

• m is the particle mass
• V₀ is the barrier height and E is the particle energy
• L is the barrier width
• κ is the decay constant below the barrier and k the wavenumber above it
• ħ is the reduced Planck constant

Worked Example

An electron with E = 0.5 eV meets a 1 eV barrier that is 0.5 nm wide. The decay constant is κ = 3.62 × 10⁹ m⁻¹, so κL = 1.81 and sinh²(1.81) = 8.83. Substituting gives T = (1 + 8.83)⁻¹ ≈ 0.102, meaning roughly a 10% chance of tunneling through.

Related tools include the Wavenumber Calculator and the Work and Power Calculator.