An electron moving with a kinetic energy E is incident upon a potential barrier

The probability of the electron appearing at the other side of the barrier is worked out from the Schrödinger equation:
 
 

The general solution:

A, B, C, D, F can be determined from the boundary conditions:

y and dy /dx are continuous at x = 0 and x = L

The quantity of interest is a transmission coefficient:

The quantum-mechanical current

Using obtained y₁ and y₂ we can get transmission probability for the current:

For typical value of U₂= 2 eV and L = 10 Å