Imagine an electron trapped in a well that has walls that are 5nm thick and 10eV high and is 10nm wide. calculate the probability of tunneling for an electron with 0.5eV of kinetic energy. Now, determine the number of times the electron would hit a wall each second. using these two pieces of information determine how long it would take for there to be a 50% probability that the electron has already tunneled out. (imagine this as if you had a large number of these wells and you observed that at time, T, 50% of them were empty.
After each collision with a wall, the electron has a probability of e^(-kappa T) of tunneling out, where
T = 5 nm is the thickness of the wall,
kappa = sqrt(2m(V0-E))/hbar
is the tunneling coefficient, m = 9.11 * 10^-31 kg is the mass of the electron, V0 = 10 eV is the height of the walls, and E = 0.5 eV is the kinetic energy of the electron. Plugging in the numbers in SI units, we have
kappa = sqrt(2(9.11 * 10^-31 kg)(9.5 eV)(1.6 * 10^-19 kg m^2 s^-2 eV^-1))/(1.055 * 10^-34 kg m^2 s^-1)
= 1.58 * 10^10 ...
We calculate the tunneling probability of an electron of given kinetic energy in a square well of given dimensions as well as the expected time for the electron to escape from the well.