Fermi’s golden rule and Bardeen’s tunneling theory

A simple way to derive Bardeen’s tunneling theory is introduced. With a low rate of tunneling of, for example, electrons from one electrode to another through a wide vacuum gap, the appearance of the second electrode can be regarded as a perturbation. Then Bardeen’s formula for tunneling follows exactly from Fermi’s golden rule, supposing that this perturbation appears suddenly at some instant of time. Bardeen’s theory is illustrated by assuming a smoothly variable barrier shape, including the rectangular one as a limiting case. This exactly solvable example clarifies the sensitivity of the tunneling probability to the barrier shape and the energy of the tunneling particle.