Quantum tunneling derived from the Schrodinger equation

Quantum tunneling is derived from the Schrodinger equation, Quantum tunneling.

With the discovery of Quantum Mechanics, strange new phenomena came about as a consequence of this new theory. One of these weird phenomena is Quantum tunneling, a new concept which admits the possibility that a particle can penetrate a potential barrier even if -classically- it would not have the right amount of energy to overcome that potential.

This short (but dense) course is about showing how to derive this quantum tunneling effect from the Schrodinger equation. In order to do that, the so-called transmission coefficient must be calculated. This coefficient will be defined and derived in the course, and a comparison between quantum and classical mechanics will be made, highlighting the impossibility that this effect can occur in the classical world.

The prerequisites required to follow the course are:

1) basics of ordinary linear differential equations with constant coefficients (1st and 2nd order). However, the calculations are done step-by-step when needed.

2) The student should be familiar with complex exponentials and hyperbolic functions because they appear in the formulae that are derived in the course.

3) the concepts of total energy, kinetic energy, potential energy are starting points in the course. Therefore, it is assumed that the student has acquired at least a comfortable degree of familiarity with these classical physics concepts.