**Required math: calculus **

**Required physics: **Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.33.

Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Section 5.4, Exercise 5.4.2 (b).

We’ve analyzed the scattering problem in the finite square well, and we can use similar techniques to analyze a finite square barrier, which has the potential

where is a positive constant energy, and is a constant location on the axis.

There are three distinct cases here:

- Energy below the barrier:
- Energy exactly equal to the barrier:
- Energy greater than the barrier:

In all three cases, the wave function away from the barrier on either side has the same form; it is only within the barrier that the three cases differ. We’ll consider first the case where .

In this case, the Schrödinger equation within the barrier is

where

This has solution

Outside the barrier, the Schrödinger equation is

Outside the barrier on the left, the solution is the sum of travelling waves (assuming particles are incident from the left only), while to the right we have right propagating waves only. Thus

where

Since the potential is finite everywhere, both and are continuous everywhere, which gives us four boundary conditions.

At we have

At :

We can solve these linear equations to get the other four constants in terms of . The results are

From here we can get the transmission coefficient as

The reciprocal of is then, substituting to get the physical quantities back:

The second case is where . In this case, the outer two solutions are the same as before, but in the barrier region we have

which has the solution

Applying the boundary conditions we have, at

At we have

Solving these equations we get

In this case the transmission coefficient is

Finally, for the Schrödinger equation within the barrier is

where

The solution within the barrier is thus

Boundary conditions give at

At :

Solving these four equations gives

The transmission coefficient is

The reciprocal is

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