# Finite double potential barrier transmission coefficient

TL;DR: I want to calculate the transmission coefficient of a particle travelling into a finite double potential barrier system and I think I've got stuck by the fact that I have 9 unknown variables (amplitudes) and only 8 equations. How do I manage to solve this?

Problem

I have a particle (an electron) with energy $$E$$ travelling in from the left into an area with two potential barriers. The potential is defined by

$$V(x) = V_1\cdot[\Theta(x)- \Theta(x-a_1)] + V_2\cdot[\Theta(x-(a_1 + L) - \Theta(x-(a_1 + a_2 + L]$$ Where $$\Theta(x)$$ is the Heaviside step function, $$a_1$$ is where the first barrier stops (i.e its length), $$L$$ is the width of the separation between the two barriers and $$a_2$$ is the width of the second barrier.

The known quantities are:

• $$V_1, V_2, E$$
• $$a_1, a_2, L$$
• The particle mass $$m$$ is not given by I assume it can be said to be the rest mass of the electron.

The goal is to calculate the transmission coefficient $$T$$.

My work

I solved the equations for the different sections and got the following solutions to the time-independent Schrödinger equation

$$\Psi_1 = Ae^{i\kappa x} + Be^{-i\kappa x}\\ \Psi_2 = Ce^{i\lambda x} + De^{i\lambda x}\\ \Psi_3 = Fe^{i\kappa x} + Ge^{-i\kappa x}\\ \Psi_4 = He^{\mu x} + Ie^{-\mu x}\\ \Psi_5 = Je^{i\kappa x}$$ Where $$\kappa = \frac{\sqrt{2mE}}{\bar{h}}, \lambda = \frac{\sqrt{2m(E-V_1)}}{\bar{h}}, \mu = \frac{\sqrt{2m(V_2-E}}{\bar{h}}$$ and $$\{A,..,J\}$$ are the amplitudes of the different waves. I have excluded the second solution to $$\Psi_5$$ as I assume there is no wave travelling in from the right.

If I apply boundary conditions to $$\Psi_i$$ and $$\Psi_i'$$ at the points $$x = \{0, a_1, a_1 + L, a1 + a2 + L\}$$ I get 8 separate equations, and the goal is to calculate $$T = \frac{|F|^2}{|A|^2}$$. As I have 9 unknown variables and 8 separate equations I do not see how I will be able to solve this. Any help is appreciated and if possible I don't want the answer outright, just some guidance. :)

Well, since scattering states are not normalizable, the wavefunction has an arbitrary overall normalization factor. The reflection coefficient $$R=\frac{|B|^2}{|A|^2}$$ and transmission coefficient $$T=\frac{|J|^2}{|A|^2}$$ depend only on the relative amplitudes. In other words, we can e.g. put the amplitude $$A=1$$ of the incoming right-mover w.l.o.g.