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?


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.

  • $\begingroup$ I'm reading Griffiths' Introduction to Quantum Mechanics and I haven't seen him do that there (as of chapter 2 at least where the potential wells/barriers are introduced). Is choosing an arbitrary amplitude the best method here? I haven't seen this done in the papers on the topic I've looked at. Can it be done in any other way? $\endgroup$ – fintallrik Oct 5 '20 at 12:22

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