0
$\begingroup$

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. :)

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.