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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. :)

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2 Answers 2

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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.

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  • $\begingroup$ 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, 2020 at 12:22
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What is missing is the normalization. The difficulty here is that early chapters of QM books often fail to make distinction between eigenvalue problems and scattering problems.

In eigenvalue problems the system is usually bound in space, the states are localized, and the normalization takes familiar form $$ \int_{-\infty}^{+\infty}dx\psi^*(x)\psi(x)=N, $$ where $N$ is the number of particles (often $N=1$).

In scattering problems one is dealing with extended states, where the normalization by the particle number defined above is meaningless or hard to implement. One therefore resorts to the normalization by particle flux, that is setting a certain value on the value of the incoming and/or outgoing particle flux/current. In your case this would amount to setting $$A=1.$$

Note that the scattering problems also have different boundary conditions. E.g., in one dimension one could consider solutions incident from the left $$ \psi(x)\sim e^{ikx}+r_{LL}e^{-ikx},\text{ when } x\rightarrow -\infty,\\ \psi(x)\sim t_{LR}e^{ikx},\text{ when } x\rightarrow +\infty, $$ as well as the solutions incident from the right) $$ \psi(x)\sim t_{RL}e^{-ikx},\text{ when } x\rightarrow -\infty,\\ \psi(x)\sim e^{ikx}+r_{RR}e^{ikx},\text{ when } x\rightarrow +\infty. $$ (One could also define instead the states outgoing to the right/left.)

The coefficients in the above conditions form what is called the scattering matrix $$ S=\begin{bmatrix}t_{LR}& r_{LL}\\r_{RR}&t_{RL}\end{bmatrix}. $$ While in the eigenvalue problems the goal is finding the eigenstates and eigenenergies, in scattering problems one is aiming to find the scattering solutions and the above scatterng matrix for given energy of the incident particles.

Scattering theory is an obligatory subject in quantum mechanics textbooks, and the basis of the quantum field theory. However, in QM its presentation is usually deferred to the later chapters, and it is usually presented for scattering in central potentials, but not in one dimension.

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