# Transmission reflection at step potential mass independent?

I was reading the Wikipedia page about transmission and reflection, and to my surprise, the transmission and reflection coefficients do not depend on the mass of the particle.

So in consequence the transmission and reflection probability would be the same, regardless of the size of the particle? What if the incoming wave was an electron beam or a proton beam? Maybe the probabilities are independent of the mass of the particles in question, but what about charge? Certainly it ought to come into play?

• You may wish to develop the question and ponder what is happening if you increase the mass $m$ from the subatomic realm, up through to atoms, molecules, ....., Bugs Bunny. I mention Bugs Bunny because of a question posed in Griffiths "Introduction to Quantum Mechanics" (Prentice Hall, 1995) Problem 2.41. – jim Sep 19 '16 at 21:34
• Right but the transmission and reflection probabilities are not, which is my question – larrydavid Sep 19 '16 at 21:51

## 1 Answer

The formula for transmission T and reflection R of a particle at a potential step derived from Schroedinger's equation depends only on the ratio of the wave vectors $k_1$ and $k_2$ in the regions of different potential, which are related to the respective kinetic energies according to $E-V= \frac {p^2}{2m}$, with the momentum $p=\frac {hk}{2\pi}$. Therefore the mass $m$ of the particle does indeed not appear in T and R. Transmission and reflection depend only on the momentum of the particle. So if you change you experiments from electrons to protons, to obtain the same reflection for a proton you would need the same momentum, so that you need a speed 1836 times slower than the electron speed. In this simple picture, the particle charge is only relevant for the potential energy $V$ in Schroedinger's equation.

• Thank you CounTo10 for the editing. I am comparatively new here and still have to become acquainted with the tools for writing formulas. – freecharly Sep 19 '16 at 23:26
• So do I , every day I learn a new bit of mathjax. I am sure you already have this link below . Sometimes its easier, when you see math on a post you know you are going to use later, to edit the post and copy the code into a text file on your pc. meta.math.stackexchange.com/questions/5020/… – user108787 Sep 20 '16 at 0:57