# Calculating the Reflection Coefficient of a Potential Step Explicitly

So I'm using the following definition for the Reflection coefficient, $R$ :

$$R=\frac{\left\rvert\ \vec{j}_{reflected}\right\rvert}{\left\rvert\ \vec{j}_{incident}\right\rvert}$$

where $j_{reflected}$ is the reflected probability current and $j_{incident}$ is the incident probability current of the given wavefunction.

Hence, since :

$$\psi_{reflected}=Be^{-ikx}$$ and $$\psi^{*}_{reflected}=B^{*}e^{ikx}$$

We can perform the usual to obtain the incident probability density current as : $$\vec{j}_{reflected} =2ik\lvert{B}\rvert^{2}$$

Now for the incident we use :

$$\psi_{incident}=e^{ikx}$$ and $$\psi^{*}_{incident}=e^{-ikx}$$

Again we obtain : $$\vec{j}_{incident} =-2ik$$

This would imply a coefficient of $-\lvert{B}\rvert^{2}$ which is incorrect.

What have I done incorrectly ?

• This whole wiki page is good, in particular the end of the section linked. en.wikipedia.org/wiki/… Commented Apr 3, 2014 at 21:49
• Of course... I simply forgot the modulus signs.... Commented Apr 4, 2014 at 4:14

## 1 Answer

My initial equation was correct by I had neglected to include the modulus signs. Thanks to @jazzwhiz for pointing it out subtly.