Probability Flux for the Potential Step *

The probability flux is given by

$$\bgroup\color{black} j(x,t)={\hbar\over 2mi} \left[\psi^*{\pa... ...\partial x}-{\partial\psi^*\over\partial x}\psi\right] .\egroup$$

We can save some effort by noticing that this contains an expression minus its complex conjugate. (This assures that term in brackets is imaginary and the flux is then real.)

$$\bgroup\color{black} j={\hbar\over 2im}\left[u^*{du\over dx}-... ...\right]={\hbar\over 2im}\left[u^*{du\over dx}-CC\right] \egroup$$

For $x<0$

$$\begin{eqnarray*} j&=&{\hbar\over 2im}[(e^{-ikx}+R^*e^{ikx})(ike^{ikx}-ikRe^{-ik... ...2ikx}-R^*R] + CC \\ j&=&[1-\vert R\vert^2]{\hbar k\over m} .\\ \end{eqnarray*}$$

The probability to be reflected is the reflected flux divided by the incident flux. In this case its easy to see that its $\vert R\vert^2$ as we said. For $x>0$

$$\bgroup\color{black} j=\vert T\vert^2{\hbar k'\over m} .\egroup$$

The probability to be transmitted is the transmitted flux divided by the incident flux.

$$\bgroup\color{black} \vert T\vert^2{\hbar k'\over m}{m\over\hbar k}={4k^2\over(k+k')^2}{k'\over k}={4kk'\over(k+k')^2} \egroup$$

again as we had calculated earlier.