What Is the Reflection Coefficient for Electrons at a Potential Step?

[nickmai123]

Homework Statement

Find the reflection coefficient for electrons traveling toward a potential change from $$V$$ to $$V_0$$ with a total energy $$E > V_0$$.
The potential diagram is just a unit step function. It goes from $$V = 0$$ to $$V = V_0$$ at $$x=0$$. In piecewise notation:
$$\begin{displaymath}<br /> V(x) = \left\{<br /> \begin{array}{lr}<br /> 0 & : x < 0 \\<br /> V_0 & : x \ge 0<br /> \end{array}<br /> \right.<br /> \end{displaymath}$$
The piecewise notation does not account for the $$V(x)$$ being continuous at $$x=0$$.

Homework Equations

a) Probability flux:
$$S\left( x,t \right)=-\frac{i\hbar}{2m}\left[ \Psi^*\left( x,t \right) \frac{\partial \Psi\left( x,t \right)}{\partial x} - \Psi\left( x,t \right) \frac{\partial \Psi^*\left( x,t \right)}{\partial x}\left]$$

b) Reflection coefficient:
$$R=\frac{S_{I}^{-x}\left( x,t \right)}{S_{I}^{+x}\left( x,t \right)}$$

The Attempt at a Solution

I've solved for the wave equations at $$x > 0$$ and $$x < 0$$. I'm stuck as far as where to go from there.

 

[nickjer]

Are you asking for help on (a)? Can you also show us your final wave function? And have you tried plugging that wavefunction into (a)?

 

[vela]

Homework Statement

Find the reflection coefficient for electrons traveling toward a potential change from $$V$$ to $$V_0$$ with a total energy $$E > V_0$$.
The potential diagram is just a unit step function. It goes from $$V = 0$$ to $$V = V_0$$ at $$x=0$$. In piecewise notation:
$$\begin{displaymath}<br /> V(x) = \left\{<br /> \begin{array}{lr}<br /> 0 & : x < 0 \\<br /> V_0 & : x \ge 0<br /> \end{array}<br /> \right.<br /> \end{displaymath}$$
The piecewise notation does not account for the $$V(x)$$ being continuous at $$x=0$$.

Homework Equations

a) Probability flux:
$$S\left( x,t \right)=-\frac{i\hbar}{2m}\left[ \Psi^*\left( x,t \right) \frac{\partial \Psi\left( x,t \right)}{\partial x} - \Psi\left( x,t \right) \frac{\partial \Psi^*\left( x,t \right)}{\partial x}\left]$$

b) Reflection coefficient:
$$R=\frac{S_{I}^{-x}\left( x,t \right)}{S_{I}^{+x}\left( x,t \right)}$$

The Attempt at a Solution

I've solved for the wave equations at $$x > 0$$ and $$x < 0$$. I'm stuck as far as where to go from there.

Require continuity of the wavefunction and its derivative at x=0. That will allow you to solve for most of the constants.