Reflection Coeff - Quantum Mech

[nickmai123]

Homework Statement

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

Homework Equations

a) Probability flux:
[tex]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][/tex]

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

The Attempt at a Solution

I've solved for the wave equations at [tex]x > 0[/tex] and [tex]x < 0[/tex]. 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 [tex]V[/tex] to [tex]V_0[/tex] with a total energy [tex]E > V_0[/tex].
The potential diagram is just a unit step function. It goes from [tex]V = 0[/tex] to [tex]V = V_0[/tex] at [tex]x=0[/tex]. In piecewise notation:
[tex]
\begin{displaymath}
V(x) = \left\{
\begin{array}{lr}
0 & : x < 0 \\
V_0 & : x \ge 0
\end{array}
\right.
\end{displaymath}
[/tex]
The piecewise notation does not account for the [tex]V(x)[/tex] being continuous at [tex]x=0[/tex].

Homework Equations

a) Probability flux:
[tex]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][/tex]

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

The Attempt at a Solution

I've solved for the wave equations at [tex]x > 0[/tex] and [tex]x < 0[/tex]. 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.