Finding R and T for double delta function potential

[jhosamelly]

Homework Statement

THIS IS THE QUESTION

V (x) = $\sqrt{((h-bar ^{2})V_{0})/2m}$ [$\delta(x-a)$+ $\delta(x+a)$]

-How do I find R and T?

-Under what condition is there resonant transmission?



2. The attempt at a solution


ok. I got these answers. Are these correct? Someone please tell me.

General Equations

$U_{I}$ = $e^{ikx}$ + R $e^{-ikx}$


$U_{II}$ = A $e^{ikx}$ + B $e^{-ikx}$


$U_{III}$ =T $e^{-ikx}$


Boundary Conditions

if a = 0

$U_{I}$ = $U_{II}$

1 + R = A + B

$U_{II}$ = $U_{III}$

A + B = T



discontinuity equation

$U'_{I}$ - $U'_{II}$ = - $\sqrt{\frac{2m V_{o}}{h-bar^{2}}}$ $U_{a}$

ik (1 - R) - ik (A - B) = - $\sqrt{\frac{2m V_{o}}{h-bar^{2}}}$ R


$U'_{II}$ - $U'_{III}$ = - $\sqrt{\frac{2m V_{o}}{h-bar^{2}}}$$U_{a}$


ik (A-B) - ikT = - $\sqrt{\frac{2m V_{o}}{h-bar^{2}}}$ T


/// i hope someone can tell me if these are correct so I can continue my calculations. Thanks.

 

[vela]

The continuity condition A+B = T isn't correct. You need to set x to the location of the second delta function.

 

[jhosamelly]

The continuity condition A+B = T isn't correct. You need to set x to the location of the second delta function.

Hmmm.. ok. Thanks. What about the discontinuity equation? are they correct?

 

[vela]

I didn't notice the other delta function is at x=-a. All four of your equations, as written, have errors in them.

 

[jhosamelly]

I didn't notice the other delta function is at x=-a. All four of your equations, as written, have errors in them.

really? how should the equation be then? Please help me. Only the signs are wrong or the whole equation?

 

[vela]

The boundary between U_I and U_II is at x=-a, not x=0, so the boundary conditions are
\begin{align*}
U_I(-a) &= U_{II}(-a) \\
U'_{II}(-a) - U'_I(-a) &= \lim_{\varepsilon \to 0^+} \frac{2m}{\hbar^2}\int_{-a-\varepsilon}^{-a+\varepsilon} V(x)\psi(x)\,dx
\end{align*}Similarly, the other boundary conditions occur at x=+a.