# Double Delta Fuinction Potential - Tell me if Im correct please

1. Oct 26, 2011

### jhosamelly

1. The problem statement, all variables and given/known data

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.

2. Oct 27, 2011

### vela

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

3. Oct 27, 2011

### jhosamelly

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

4. Oct 27, 2011

### vela

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

5. Oct 27, 2011

### jhosamelly

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

6. Oct 27, 2011

### vela

Staff Emeritus
The boundary between UI and UII 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.