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

• 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.

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

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?

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

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?

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.