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.

 

[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.