So just in case people find this and want an answer. Yes, this is right and by "algebraic equations" the question didn't mean analytic so a relation that can be solved graphically is fine. Also, if you find that the odd solution ignores the central delta function completely, then you're headed...
I've been trying to solve it graphically but then I need values for \beta,\gamma and a and the functions change drastically depending on the values of these.
I'm also unsure about this solution since it's currently:
Be^{kx} for x<-a
De^{2ka}(\frac{2k}{\gamma}-1)e^{kx}+De^{-kx} for -a<x<0...
Well, I get C=De^{2ka}(\frac{2k}{\gamma}-1) as before
Which I put into k=\frac{\beta}{2}\frac{D+C}{D-C}
Which comes out with k=\frac{\beta}{2}\frac{1+e^{2ka}(\frac{2k}{\gamma}-1)}{1-e^{2ka}(\frac{2k}{\gamma}-1)}
I'm having a hard time solving this for k, almost every term has a k in it...
I understand that if this were a non-delta well, I could compare k=\sqrt{-E} at the V=0 portion to \kappa =\sqrt{E+V_0} inside the well by forming an equation of k as a function of \kappa and finding the intersect of the two functions. But here that isn't possible here because there is no...
Ok, going through the calculations again, I've arrived at 3 equations for k, 1 for each boundary:
For even symmetry
Boundary 1: k=-k\frac{De^{ka}-Ce^{-ka}}{De^{ka}+Ce^{-ka}} +\gamma
Boundary 2: k=\frac{\beta}{2} \frac{D+C}{D-C}
Boundary 3: k=-k\frac{De^{ka}-Ce^{-ka}}{De^{ka}+Ce^{-ka}}...
Yes. Yes it should.
If I use the differential continuity at the barrier I come up with a different relation between C and D:
D=C\frac{2+\beta/k}{2-\beta/k}
These seems at odds with the relation from the first well:
C=D(\frac{2k}{\gamma}-1)e^{2ka}
Mathmatically they both seem correct but...
So doing that, I get
C=D\frac{2k}{\gamma} e^{2ka} (\frac{2k}{\gamma}-1) from the first well
G=F\frac{2k}{\gamma} e^{2ka} (\frac{2k}{\gamma}-1) from the second well
And then I use these with C+D=F+G from the barrier to get:
D=F and C=G
So that I have:
Be^{kx} for x<-a...
I know that but I was trying to solve it generically first because I don't understand what happens to an exponential at a barrier. I know that for a traveling wave, it would exponentially decrease throughout the barrier until it got to the other side with a lower amplitude. But if it's already...
Homework Statement
Consider a one-dimensional time-independent Schrodinger equation for an electron in a double quantum well separated by an additional barrier. The potential is modeled by:
V (x) = -γδ(x - a) -γδ(x + a) + βδ(x)
Find algebraic equations which determine the energies (or...