hmparticle9
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I am solving Schrödingers equation with a potential with two "delta" spikes: One at ##a## and another at ##-a##, with magnitude ##-\alpha##. In the process of looking for scattering solutions I have come across the following four equations:
$$A + B \beta = C + D \beta$$
$$C \beta + D = F \beta$$
$$ik(C - D\beta - A + B\beta) = \delta (A+B\beta)$$
$$ik(F\beta - C\beta + D) = \delta F \beta$$
where ##\beta = e^{2ika}## and ##\delta = -\frac{2m\alpha}{h^2}##.
##A,B,C,D,F## are (possibly) complex numbers. ##k## is a constant. ##i## is the square root of ##-1##.
I need to find the transmission coefficient of the problem, which basically means I need the quantity:
$$\frac{|F|^2}{|A|^2}$$
which means I need to find ##A## in terms of ##F##, or more so the other way around. ##F## in terms of ##A##.
I have ##F## in terms of ##C## and ##D##:
$$C \beta + D = F \beta$$
I have found $$C = (\frac{\delta}{2ik} +1) A + \frac{\delta \beta}{2ik}B$$
and $$D = -\frac{\delta}{2ik\beta} A + (1-\frac{\delta}{2ik}) B$$
I have put these values in the equation
$$A + B \beta = C + D \beta$$
to verify correctness. All I need to do now is find a relationship between ##A## and ##B##.
This I am struggling with. Could someone give me a hand?
$$A + B \beta = C + D \beta$$
$$C \beta + D = F \beta$$
$$ik(C - D\beta - A + B\beta) = \delta (A+B\beta)$$
$$ik(F\beta - C\beta + D) = \delta F \beta$$
where ##\beta = e^{2ika}## and ##\delta = -\frac{2m\alpha}{h^2}##.
##A,B,C,D,F## are (possibly) complex numbers. ##k## is a constant. ##i## is the square root of ##-1##.
I need to find the transmission coefficient of the problem, which basically means I need the quantity:
$$\frac{|F|^2}{|A|^2}$$
which means I need to find ##A## in terms of ##F##, or more so the other way around. ##F## in terms of ##A##.
I have ##F## in terms of ##C## and ##D##:
$$C \beta + D = F \beta$$
I have found $$C = (\frac{\delta}{2ik} +1) A + \frac{\delta \beta}{2ik}B$$
and $$D = -\frac{\delta}{2ik\beta} A + (1-\frac{\delta}{2ik}) B$$
I have put these values in the equation
$$A + B \beta = C + D \beta$$
to verify correctness. All I need to do now is find a relationship between ##A## and ##B##.
This I am struggling with. Could someone give me a hand?
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