Transmission and reflection coefficients

[pleasehelpmeno]

Hi when using the WKB approx, is there a general method to find these Refelction and Transmission coefficients, I have tried looking in books and on the net and I can't find a 'general' formula, they tend tjust to be quoted. I know that $|T|^{2}+|R|^{2}=1$.

And generally that $T= \frac{j_{trans}}{j_{trans}}$ and vice versa, but for something like, $\ddot{X} + (y^{2}+t^{2})X=0$

The coeficients are given by:
Reflection =$\frac{-ie^{i\theta}}{\sqrt{1+e^{\pi y^{2}}}}$
Transmission= $\frac{e^{-i\theta}}{\sqrt{1+e^{-\pi y^{2}}}}$

but am largely unsure as how to calculate it?

 

[fzero]

Suppose there is a potential barrier centered at $x=0$ (the precise location doesn't matter for the below description). Let us take the convention that our incident wave comes in from $x=-\infty$ and is right-moving toward the barrier. The reflected wave is left-moving toward $x=-\infty$, while the transmitted wave is right-moving toward $x=\infty$.

Now, very far from the barrier, the potential is approximately zero, so we have approximately free-particle wave solutions. We can write this as

$$ \psi_- \approx A_i e^{-ikx} + A_r e^{ikx}, ~~~x\rightarrow -\infty,$$

$$ \psi_+ \approx A_t e^{-ikx} , ~~~x\rightarrow \infty.$$

Note that there is no left-moving wave to the right of the barrier, since we chose the convention that the incident wave comes in from the left.

The reflection coefficient is the fraction of the incident wave that is reflected, this is

$$ R = \frac{|A_r|^2}{|A_i|^2}, $$

while the transmission coefficient is the fraction that is transmitted,

$$ T = \frac{|A_t|^2}{|A_i|^2}. $$

In order to say anything more specific, one would need to know the detailed form of the potential specifying the barrier. Exactly what potential corresponds to your example is unclear from the equation you wrote down. Given the potential, we solve for the wavefunctions as precisely as possible in the region of the potential and then match on to the asymptotic wave solutions in order to define the transmission and reflection coefficients.

 

[pleasehelpmeno]

Could you clarify again the sign choices, am I correct:

1)is $\psi_-$ and$\psi_{+}$ inside the barrier, hence why only transitted wave functions?

2)- means going to $-\infty$ + mean going to $+\infty$ That I was hoping it would be a known example. One has to solve the above equation giving parabolic cylinder functions, which then have to be combined giving those. Il have to have another look at it, thanks again for your help.