Solving recurrence equations in QM tunneling example

[Poirot]

I am ultimately trying to get Mathematica to solve some equations which are related to a tunnelling problem with an oscillating delta potential.

I have the coefficients for absorption and emission:

$$ T_n = \frac{sm}{i\hbar^2} (T_{n-1} + T_{n+1})$$

$$T_q = \frac{sm}{2ik_q \hbar^2} (T_{q-1} +T_{q+1}) +1$$

where $$k_{q/n} =(\frac{2m(E-\hbar \omega q(or \ n))}{\hbar^2})^{1/2}$$

So I'm trying to do this for a finite number of n, such as -2 to 2, and have given the seed values T[-2]=T[2]=1 (as far as I'm aware these can be whatever I want).

I've tried using RSolve but it just pops out the input again as the output, and also won't allow me to specify the particular range for n. I was told that RSolve probably won't be able to do this as they "won't have a closed form" and so to use the "Do" function. I tried this but all it spits out is the 5 equations for n=-2,-1... 2 etc and doesn't solve them.

I'm setting all the constants out the front to 1. For some context, I'm trying to build a programme that shows a gaussian wave packet entering a delta potential which is oscillating in strength ($$V(x) =\delta(x) scos(\omega t)$$) and the initial wave is monochromatic (which gives the transmission coefficient Tq, taking q to be effectively anything). The Tn part is from the Fourier transform that allows all n not equal to q which has no incident part, only reflection and transmission (I'll deal with the reflection once I get somewhere with the transmission). Any help would be very much appreciated, I'm quite a newbie with Mathematica and even the physics is beyond me for the most part.

Thanks!

 

[Valkarie]

One possible solution is to use the NDSolve command. This command allows you to specify the initial conditions and range of n and then solve the equations for the T_n coefficients. You can also specify additional parameters such as the energy and frequency of the delta potential. For example, in this case, you can use the following code to solve the equations:NDSolve[{T[n] == sm/(I*hbar^2)*(T[n - 1] + T[n + 1]), T[-2] == 1, T[2] == 1, k[q] == Sqrt[2*m*(E - hbar*omega*q)/hbar^2], k[n] == Sqrt[2*m*(E - hbar*omega*n)/hbar^2], n ∈ Range[-2, 2]}, T, {n, -2, 2}]This will return a list of solutions which you can then use to plot the transmission coefficient as a function of n.Hope this helps!