Efficiently Solving Large Equation Systems in Particle Penetration Calculations

[nille40]

Hi!
I'm trying to calculate the probability for a particle to penetrate a potential well. The problem is that to calculate this, one must solve an equation system that becomes to big. I tried solving it using a matrix and inverting that, but half way, the matrix didn't even fit on my computer screen...

These are the equations:

$$e^{ika} + Re^{-ika} = Ae^{ima} + Be^{-ima}$$
$$Te^{ikb} = Ae^{imb} + Be^{-imb}$$
$$ke^{ika} - Rkae^{-ika} = mAe^{ima} - mBe^{-ima}$$
$$kTe^{ikb} = imAe^{imb} - mBe^{-imb}$$

I tried substituting some common parts, but that didn't help much. How can I solve this? What method should I use?

Thanks in advance,
Nille

 

[turin]

Originally posted by nille40
$$e^{ika} + Re^{-ika} = Ae^{ima} + Be^{-ima}$$
$$Te^{ikb} = Ae^{imb} + Be^{-imb}$$
$$ke^{ika} - Rkae^{-ika} = mAe^{ima} - mBe^{-ima}$$
$$kTe^{ikb} = imAe^{imb} - mBe^{-imb}$$

I'm counting eight unkowns and only 4 equations. Are the m, k, a, and b given? If so, then this system is straightforward: just use either the first and third or second and last equations to solve for A and B in terms of either R or T. Then, use this expression in the remaining two equations to solve for R and T.

I'm not so sure I understand the physical situation here. Are you approximating the wavefunction in a momentum eigenstate that has a positive energy with respect to the potential at infinity?

 

[nille40]

Thanks for responding!
There are 4 unknowns.

k and m is given by a differential equation (schrödingers time independent equation)

a and b are limits for a barrier (start and end).

What I need is R, T, A and B.
Generally, it is not very hard to solve a system of 4 equations. The problem with this one is that the expressions become to big to handle.

The physical aspect of this problem is to calculate the probability for a particle to pass through a barrier (a potential well). There's a probability to find the particle before the barrier, given by Schrödingers time independent equation:

$$\psi (x) = e^{ikx} + R e^{-ikx}$$ (incoming and reflected wave)

There's a probability to find the particle in the barrier, given by $$\psi (x) = Ae^{ikx} + Be^{-ikx}$$

And there's a probability to find the particle after the barrier, given by

$$\psi (x) = Te^{ikx}$$ (One direction only).

These three functions should be connected in x=a and x=b, so it gives a continginous function. This gives 2 equations. 2 more equations can be derived, giving 4 functions and 4 unknowns - an equation system.

So the variables that should be calculated are R, T, A and B, where $$|T|^2$$ is the probability for the particle to penetrate the barrier.

It shouldn't be that hard to solve this, but I haven't done equation systems for a while. I would really appreciate some help...

Thanks in advance,
Nille

 

[turin]

OK, what I said in my previous post should solve the equation, but there seems to be something missing. I am suspicious that the incident wave is not properly normalized. Shouldn't you leave the coefficient of the incident part arbitrary at this point, and then include a fifth equation:

|I|² = |R|² + |T|²?

I don't remember this problem exactly, and could certainly be wrong about this, so please don't take this the wrong way. I just don't want you to do the work for nothing.

 

[michaelc187]

YOu only need those three equations.

such that each is < 1, (for probability)

remember... different mediums gives you independent probs and you can say (prob1)*(prob2)*Prob3 gives you total probability (5th equation)

such that total prob < 1

through out the complex plane, sum of forces (from physics) (its all decaying or reverse anyway, regardless of the incendent angle, remember... i haven't done the problem.

through out e, maybe use 2.5 or something -to see if the computer is messing you up.

 

[michaelc187]

I'd hate tot take to prob eq form you , but if yo can figure out how to thorw th complex plane away, it's probably all you need.