Using Runge Kutta Method for T.I.S.E in Electron Motion Approximation

[Cinimod]

Homework Statement

I really am desperate for help on this one.
I need to use the runge kutta method to approximate the motion of an electron in the potential below. The T.I.S.E is known, and I have to try and use the runge kutta method to find the wavefunction of the particle.

V(x) = infinity for |x|>1
V(x) = 0 for 0.2<|x|<1
V(x) = 50 for |x|<0.2

Homework Equations

The time independant schrodinger equation.

The Attempt at a Solution

I don't even know where to start. Any form of help would be appreciated. I have found examples of its implementation from google, but all of the websites found only involve first order differential equations.

 

[Marco_84]

Homework Statement

I really am desperate for help on this one.
I need to use the runge kutta method to approximate the motion of an electron in the potential below. The T.I.S.E is known, and I have to try and use the runge kutta method to find the wavefunction of the particle.

V(x) = infinity for |x|>1
V(x) = 0 for 0.2<|x|<1
V(x) = 50 for |x|<0.2

Homework Equations

The time independant schrodinger equation.

The Attempt at a Solution

I don't even know where to start. Any form of help would be appreciated. I have found examples of its implementation from google, but all of the websites found only involve first order differential equations.

I have made last year a program to solve diff equations, if u want i can send it to u.
But i firts suggest u to start with eulers method than it would be easier to get the runge kutta one.

 

[Cinimod]

I would be very very grateful to have a look at your program, I will study euler's method, see if that helps. I understand the principle of how the ruger kutta method works, but all the examples I have involve 1st order differential equations, I have a second order, and I'm not sure how you deal with coupled differential equations. If I understood that, then the task wouldn't be any where as difficult.

 

[siddharth]

I understand the principle of how the ruger kutta method works, but all the examples I have involve 1st order differential equations, I have a second order, and I'm not sure how you deal with coupled differential equations. If I understood that, then the task wouldn't be any where as difficult.

You can reduce higher order diff eqns to a system of first order eqns by an appropriate change of variables. For example, if the equation is,

\psi &#039;&#039; + A(x) \psi = B \psi

set ,

\psi = x1
\psi &#039; = x2

So that, your system of equations is now

x1&#039;=x2
x2&#039; = \left(B-A(x)\right)x1

If,

X = \left(\begin{array}{c}x1 \\ x2\end{array}\right)You need to solve,

\frac{d}{dt} \left(\begin{array}{c}x1 \\ x2\end{array}\right) = \left(\begin{array}{cc} 0&amp;1 \\ B-A(x)&amp;0\end{array}\right) \left( \begin{array}{c}x1 \\ x2\end{array}\right)

If you know the initial values \psi(0) and \psi&#039;(0), you can use any runge kutta method to the above system. The only difference is that, in this case the values of k1,k2,etc in the rk schemes will be matrices.

 

[Cinimod]

Siddharth. Once again, you have no idea how useful your posts have been. Thank you. That cleared up a lot of problems I had.