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 independent Schrödinger 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 independent Schrödinger 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,

$$\psi '' + A(x) \psi = B \psi$$

set ,

$$\psi = x1$$
$$\psi ' = x2$$

So that, your system of equations is now

$$x1'=x2$$
$$x2' = \left(B-A(x)\right)x1$$

If,

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

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

If you know the initial values $$\psi(0)$$ and $$\psi'(0)$$, 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.