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

In summary, the conversation discusses the use of the Runge Kutta method to approximate the motion of an electron in a given potential and find its wavefunction. The homework equation is the time independent Schrodinger equation and the attempt at a solution involves finding a way to deal with a second order differential equation using Euler's method and the Runge Kutta method. The conversation also mentions reducing higher order differential equations to a system of first order equations to solve using the Runge Kutta method.
  • #1
Cinimod
34
0

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.

[tex] V(x) = infinity [/tex] for |x|>1
[tex] V(x) = 0 [/tex] for 0.2<|x|<1
[tex] V(x) = 50 [/tex] 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.
 
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  • #2
Cinimod said:

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.

[tex] V(x) = infinity [/tex] for |x|>1
[tex] V(x) = 0 [/tex] for 0.2<|x|<1
[tex] V(x) = 50 [/tex] 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.
 
  • #3
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.
 
  • #4
Cinimod said:
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,

[tex]\psi '' + A(x) \psi = B \psi[/tex]

set ,

[tex]\psi = x1[/tex]
[tex]\psi ' = x2[/tex]

So that, your system of equations is now

[tex]x1'=x2[/tex]
[tex] x2' = \left(B-A(x)\right)x1[/tex]

If,

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

[tex]\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)[/tex]

If you know the initial values [tex]\psi(0)[/tex] and [tex]\psi'(0)[/tex], 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.
 
Last edited:
  • #5
Siddharth. Once again, you have no idea how useful your posts have been. Thank you. That cleared up a lot of problems I had.
 

1. What is the Runge Kutta method?

The Runge Kutta method is a numerical algorithm used for solving ordinary differential equations. It is a popular method for approximating the solutions of a wide range of differential equations in various scientific fields.

2. How does the Runge Kutta method work?

The Runge Kutta method works by calculating the slope of the function at different points in the interval of the differential equation. It then uses this slope to estimate the value of the function at the next point. This process is repeated until the desired accuracy is achieved.

3. What are the advantages of using the Runge Kutta method?

The Runge Kutta method is known for its high accuracy and stability. It is also a versatile method that can handle a wide range of differential equations, including stiff equations. It is also computationally efficient and can handle large systems of equations.

4. Can the Runge Kutta method be used for all differential equations?

No, the Runge Kutta method is most suitable for solving nonlinear differential equations. It may not be the best choice for linear equations or equations with a constant solution. In these cases, other methods such as Euler's method may be more appropriate.

5. Are there any limitations of the Runge Kutta method?

One limitation of the Runge Kutta method is that it requires the function's derivative to be continuous. It may fail to produce accurate results for equations with discontinuous derivatives. Additionally, the accuracy of the method decreases as the step size increases, so it may not be suitable for problems with very small or large intervals.

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