Solving a QM problem numerically

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In summary, - Python is ok for this job- To cancel out solutions that blow up at infinity, you need to know how to solve the equations numerically.
  • #1
rahuldandekar
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I've been trying to solve the Deuteron Tensor polarisation equations on the PC, in Python, using a fourth-order Runge-Kutta method. I have a couple of questions:

1) Is Python ok for this job, in that does it give the right answers? The speed is not an issue, it's fast enough for me (I use Numpy and psyco).

2) (which is much more important) How do I cancel out the solutions that "blow up" at infinity? When the computer solves the equations, it gives the most general solution, and doesn't care that the solution blows up at infinity. I DO care ;) , and would like to only get the solution which doesn't blow up... and that would give me th right answer... how do I do it?
 
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  • #2
1. Python uses double precision floating point to represent numbers. Only you can decide if that meets your needs. Double precision gives 15 signficant digits in a result.
See:
http://steve.hollasch.net/cgindex/coding/ieeefloat.html

2. I don't understand what you mean by cancel out -discard the result?
 
  • #3
rahuldandekar said:
2) (which is much more important) How do I cancel out the solutions that "blow up" at infinity? When the computer solves the equations, it gives the most general solution, and doesn't care that the solution blows up at infinity. I DO care ;) , and would like to only get the solution which doesn't blow up... and that would give me th right answer... how do I do it?

A numerical solution to a (system of) differential equation(s), is not a general solution, it is (a close approximation to) the *unique* solution for the initial values that you specify.

If your numerical solution is blowing up, possibilities include:

1) you have specified the wrong initial conditions;

2) you haven't coded things correctly;

3) the solution depends critically on initial values, with very small variations in initial conditions producing wildly different solutions.
 
  • #4
Thanks for the help. :) In fact double precision is what the book says is best... it allows one to use smaller intervals.

About blowing up, what I mean is this:

Usually, you get two solutions to an equation... say e^x + e^-x. Now, in QM, I want the solution to be normalisable, and thus I discard the e^x solution because it doesn't go to 0 at +infinity. However, when you solve numerically, the computer will see the initial conditions, and give an appropriate complete solution, NOT discarding the e^x term. Thus the numerical solution is of no use to me if it's not normalisable.

There must be a way around this... I've tried some trickery, but it doesn't work.
 
  • #5
I don't quite see what you're after, so let me try out a toy example.

The differential equation

y'' - y = 0

has general solution

y(x) = a e^x + b e^{-x}.

Suppose the initial conditions are y(0) = 1 and y'(0) = 0.

This implies a = b = 1/2, so, for these initial conditions,

y(x) = 1/2 (e^x + e^{-x}).

Throwing away the e^x term as unphysical gives

y(x) = 1/2 e^{-x},

but this isn't a solution of the given differential equation for the given initial conditions.

For this to be a solution, either the differential equation or the initial conditions, or both, has to change.

For example, keeping the original DE, but changing the initial conditions to y(0) = 1/2 and y'(0) = -1/2 results in the solution

y(x) = 1/2 e^{-x}.

Maybe I'm way off base, but I think choosing appropriate initial conditions might eliminate exponentially growing solutions.
 
  • #6
George Jones said:
Maybe I'm way off base, but I think choosing appropriate initial conditions might eliminate exponentially growing solutions.

I tend to agree that the boundary conditions are important here.

I'm not familiar with the particular system under study, but I'd suggest that systems in the spatial domain (including QM) should be initially cast as BVP not IVP.
In practice this means approximating the normalisation condition by explicitly requiring the function at large distance to be zero. As a starting point investigate the 'shooting method'.
 

1. What is the first step in solving a QM problem numerically?

The first step in solving a QM problem numerically is to define the problem and determine the appropriate mathematical model to use. This involves identifying the relevant physical quantities and their relationships, and selecting an appropriate numerical method for solving the problem.

2. What are the most commonly used numerical methods for solving QM problems?

The most commonly used numerical methods for solving QM problems include finite difference methods, finite element methods, and variational methods. These methods involve discretizing the problem into smaller, solvable parts and then using iterative techniques to obtain a numerical solution.

3. How do numerical solutions compare to analytical solutions in QM problems?

Numerical solutions in QM problems are generally considered to be approximations of the exact analytical solutions. However, with increasingly powerful computers and more sophisticated numerical methods, these solutions can often be very accurate and provide valuable insights into the behavior of quantum systems.

4. What are some common challenges in solving QM problems numerically?

One common challenge in solving QM problems numerically is the issue of numerical stability. This refers to the accuracy and reliability of the solution, which can be affected by factors such as the choice of numerical method and the precision of the computer used. Another challenge is the sensitivity of the solution to changes in the input parameters, which can lead to difficulties in obtaining a consistent and reliable solution.

5. What are some tips for improving the accuracy of numerical solutions in QM problems?

To improve the accuracy of numerical solutions in QM problems, it is important to carefully select and validate the numerical method used. This may involve comparing the results with analytical solutions or using multiple methods and comparing the results to ensure consistency. Additionally, increasing the precision of the computer and using smaller time and space intervals can also improve the accuracy of the solution.

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