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Numerical simulation of the Schrödinger equation

This repository shows various methods of numerical integration of the Schrödinger equation in the positional representation

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in either one or two dimensions with various potentials. The whole system is an infinite well, so the boundary conditions are zeros.

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Content

One-dimensional simulations

The code files for the one-dimensional case are schrod_*_1d.c. They utilize various integrating methods, including:

  • schrod_a_1d.c: model 1 – approximating evolution using a sort of canonical analytical solution. Rather something to start from than the generic method of numerical integration of any differential equations (I stumbled across this method in literature and just wanted to try it). Anyway, this method is stable for a reasonably small time step.
  • schrod_e_1d.c: model 2 – Euler's method. This method is unstable when you use any non-zero potential.
  • schrod_2s_1d.c: model 3 – two-step explicit method. This method is stable for a reasonable time step.
  • schrod_2s_5p_1d.c: model 3a – two-step explicit method with five-point formula for Laplacian. This method is stable but has higher numerical diffusion than the previous method with the three-point formula.
  • schrod_rk4_1d.c: model 4 – fourth-order Runge-Kutta method. This method is crazy because of having very long expressions for coefficients, which I derived in Mathematica (the notebook is printed to wsp_rk4.pdf and the final formulas are in wzory.txt and wzory_kopia.txt). This method is stable for a reasonable time step.
  • schrod_lw_1d.c: model 5 – Lax-Wendroff method. This method has very high dispersion in this case and the solution behaves differently than the others (because it was a bit “over-engineered”) – unstable.
  • schrod_cn_1d.c: model 6 – second-order implicit method (Crack-Nicholson) with solution of the system of equations using the Jacobi iterative method. This method is stable for the reasonable time step (e.g., when dx==1, dt<0.03).

Two-dimensional simulations

The code files for the two-dimensional simulations are schrod_*_2d.c. They are the two-dimensional extensions of the successful one-dimensional models. They include:

  • schrod_a_2d.c: model 1 – approximating evolution using a sort of canonical analytical solution. This method is unstable for reasonable time steps (e.g., dt>0.01 with dx==1).
  • schrod_2s_2d.c: model 3 – two-step explicit method. This method is unstable for dt>0.01 with dx==1.
  • schrod_2s_2d_9p.c: model 3a – two-step explicit method using the nine-point formula for Laplacian. Similarly to the five-point formula (model 3), this method is unstable for dt>0.01 with dx==1.
  • schrod_cn_2d.c: model 6 – second-order implicit method (Cranck-Nicholson) with solution of the system of equations using the Jacobi iterative method. This is the only 2D method here which is stable.
  • schrod_cn2_2d.c: model 6a – second-order implicit method (Cranck-Nicholson) with solution of the system of equations using the Jacobi iterative method. This is some experimental form of model 6.

Potential generating programs

The programs which generate chosen potential are potencjal_1d.c and potencjal_2d.c, respectively, for the 1D and 2D cases.

Initial state generating programs

The programs generating the initial states for all simulation models are:

  • schrod_init_1_1d.c: generates one Gauss packet for the 1D case.
  • schrod_init_2_1d.c: generates two Gauss packets for the 1D case.
  • schrod_init_2d.c: generates one, two or three Gauss packets for the 2D case, depending on the number passed as the command-line argument argv[1].

Gnuplot scripts for running and visualizing the simulation

The scripts run.gnu and run2.gnu (for 1D and 2D, respectively) run the loops in which consecutive iterational steps of a chosen simulation model are performed and plotted. The 1D script plots the real and imaginary parts of the wave function and the probability amplitude along with the potential. The 2D script plots the probability amplitude along with the potential.

Usage

To run the simulation, first run the compiled potencjal_2d.out program and, following the instructions displayed on the screen, select the appropriate potential, which will be saved in the file potencjal_2d.dat. Then, run Gnuplot and plot the result of compiled schrod_init_2d.out with the command-line argument 1, 2 or 3, indicating the number of wave packets. You are going to be interested in the 1:2:5 output columns to plot the probability amplitude. It will plot the initial state of the system and save it in the files schrod_*_2d.dat so that all the simulation models can use it. Finally, set the number of repetition of iterations n and run the run.gnu script which does the loop executing the chosen model of simulation until the given number of repetition is satisfied. The simulation models programs, at each iteration, read the state of the system from the schrod_*_2d.dat files, calculate the evolution of the system in the current time step and save the result to the same file.

So, in short, after generating the potential with potencjal_2d.out and saving the desired model within run.gnu, run Gnuplot and type, for example:

splot "< ./schrod_init_2d.out 3" u 1:2:5 pt -1 w pm3d
n=200
load "run.gnu"

This describes the usage in the two-dimensional case. For the one-dimensional models you should proceed in an analogous way.

Discussion

The Schrödinger equation has, excluding the term with potential and the imaginary unit, similar form to the difussion equation, and the diffusion is visible in the simulation. This seems to be the manifestation of the Heisenberg's indeterminacy principle.