Numerically solutions with periodic boundary conditions

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Discussion Overview

The discussion centers on methods for numerically solving the one-dimensional Schrödinger Equation (SE) with periodic boundary conditions. Participants explore different approaches applicable to both time-dependent and time-independent scenarios, focusing on the ground state solution.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the numerical solution of the 1D Schrödinger Equation with periodic boundary conditions.
  • Another participant asks whether the equation is time-dependent or stationary, and notes the need for either the ground state or an excited state solution if it is stationary.
  • A clarification is provided that the focus is on the time-independent Schrödinger Equation and specifically the ground state solution.
  • A suggestion is made to use the split-step method for the time-dependent equation, highlighting that Fourier Transform inherently accommodates periodic boundary conditions. Alternative methods like Euler or Crank-Nicholson are mentioned for propagation if not using Fourier Transform.
  • For stationary solutions, it is proposed to utilize the symmetry of the solution to rewrite it on a closed interval and apply methods such as the shooting method.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate methods for solving the equations, with no consensus reached on a single approach or technique.

Contextual Notes

Participants do not fully resolve the implications of using different numerical methods or the specific conditions under which each method is applicable.

jimmy neutron
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Is anyone aware of how to numerically solve the (1D) SE with periodic boundary conditions?
 
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It is time dependent SE or stationay SE? If it is stationary you need the ground solution or some excited state solution?
 
soarce said:
It is time dependent SE or stationay SE? If it is stationary you need the ground solution or some excited state solution?
It is for the time independent Schrödinger Equation and for ground state solution
 
For time dependent equation I would start by split-step method, when using Fourier Transform you get by default the periodic boundary conditions: http://en.wikipedia.org/wiki/Split-step_method otherwise you would need to implement a propagation method (Euler, Crank-Nicholson etc).
As for stationary solutions you can you the symmetry of your solution, i.e. translation invariant, to rewrite your solution on a closed interval [a,b] and the use, for instance, shooting method http://en.wikipedia.org/wiki/Shooting_method
 

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