Tutorial:Particle in a box

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In the previous tutorial, we considered applying a user-defined potential. What if we wanted to do the classic quantum problem of a particle in a box, i.e. an infinite square well?

V(x) =
\begin{cases}
0, & \tfrac{L}{2} < x <\tfrac{L}{2}\\
\infty, & \text{otherwise}
\end{cases}

There is no meaningful way to set an infinite value of the potential for a numerical calculation. However, we can instead use the boundary conditions to set up this problem. In the locations where the potential is infinite, the wavefunctions must be zero. Therefore, it is equivalent to solve for an electron in the potential above in all space, or to solve for an electron just in the domain x \in [-\tfrac{L}{2}, \tfrac{L}{2}] with zero boundary conditions on the edges. In the following input file, we can accomplish this by setting the "radius" to \tfrac{L}{2}, for the default box shape of "sphere" which means a line in 1D.

inp

FromScratch = yes
CalculationMode = gs

Dimensions = 1 
TheoryLevel = independent_particles

Radius = 5
Spacing = 0.01

%Species
  "null" | species_user_defined | potential_formula | "0" | valence | 1
%

%Coordinates
  "null" | 0
%

EigensolverMaxIter = 100
ConvEigenError = yes

The last two variables are added to ensure sufficient convergence of the wavefunctions. Without them, you may see warnings about lack of convergence.

Run this input file and look at the ground-state energy and the eigenvalue of the single state.

Exercises

  • Calculate unoccupied states and check that they obey the expected relation.
  • Vary the box size and check that the energy has the correct dependence.
  • Plot the wavefunctions and compare to your expectation.
  • Set up a calculation of a finite square well and compare results to the infinite one as a function of potential step. (Hint: along with the usual arithmetic operations, you may also use logical operators to create a piecewise-defined expression. See Manual:Input file).
  • Try a 2D or 3D infinite square well.

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