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Particle in a box

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?

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

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.

Input

As usual, we will need to write an input file describing the system we want to calculate:


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
%

LCAOStart = lcao_states

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

Exercises