From time to time I get asked about the implications of the Pauli exclusion principle for quantum mechanical wave-packet simulations.

I start with the simplest antisymmetric case: a two particle state given by the Slater determinant of two Gaussian wave packets with perpendicular directions of the momentum:

φ_{a}(x,y)=e^{-[(x-o)2+(y-o)2]/(2a2)-ikx+iky} and φ_{b}(x,y)=e^{-[(x+o)2+(y-o)2]/(2a2)+ikx+iky}

This yields the two-electron wave function

ψ_{a}(x1,y1,x2,y2)=φ_{a}(x1,y1)*φ_{b}(x2,y2)-φ_{a}(x2,y2)*φ_{b}(x1,y1)

The probability to find one of the two electrons at a specific point in space is given by integrating the absolute value squared wave function over one coordinate set.

The resulting single particle density (snapshots at specific values of the displacement o) is shown in the animation for two different values of the momentum k (we assume that both electrons are in the same spin state).

For small values of k the two electrons get close in phase space (that is in momentum and position). The animation shows how the density deviates from a simple addition of the probabilities of two independent electrons.

If the two electrons differ already by a large relative momentum, the distance in phase space is large even if they get close in position space. Then, the resulting single particle density looks similar to the sum of two independent probabilities.

The probability to find the two electrons simultaneously at the same place is zero in both cases, but this is not directly visible by looking at the single particle density (which reflects the probability to find any of the electrons at a specific position).

For further reading, see this article [arxiv version].

# When two electrons collide. Visualizing the Pauli blockade.

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