The detailed structure of many-particle wave functions of electrons (such as in any conductor, semiconductor, or superconductor) is a fascinating current physics problem that still contains many unexplored aspects. For noninteracting fermions, the stationary wave function can be represented as a Slater determinant
denotes the number of electrons and the
are the wave vectors from within the (discretized) Fermi surface/sphere. Many-particle effects (due to the electron–electron Coulomb interaction) can be taken into account semi-phenomenologically through backflow effects, realized through a change to collective variables
Here, the simple parametrization
has desirable asymptotic properties.
If in the resulting function
of the first
particles are fixed, a real-valued reduced wave function of
results that can be visualized in the 2D case as a contour plot.
The wave vectors within the Fermi sphere are discretized through Born–von Karman boundary conditions applied to the wave function. As a result, there is a one-to-one correspondence between the number of electrons and the Fermi wave vector and at certain
the wave functions will change their appearance due to the addition of new electrons.
This Demonstration visualizes the square of the absolute value of the reduced wave function as a contour plot on the regions where the reduced wave function is positive or negative. The nodal lines of the resulting one-particle function are shown as blue curves and the movable white points indicate the fixed electrons at position
. Depending on the backflow parameters, the resulting wave function can develop regions with intricate fractal structure.