report1 - rht/fermionnode GitHub Wiki

#Report 1: Plotting the wavefunction cross section of an electron in N electron gas

The goal is to characterize the zeros of the wavefunctions...

###What are we trying to do here? What is the content of the plot? Basically we want to know the shape of the wavefunction cross section of an electron in a free N-electron system with periodic boundary condition. Specifically, we fix the location of the other N-1 electrons. So the wavefunction is reduced from N position variable into a single position variable. [\psi(x_1,\ldots,x_N) = \psi'(\x_1)]

###Uh, aren't electrons indistinguishable? Sure, it may look as if we force an electron to be distinguishable, but what we are doing here is just that we are sampling a specific subspace of all the possible coordinate of our wavefunctions (look at the goal of this project). There is no sin in plotting them.

Of course, you can have a different point of view: you force 1 electron to be distinguishable by fixing the N-1 electrons as background field. Your electron will move according to the constraint by the "effective potential" caused by the exchange interactions from the N-1 electrons, I mean, pauli repulsion force ("force": classical mechanics jargon...). Actually, this pov is still meaningless because it is just not possible to represent antisymmetrization as an interaction hamiltonian.

More concretely, consider the example here: Consider a scattering process between two noninteracting fermions, e.g. low energy neutrons (you force the initial state to be distinguishable by firing them from different sources): in the classical case, the cross section is basically zero, however in the quantum case, the neutrons mix together, creating an antisymmetrized outgoing state (tangential, because during the scattering process, the real eigenstate is the quantum field eigenstate, hence both electrons become virtual particles until they are captured by the detectors -- note: be careful, this sentence is only needed in the interacting theory).

What are you talking about? There is no internal scattering term in your diagram, at all. So of course the distinguishable neutrons will remain distinguishable. And also, this is not real life, because of course there will be photon/gluon exchange no matter how small, which will always result in an antisymmetrized final state. An explicit example: an electron with state px and another electron with state py are distinguishable. see http://www.av8n.com/physics/exchange.htm

http://physics.stackexchange.com/questions/18595/is-decoherence-even-possible-in-anti-de-sitter-space http://physics.stackexchange.com/questions/79/is-quantum-entanglement-mediated-by-an-interaction

#Slater determinant update [|n_1 n_2 \cdots n_N; A\rangle = \frac{1}{\sqrt{N!}} \sum_p \mathrm{sgn}(p) |n_{p(1)}\rangle |n_{p(2)}\rangle \cdots |n_{p(N)}\rangle]

Milestone:

  1. Plot the slater-det gs wavefunction when N-1 electrons are fixed nodes
  2. Performance improvement of 5 fold, currently 6 electrons require 2.73s. Here I precompute the slater determinant of N-1 electrons to reduce redundant computations when plotting the wavefunction cross section.