Do electrons in an electron beam (cathode ray) follow the Pauli exclusion principle? Or in other words, does the Pauli exclusion principle apply for the beam of electrons?
A cathode ray tube where the electrons are randomly ejected and are not constrained in space the Pauli exclusion seems irrelevant..
In structured beams though, things may be different:
The Pauli Exclusion Principle places a fundamental limit on the brightness of an electron beam. Developing a cathode which can reach this limit is useful for achieving maximum operation in current applications of electron beams, but also opens new areas of physics to be explored. When the phase space of the electron beam is filled to the maximum density, the electrons will experience a degeneracy
pressure, similar to that which keeps a neutron star from collapsing. One promising source or a quantum degenerate beam is field emission from adsorbates on carbon nanotubes. Adsorbates have been shown to provide several orders of magnitude enhancement to emission brightness, which approaches the degeneracy limit. We have developed experiments to test various adsorbates, in order to find those which bind tightest and provide the largest enhancement in brightness. Continuing work to discover better adsorbates should soon allow for the generation of a quantum degenerate electron beam .
So it seems to be an aim, to reach the degeneracy limit given by the Pauli exclusion principle.
If by the exclusion principle you mean that the total wave function is anti-symmetric under particle exchange, then yes.