Free particle in quantum mechanics (Pauli principle?) By what is a state of a free particle defined? For a bound electron for instance it is fully described by the quantum numbers $n,l,m$ and the respective state $|n,l,m\rangle$. But by what is the free particle (electron) defined? I'd guess energy and spin maybe?
From that point on I ask myself: Since a bound electron is defined by $|n,l,m\rangle$ there is the Pauli exclusion principle excluding two or more electrons with the same $n,l,m$. Is it also possible to formulate a similar Pauli exclusion for free particles?
Thanks!
 A: The energy eigenstates for a free particle are simply given by their momentum vector $\vec{k}$ (or $\vec{p}=\hbar \vec{k}$ if you prefer). These states are all orthogonal in the sense that states of different $\vec{k}$ have zero overlap and any wavefunction can be expressed as a linear combination of them. Thus for fermions with no internal degrees of freedom, what Pauli would forbid is two fermions having exactly the same momentum.
For the case of electrons, they do also have an additional 'internal' degree of freedom due to their spin. So you can have two electrons in a spin singlet state with the same momentum, but not 3.
(Note: I am intentionally ignoring any subtleties about the fact that momentum eigenstates are not normalisable, which is a much longer story.)
A: Certainly!!
If you have solved, the free particle problem in quantum mechanics. We the eigen state look like
$$|E,+\rangle =|p=\sqrt{2mE}\rangle$$
$$|E,-\rangle =|p=-\sqrt{2mE}\rangle $$
In the system of two electrons, you need to take account of  Coulomb potential. So the eigenstate would look pretty different (further it's 3D problem).
But still, they have Pauli's principle (particle should be fermion). Not just free particle, any system of particles obeys Pauli exclusion principle.
A: The same question can be raised for the conduction band in solids, where the electrons are "free" all over the lattice. The obvious answer is that there is practically  a continuum in the energy levels available to conduction band electrons, so there is no real  exclusion as far as the accuracy of energy levels goes. Of course there will be a small delta(E) that will keep the Pauli exclusion principle.
The same is true for free particles in space,their wavefunction has to be modeled as a wavepacket of frequencies which means , if two same fermions meet in space (even if it is two electrons in an accelerator beam) their wavefunctions can easily occupy other energy levels, and obey the Pauli exclusion ( not to forget how hard it is to get small errors on beams).
So no measurable effects can come from the exclusion, the way they can in atomic construction.
