I know that the Pauli Exclusion Principle does not allow two identical fermions to have the same set of quantum numbers. But can they share the same location in space if they are uncharged such as two neutrons? If not, why not because there is no quantum number associated with position. Thanks

The wavefunction for fermions has a spatial part too, and this is what describes the location of a particle at best in QM. Pauli Exclusion Principle forbids two identical fermions from having the very same wavefunction. For example, you can have spin-up fermions both in s- and p-waves, but of course no two spin-up fermions in an s-wave.

  • But does it forbid two fermions to share the same position? For example, if the spin part of the wavefunction is antisymmetric singlet, the spatial part is symmetric. Does the principle forbid them to share the same coordinate? Thanks – mithusengupta123 Aug 12 '17 at 13:14
  • I think the confusing bit here is the meaning of "coordinate". How can you assign a coordinate to a quantum particle? Surely you can take the expectation value for $x$, but two different spatial wavefunctions might yield the same average, still allowing you to have two collinear fermions. So, instead of thinking in terms of "same coordinate" (as in the case of a classical particle), you really should think in terms of "same spatial wavefunction". – Phoenix87 Aug 12 '17 at 13:32
  • You haven't really completed this answer until you explain that they can have the same position so long as the momentum are sufficiently different. Pauli-exclusion lives in phase-space. – dmckee Aug 12 '17 at 17:38
  • @Phoenix87 But what causes the Paul repulsion in degenerate fermi gas as in neutron star? What forbids the neutrons to come closer in a neutron star? What causes the degeneracy pressure? – mithusengupta123 Aug 12 '17 at 19:01

The principle forbids two fermions of the same species from sharing the same quantum state. Let's first carefully look at where the idea comes from.

This principle arises simply because the spin statistics theorem implies that half integer spin particles - i.e. fermions - have antisymmetric multiparticle quantum states, i.e. that such states undergo a sign change under the swapping of any pair of members of the multipartite state. That needfully implies that a two-particle state, for example, must be of the form $\sum\limits_{i,j} \alpha_{i\,j}|i\rangle\otimes|j\rangle$, with $i,\,j$ ranging over all basis states for the individual particles, where $\alpha_{i\,j}=-\alpha_{j\,i}$ which immediately implies $\alpha_{i\,i}=0$: the bipartite state can never have the two members in the same state.

But state of course is the full quantum state. Therefore, identical fermions can perfectly well be at the same position if their other quantum numbers differ. On the other hand, the position is most definitely part of the quantum state, so that two fermions of the same species with all other quantum numbers being equal cannot be at the same position.

Now, of course, position is an observable with a continuous spectrum, so:

  1. One doesn't wontedly think of it as a quantum number, but it is part of the full state nonetheless and so you will have an incorrect understanding of the principle if you think it says that positions "don't count because they have no quantum numner"and
  2. Even though one might argue that the probability to measure two particles at exactly the same value of $x$ when $x$ is a continuous variable is nought, nonetheless the exclusion principle is still highly significant because it means that the amplitude must smoothly approach 0 as the position co-ordinates for the two particles approach equality. Thus, the requirement deeply influences the equation of state for a Fermi gas and thus the principle applied to position alone makes itself felt widely over the field of condensed matter physics.

Position is an undefined concept for a Particle in Quantum Mechanics. The thing which describes the Particle is the state-vector in Hilbert space and the position wavefunction can be considered as the position component of that vector.

So like the previous answer says, all the principle says is the impossibility of having the same wavefunction.

Two wavefunctions can overlap at a point is that what you mean by position? There is no meaning to position of a particle till you measure it. All you get is the probability distribution.

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