When are two fermions considered identical?

Question

I am a bit confused about the terminology concerning identical fermions. In quantum mechanics, identical fermions need to obey certain anticommutation relations i.e. have an antisymmetric total wavefunction (product of spatial and spin parts) under the exchange of two particles. But when exactly are two fermions considered to be 'identical'?

As far as I understand, two electrons are identical in the sense that they for example have the same electric charge, mass and spin quantum number $s=\frac{1}{2}$. (These are shared by all electrons.) How about the secondary spin (or spin projection) quantum number $m_s=\pm \frac{1}{2}$? If in an atom we have two electrons with the same principal quantum number, orbital quantum number and magnetic quantum number, but different $m_s$, are these two electrons 'identical' or not? The electrons are definitely identical if they additionally have the same $m_s$, but the Pauli exclusion principle forbids that. How does the consideration of the quantum numbers fit with the electrons having the same electric charge, mass, and spin $s$, which are independent of atomic orbitals?

Furthermore, are 'identical' and 'indistinguishable' fermions different things or not? If 'identical' particles cannot be told apart by an experiment, how about the Stern-Gerlach experiment? This links back to the above question whether $m_s=\pm \frac{1}{2}$ electrons are identical. My partial answer is that if we have electrons labelled 1 and 2, we cannot say which of the electrons has $m_s=-\frac{1}{2}$ and which has $m_s=\frac{1}{2}$. So in an experiment, you cannot say that 'The electron with $m_s=-\frac{1}{2}$ is electron 1' even if you were able to separate the $m_s=\pm \frac{1}{2}$ states. Or something along those lines. In other words, electrons are identical regardless of the quantum numbers.

Further confusion arises when we go beyond electrons. I have learned that atoms with an odd number of neutrons in an 'optical lattice' can be used to mimic electrons in a metal. Then the role of $m_s=\pm \frac{1}{2}$ is played by different hyperfine states of the atom as far as I can tell. If I now have atoms of the same mass (same species) in two different hyperfine states corresponding to spin-up and spin-down electrons, how are these atoms classified? Are they 'identical', 'indistinguishable', 'distinguishable', or what? Do these atoms obey the fermionic anticommutation relations?

I thought I understood what's going on i.e. two atoms of the same species are identical but can be in different internal states representing the spin projections and thus the atoms obey the usual fermionic anticommutation relations of spin-$\frac{1}{2}$ electrons. But then I found this paper (https://arxiv.org/abs/1111.2727) which says that atoms in the $| F=\frac{1}{2}, m_F=\pm \frac{1}{2} \rangle$ states of $^6$Li are distinguishable. How can distinguishable atoms play the role of electrons? How can they obey the anticommutation relations of spin-$\frac{1}{2}$ electrons? If I have understood correctly, there are no symmetry requirements for distinguishable particles in quantum mechanics. Or are the authors just possibly using sloppy language by calling two identical atoms in different internal states distinguishable? I would call two atoms of different species, e.g. $^6$Li and $^{40}$K, distinguishable.

Sorry for the long text, but hopefully I made my confusion sufficiently clear. Any clarifications are appreciated.

Answer 1

“Two particles are said to be identical if all their intrinsic properties (mass, spin, charge, etc.) are exactly the same.” (Cohen-Tannoudji et al.) Two particles are indistinguishable if you can’t tell which is which, and that depends on the system. In classical mechanics, two identical particles may be distinguishable by their past histories, e.g. you can follow their paths without losing track of which is which. In quantum mechanics, identical particles are indistinguishable, but identical particles may be in distinguishable states, and it is easy to confuse particles and states. Two neutral ⁶Li atoms each have identical constituents (3 electrons, 3 neutrons, 3 protons) but these constituents can be arranged in different internal states that do not all have the same spin or mass. Two ⁶Li atoms are only identical if they are in the same internal state.

Even if particles have identical intrinsic properties, they can be in different external states as part of larger system. In a magnetic field, identical ⁶Li atoms will be in one of two distinguishable states with different energies, depending on whether the atom spin is aligned or anti-aligned with the magnetic field. I believe that in the paper you reference, the “indistinguishable” pairs of atoms are those with aligned spins, and “distinguishable “ pairs are those with opposite spins. This is compatible with how we might think about the two electrons in a neutral helium atom. If they have different spins then they can both be in the 1S level since the spin-up state is distinguishable from the spin-down state. If they have the same spin then they are indistinguishable in the 1S level, so by the Pauli exclusion principle one electron must be in a higher energy level.

Spin-projection to an outside axis is not an intrinsic property of a particle, it is a parameter describing its external quantum state. Internal parameters such as the rest mass, charge, and spin of all electrons are always the same for identical particles, but external state parameters can change. Just because two electrons can have different positions, momenta, or spin-projections does not mean that all electrons are not identical, it just means they can be in different external states. $F$ is an internal quantum number, and the ⁶Li $F=\frac{1}{2}$ and $\frac{3}{2}$ states have different masses and total angular momenta and hence are different particles. Unlike the $m_s$ and $m_l$ quantum numbers that parameterize differences in internal atomic structure due to the possible relative orientations of nuclear and electron orbits and spins, $m_F$ can only be defined relative to an external axis, and the rest mass, charge, and spin of the atom are independent of $m_F$. So two $2 S_{1/2}$ $F=\frac{1}{2}$ ⁶Li atoms are identical particles, but they may be in different $m_F$ states.

The answers to What are the differences between indistinguishable and identical? and Distinguishing identical particles, may also be helpful.