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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/pdf/1111.2727.pdf) 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.

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  • $\begingroup$ Identical is the the same as indistinguishable. All quantum numbers are the same. No two electrons can have identical quantum numbers. In an atom, those are principle, orbital, magnetic, and spin projection quantum numbers. Hence the Pauli exclusion principle. Sorry I can't comment on your last paragraph. $\endgroup$ – garyp Apr 1 '17 at 13:47
  • $\begingroup$ @garyp I'm still confused. It seems that you are identifying particles with the state they occupy. Wikipedia says: "There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which do not share quantum states as described by the Pauli exclusion principle." So particles can be identical even if all their quantum numbers are not the same? And if two particles have the same quantum numbers, are they always identical? E.g. an electron vs a hypothetical spin-$\frac{3}{2}$ particle. They can both have $n=1$, $l=0$, $m_l=0$, and $m_s=-\frac{1}{2}$. $\endgroup$ – JayKay Apr 1 '17 at 14:31
  • $\begingroup$ Sorry. Electrons are indistinguishable. Only one electron can have a given complete set of quantum numbers. (Pauli principle) On the other hand, bosons can have any number of particles with a given complete set of quantum numbers. (Hence the possibility of Bose-Einstein condensation.) $\endgroup$ – garyp Apr 1 '17 at 15:57
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“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.

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  • $\begingroup$ Thank you! I need to understand the role of spin projection in the context of identical particles. Books say two electrons are identical, but they can have different spin projections. If $m_s$ is different, why are the electrons identical? Then take two $^6$Li atoms in the level $^2S_{1/2}$. This level is split into the $F=1/2$ and $F=3/2$ states. $F$ is the total spin. If $F$ is different, the atoms are not identical. Assume both have $F=1/2$. Are they identical? If yes, why, as $m_F$ can be different (singlet and triplet states)? Why is 'spin' the relevant thing, not 'spin projection'? $\endgroup$ – JayKay Apr 2 '17 at 17:20
  • $\begingroup$ @JayKay I added a paragraph that I hope helps. $\endgroup$ – David Bailey Apr 4 '17 at 6:44
  • $\begingroup$ @DavidBailey Hi. I read your answer but I still do not understand why spin projection being an external parameter of an electron prevents us from distinguishing 2 electrons with opposite $m_s$ values. I feel like if we had 2 electrons, initially measured in 2 different positions and with 2 different $m_s$ values, and if we allowed them to evolve for a time and then measured the location of one of them, we'd be able to say which one of the initially measured pair we had based on the finally measured $m_s$ value. Could you tell me where the mistake in my logic is? $\endgroup$ – user2582713 May 9 '17 at 15:16
  • $\begingroup$ @user2582713 Let’s say we call the electron that initially has $m_s=+1/2$ “Alice” and call the one with $m_s=-1/2$ “Bob”. If you later measure an electron with $m_s=-1/2$, it might be Bob, or it might be Alice if the two electrons have interacted and swapped spins. The only way to be sure such a spin exchange has not happened is to keep Alice and Bob well separated, e.g. put a thick wall between them, but in that case they can be distinguished by which side of the wall they are on, independent of their $m_s$ value. $\endgroup$ – David Bailey May 13 '17 at 1:46

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