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Suppose in the system of two electrons, can you use the electron spin to distinguish the electrons. In the whole system of particles,where the electrons are swapped,the quantum state of the system changes, why is the electron still indistinguishable?

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Two electrons are indistinguishable if all of their quantum numbers are equal. Therefore, if you apply a magnetic field and prepare one electron in the $m_s = +\frac{1}{2}$ state and one in the $m_s = -\frac{1}{2}$ state, their quantum numbers are not equal. Hence, they are distinguishable, if you do not allow spin-flipping collisions.

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As you stated corretly, electrons are indistinquishable, but we have to be very precise what this means.

The complete wavefunction has to be antisymmetric (or symmetric for bosons) when you exchange the particles. Lets look at the Helium-Atom as example. Suppose one of the electrons is in an excited state, and you ionize the atom with a laser, so that the energy is just enough so the excited electron can escape.

Naturally you would say, you lost the electron which was previously excited, and there is nothing wrong about that. However, when you want to calculate the energy levels of the described excited state, you have to take into account, that the wavefunction needs to be assymetric, otherwise getting false results. So you cannot say, electron 1 was in excited, electron 2 was in the ground state. What you instead have to say, is the system was in an excited state, with one electron being at a different energy level.

I hope i put this thought in an understandable way. Also, i am pretty sure previous answer by Semoi is wrong. As Fermions, electrons can never have identical quantum numbers

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  • $\begingroup$ " As Fermions, electrons can never have identical quantum numbers" Only if they are in a quantized system with energy levels, fermions have to occupy different energy levels ( the Pauli exclusion principle) $\endgroup$ – anna v Sep 20 '17 at 16:29

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