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Pauli's EP says that electrons in a subshell will have opposing spins.

Now, lets say I prepare two electrons with spin up. If I took an alpha particle ($\mathrm{He^{2+}}$) and added one of these electrons I would have a $\mathrm{He^+}$ ion with an electron with a spin up. If I were to add the second electron to this ion which of the following would happen?

  1. The electron being added would change to spin down and become part of the atom forming $\mathrm{He}$.

  2. The electron previously added would change to spin down and the new one would stay spin up, forming $\mathrm{He}$.

  3. The new electron would get repelled and the ion would remain as $\mathrm{He^+}$.

I'm really curious as to what would happen! :)

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1 Answer 1

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The options 1,2 are actually physically identical because the electrons are identical particles. Once we have two electrons, we can't say which of them is "Paul" and which of them is "Peter".

When the addition is slow etc., the option 1=2 violates the conservation law for the angular momentum. So it is indeed 3 that has to happen: the ion will refuse to "accept" another electron with the same spin.

The reason why the electron won't be accepted is sometimes called a force, the Pauli force, although it is not a genuine force in the sense that it would be given by an extra term in the Hamiltonian.

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  • $\begingroup$ So if the addition is fast enough then the spin should change? $\endgroup$
    – ashiswin
    Commented Nov 7, 2015 at 11:25
  • $\begingroup$ If the addition is such that there's a significant probability that the new electron carries some (orbital) angular momentum, then the absorption may occur, and the change the two electrons' spin may be attributed to the change of the angular momentum. Also note that there's an energy conservation law. So the new electron has to have the right energy to be absorbed at any case. If it has too big an energy, the absorption of the new electron may occur, but the newly born helium atom has to emit a particle, most likely a photon. The photon may carry the angular momentum as well. $\endgroup$ Commented Nov 7, 2015 at 11:33
  • $\begingroup$ So with the emission of the extra photon that carries the excessive energy away, one may have a process that deals with the problem of the wrong angular momentum, too. $\endgroup$ Commented Nov 7, 2015 at 11:34
  • $\begingroup$ So then it is not possible to control which electron is the one that changes its spin? $\endgroup$
    – ashiswin
    Commented Nov 7, 2015 at 11:36
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    $\begingroup$ It is not possible to control "which" electron does anything because it is not possible to determine which electron is which in general, so claiming that you did something to a particular one and not the other would be physically meaningless. When you calculate the probability of processes, you must include histories without as well as with the permutations of the electrons, and only the sum may be interpreted as a probability amplitude that something happens. If the electrons are in the same atom, they are indistinguishable even in practice "by location", which would be possible otherwise. $\endgroup$ Commented Nov 7, 2015 at 12:59

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