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I have a problem regarding the Pauli exclusion principle which as far as I understand states that two or more identical fermions cannot occupy the same quantum state.

So is the position ($r_1$) of an electron in an atom part of its quantum state ?

  1. If yes, why do two electrons in a $ 1s$ orbital have opposite spins despite having different positions?

I am guessing that its related to Heisenberg uncertainty principle as follows: since the electrons' position's aren't known with complete precision, they may coincide.

lineage
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Salmon
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  • Who says two elections in a 1s orbital have different positions? Their position space wave functions are identical. – Jahan Claes Apr 29 '20 at 15:42

2 Answers2

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Electrons don't have a single well-defined position. They have a position wavefunction, which is related to a probability distribution that an electron will be at a particular position. The Pauli exclusion principle says that two fermions cannot have the same state. The state includes the particle identity, the spin, and, among (possibly) other things, the wavefunction. Two electrons with the same spin and the same wavefunction are in the same state, so this is forbidden by the Pauli exclusion principle.

For electrons in an atom, or subject to some other time-independent potential, electrons in their lowest-energy state occupy one of a discrete set of possible wavefunctions. Since they are discrete, we can number them; these quantum numbers are a shorthand for referencing a particular state compatible with a particular potential. This means that the Pauli exclusion principle can also be restated as: two identical fermions cannot have the same set of quantum numbers.

probably_someone
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  • Hello, thank you but i still can't understand. The wavefunction is the description of a quantum state but the square modulus of the wave function gives me only informations about the position, so who gives me all the other properties of my particle as spin ecc.?Thanks – Salmon Apr 29 '20 at 16:33
  • @Salmone The wavefunction is not the same as the full state of the particle. You could say that it's the "spatial part" of the state. The full state for an electron looks something like: $\text{Electron state} = (\text{Spin state})\otimes (\text{Wavefunction})$. If you're dealing with things like nucleons, other properties can be included in the state: for example, a nucleon has isospin as well as spin, so its state is the combination of a spin state, an isospin state, and a wavefunction. – probably_someone Apr 29 '20 at 16:35
  • Ok now i've understand, thank you so much. So if I have 1 electron in position r1 and another electron in position r2 the $\psi_A(r1)$=$\psi_B(r2)$ right? – Salmon Apr 29 '20 at 16:49
  • This is actually a bit trickier to answer than it may seem at first glance. Electrons are identical particles and so we cannot distinguish between the A particle having a wave function at r1 and the B particle having a wave function at r2 on the one hand from A particle having a wave function at r2 and the B particle having a wave function at r1 on the other – Metropolis Apr 29 '20 at 19:05
  • Ok but in Pauli's principle aren't we assuming that 2 fermions in the same state have also the same wavefunction? So $\psi_A(r1)=\psi_B(r2)$? – Salmon Apr 29 '20 at 19:19
  • @Salmone You need to get rid of this notion that you can have "an electron in position $r_1$". That's not a thing that makes sense in quantum mechanics. Electrons don't have a well-defined position. – probably_someone Apr 29 '20 at 19:23
  • Right, so I can say, I have 2 fermions each one described by a $\psi(r)$ and these two wavefunctions, for example 2 electrons in the Helium atom, could be the same? – Salmon Apr 29 '20 at 19:30
  • @Salmone Yeah, the wavefunction for two different electrons can be the same as long as their spins are different. – probably_someone Apr 29 '20 at 19:40
  • Thank you so much! – Salmon Apr 29 '20 at 19:41
  • Hi I want to ask you 2 other things if possible:1-you told me that the wavefunction is the "spatial part" of the state, if I put the wavefunction in this way: $|\psi_a>$, does this mean I'm considering the whole state? 2-How information of the momentum is contained in the wavefunction? – Salmon May 02 '20 at 23:10
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    @Salmone $|\psi_a\rangle$ is not a wavefunction. $|\psi_a\rangle$ is a state. A position wavefunction is a function that describes the overlap of a state with a position eigenstate: $\psi_a(x)=\langle x|\psi_a\rangle$. Similarly, a momentum wavefunctoin is a function that describes the overlap of a state with a momentum eigenstate: $\psi_a(p)=\langle p|\psi_a\rangle$. The position and momentum wavefunctions are interconvertible via Fourier transform, so they both contain the same information. – probably_someone May 02 '20 at 23:41
  • @probably_someone If I'm understanding you correctly, then the state of an electron can either be given by position and spin, or by its four quantum numbers? If the first three quantum numbers define the orbital, then this means that the orbital uniquely defines the wave function (and vice versa)? – A_P Sep 02 '21 at 16:54
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If you look at the periodic table, which has ordered the elements according to their behaviour, you can see that there are 2, 2+8 and 2+8+8 electrons in the first three periods. The amazing thing is that you can imagine (please only for yourself) the electrons as bar magnets. In helium they are standing head to foot opposite each other and in 8 electrons they are arranged in the edges of a cube, 4 with the north pole to the nucleus and 4 with the south pole.

This image is helpful for me for understanding the founded empirically Pauli principle or for example the symmetrical behavior of methan CH4.

HolgerFiedler
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  • If I imagine electrons as bar magnets there is a dipole magnetic moment associated with each electrons? And the fact that I can imagine electrons as bar magnets is due to their spin? – Salmon Apr 30 '20 at 14:28
  • Electrons have a magnetic dipole moment https://en.wikipedia.org/wiki/Electron_magnetic_moment, so to your first question: yes. To the second question: What was the first one! The egg or the chicken? (For me (and only for my imagination) the spin is the result of the magnetic dipole moment that is created when moving electrons are deflected in an external magnetic field). – HolgerFiedler Apr 30 '20 at 15:10
  • I don’t know your background. But the best way to lern something is to read about Stern-Gerlach, about introduction of the magnetic dipole moment and the spin. – HolgerFiedler Apr 30 '20 at 15:12
  • I can't understand how a single electron could act like a dipole if it's just a single negative charge? There's an answer for this? – Salmon Apr 30 '20 at 15:44
  • That is the point. The characteristic of a charge (for the electron) was found long before the characteristic of its magnetic dipole moment. But does this mean that one property is prevailing the other? Maybe the magnetic dipole is more important inside an atom? I can’t do more for you, you have to read the sources in its chronological order. Are you studying physics? – HolgerFiedler Apr 30 '20 at 15:55
  • Thank you, i will try to understand better – Salmon Apr 30 '20 at 16:01