Pauli exclusion principle and parallel spin.

[Silversonic]

Homework Statement

It's not a homework question. It's a piece of my textbook I don't understand.

Here's what it says

In a two electron atom, taking the orbital states of two electrons to be the same, then the antisymmetric wavefunction tends to zero, as well as the quantum numbers n, l and m(l) are the same for the electrons. Therefore, the possible spin wavefunctions are;

\chi_{a}, S = 0 (m_{s} = 1/2 for one electron and m_{s} = -1/2 for the other.\chi_{s}, S = 1 (electrons have the same (parallel) spin, i.e. both m_{s} = 1/2 or -1/2), Since this combines with the antisymmetric orbital wavefunction which is zero, the total combined wavefunction of the orbital does not exist.

So we must have that an electron cannot exist with all the same four quantum numbers.

What I don't under is the bolded bit. It says that for S = 1 the electrons have the same spin. But surely this is untrue, it's possible to have m_{s} = 1/2 for one electron and m_{s} = -1/2 for another, but S = 1 still, as shown by this diagram;

http://img21.imageshack.us/img21/4033/12986325.png

i.e the M_{s} = 0 in the middle (capitalised M) gives a state with S = 1 but m_{s} = 1/2 for one and -1/2 for the other. Admittedly it doesn't matter since it combines with a 0 antisymmetric wavefunction, but still.

This same concept that I don't understand is used in the description of orthohelium. I'm not sure if orthohelium is defined as a helium atom where two electrons have the same spin, because as far as I see and anti-symmetric orbital wavefunction paired with a symmetric spin wavefunction (S=1) (which is orthohelium's wavefunction) can have electrons with different m_{s}.

 

[vela]

Your textbook is misleading, if not wrong. I think what it meant to say is that:

1. If the spins are parallel, the electrons are in a symmetric total spin state.
2. If the electron spins are antiparallel, then it's possible the electrons are in the antisymmetric total spin state.

The converse of (1) isn't true, as you noted, and the book unfortunately makes it sound like antiparallel spins imply an antisymmetric state, which isn't correct either.

The consequence of (1) is that the spins of the electrons can't be parallel they have the same n, l, and m_l quantum numbers. However, this doesn't necessarily mean that states with antiparallel spin exist. If all the antiparallel states were also symmetric, then you'd have to conclude there are no acceptable states where the two electrons have the same spatial quantum numbers.

Statement (2), however, says that an antisymmetric spin state does exist, so it is possible for an atom to have two electrons with the same spatial wave function. In that case, the spins must be antiparallel.