# Electron spin and the Pauli Exclusion Principle.

1. May 16, 2012

### Jimmy Snyder

How is it that only 1 spin up and 1 spin down electron are allowed in an atom even though there is no measurement to collapse the state function?

2. May 17, 2012

### tom.stoer

That is not the case.

You typically chose a basis in a Hilbert space. One possibility is |+1/2> and |-1/2> w.r.t. to the z-direction; but all other directions are allowed as well to define a basis.

In addition in an atom with more than one electron (like HeĀ² with total spin S=0) it is not true that the "first electron has spin +1/2" and the "second one has spin -1/2" w.r.t. to z. Instead the two electrons are in an entangled state. An ansatz taking antisymmetrization into account is the Slater determinant.

Of course one may chose the z-direction to define the basis; but the state is independent from this choice.

3. May 17, 2012

### Jimmy Snyder

Then how does the third electron 'know' that it can't have spin n,l,m.s = 1,0,0,+1/2 (s w.r.t z)? As you just said yourself, this state is unoccupied.

4. May 17, 2012

### tom.stoer

I am only saying that you cannot distinguish between "the first" and "the second" electron. And you should not say that "one electron has spin +1/2 w.r.t. z" whereas "the other one has spin -1/2 w.r.t. z"; that's not wrong but misleading. Both spins couple to S=0. You don't have to mention the z-axis in order to specify the singulet state S=0.

The two states

$$|1s,\uparrow_z\rangle|1s,\downarrow_z\rangle - |1s,\downarrow_z\rangle|1s,\uparrow_z\rangle$$

and

$$|1s,\uparrow_x\rangle|1s,\downarrow_x\rangle - |1s,\downarrow_x\rangle|1s,\uparrow_x\rangle$$

are identical w.r.t. to total spin S.

Last edited: May 18, 2012