Can't understand ket notation for spin 1/2

[pepediaz]

Homework Statement

Each particle can have spin up or spin down, so there are four possibilities.

Relevant Equations

It's from Example 4.5, from Griffiths (Quantum Mechanics)

I can't why there are four elements in each ket instead of only two

 

[nrqed]

Homework Statement:: Each particle can have spin up or spin down, so there are four possibilities.
Relevant Equations:: It's from Example 4.5, from Griffiths (Quantum Mechanics)

I can't why there are four elements in each ket instead of only two View attachment 279407

The notation is that the first two 1/2 just indicated the spin of each particle. The last two entries give the z component of the spin, $S_z$ of each particle.

 

[kuruman]

With electrons it is understood that the spin is 1/2 and is often omitted for compactness. In that case instead of $|\frac{1}{2},\frac{1}{2},s_{1z},s_{2z}\rangle$ one can write $|s_{1z},s_{2z}\rangle$ where $s_{1z},s_{2z}=\pm \frac{1}{2}$ or use $\uparrow\downarrow$ as Griffiths does.

If the two spins, or more generally angular momenta, are greater than 1/2, one must be clear about what one means and write the full-blown $|S_1,S_2,S_{1z},S_{2z}\rangle$ to avoid confusion.

 

[pepediaz]

Thanks! I see it now!

 

[PeroK]

Thanks! I see it now!

In the later edition he abandons that notation, in fact. Instead, he has:
$$\uparrow \downarrow \ = \ |\frac 1 2 \frac 1 2 \rangle |\frac 1 2 -\frac 1 2 \rangle $$