Are Spin and Quantum States Independent in Wave Functions?

[hokhani]

When we speak about wave function of an electron, we write it as ψ[itex]_{n,σ}[/itex] (x,ζ) so that we specify here the orbital quantum number by n and spin quantum number by σ. σ can take two values according to spin up or down. x is space position and ζ has two discrete values related to spin up and down.
Now my question:
Is it possible to have σ related to spin up and ζ related to spin down simultaneously? In other words are σ and ζ independent (like n and x that are independent)?

 

[dextercioby]

It's not correct, there's only a continuous (space, momentum, space-time, momentum-time) functional dependence of the electron/spin σ particle. * The "ζ" should be deleted, the spin character of the wavefunction appears only as a counting label on ψ, just like other labels (total angular momentum, electric charge, parity).

* Mathematically speaking ψ is a mapping from R³ (disregard time) to [itex] \otimes_{\sigma} L^2 (R^3) [/itex], where sigma takes 2s+1 = 2 (s=1/2) values in the Pauli theory and 2(2s+1) =4 (again s=1/2) values in the Dirac theory.

 

[hokhani]

Thanks Mr/Mis Kurt Lewin
I agree, but there is such a statement in the book "Nanostructures; Theory and modeling" by C. Delerue & M. Lannoo, chapter1, formula (1.6), that had made me confused.