Question

I am kind of confused about the operator for spin, all the ones Ive seen are written in terms of matrix, but this form can not be applied to things like is orbital of a hydrogen atom, which is e^-(r/a). Or is there actually a way to multiply matirx with functions that Im not aware of

Answer 1

Well, how would you apply a spin operator to what seems to be a part of radial solution for the position of the electron?

If you want to apply the spin operator you need to represent the state of the particle over/including the spin eigenvectors. Right now in Dirac's notation all you have is:
|n> and what you need to have is |nlms>.

Post the actual task and I'll be able to help you more. I'm not overly sure what you're asking about.

Answer 2

So basically my questions is that does spin absolutely have to operate on dirac notation. Is there a way to extract the spin out of a wave function, not a wave vector

Answer 3

@ljetibo

Answer 4

Spin doesn't have to act just on bra or kets. Matrices are just representations of operators in a chosen basis. I figure you're asking if spin is ever expressed as an analytical function.

I've never seen a function like that, but it's not impossible. The question becomes how practical would it be to have it. Unlike operators of position or impulse, spin has very discrete end values. I.e. all positions are allowed, but only some have high enough probability of being possible so the space is restricted. The edges between possible and forbidden areas are smooth so it makes sense to describe that with a analytical function. Spin only has + or -.

Even when compared to let's say, angular momentum spin is "discreeter". Angular momentum, as you hopefully know, also has discrete solutions. The solutions for L^2 are something like hbar*l(l+1). But with l being from 0 to n. As n gets bigger there are more and more solutions possible, up to infinity as n-> infinity. Spin still has + or -.

In many ways, whether you write bra or kets or just act directly on wavefunctions, it's just easier to just write the eigenvalues out because you know you just can do it. To see examples check out total angular momenta, zeeman effect (spin and angular mometum coupling), fine and hyperfine structures. You operate with S on Psi in there but never write out S as an analytic function.

I'm not sure what analytical function would be able to generate correct equations for spin in any rotation and for any spin number (remember that it's only for electron that the solutions are +-1/2, for multi particle systems you can have 3/2, 5/2...) apart from constructing the operators over the ladder operator. For the simple case of +-1/2 you could use kronecker delta to denote the |+> and |->, probably. You would need higher order tensors, like levi-civita the more basis states you can have.

Answer 5

Maybe someone here has met a function like that, something that would be able to get the discrete solutions for general case of orientation and spin number like the spherical harmonics give you for angular momentum.