Okay so I see the question was bit unclear. I know about matrices as far as we have used them but we never used any group theory. Determinants like slater determinants are used in chemistry, too. So the link that was provided here was on my list already and has one of these examples that I don't understand:
So the wave function is a vector in a representation of Lie(SO(3)) = Lie(SU(2)). "Spin" is the label of precisely which representation this is. Note that while SO(3) and SU(2) share a Lie algebra, they are different as groups, and it is a fact of life ("the connection between spin and statistics") that some particles -- fermions, with half-integral spin -- transform under representations of SU(2) while others -- bosons, with integral spin -- transform under SO(3). Blockquote
So spins have their own group and transform with this group ? Because I thought that spin are like SO(3) or SU(2) but our wold is something different and in order to connect both the spin is used. But that seems to be wrong then. So I understand that spin is just a postulate and it will work but is there anything to proof spin where I could say without it this equation wouldn't work. I mean I can just postulate something to be there but there should be at least one thing where it's needed and without it it wouldn't make sense so the role of the spin would become obvious. Or to make it more clear: If I understood the basics of all these groups SO(3) and SU(2) would this make it clear to me why spin is there or can the spin just be demonstrated using these groups ?
So first I should probably introduce myself. I'm Martin and I study Chemistry. I've already had the lectures on theoretical chemistry and now the advanced physical chemistry lecture on spectrocopy and use the particle spin more and more often.
I've always liked particle physics but I'm not good in mathematics unfortunately nor do we learn any relativistic stuff. We had 2 semesters mathematics but never did any group theory and that's the basic problem here. I have no idea what an SO(3) Group means.
That is why I was checking this forum and other books like the Sakurai as well as many papers for good explanations and found some stuff that seemed logic to me and I wanted to ask if I got them correct or if there is any simple explanation. I know about the spin and spin operator and that it is not a rotation but if it would be it's a 720° rotation for an electron and spinors, .... We used spins quite often, now in spectrocopy but also with wave functions, quantum mechanics and so on. What I wanted to know is WHY there needs to be a spin.
So this are some facts I got from various sites which might be totally wrong. Note: I don't need any mathematical explanations, it doesn't need to be that perfect either but I need an example for some things I don't get. So let's start:
- I think this one came from this site here: Spin is a symmetry manifestation. So liked gauge theory (the principal idea behind it), how local disturbance on the gauge field cause a force like electromagnetic interaction, the mexican hat potentials and so on. I think symmetry is behind everything in relativstics and particle physics and is like the force that causes things to happen. Much like Entropy for us chemists but the opposite. And I always imagined things like Lorentz-Symmetry or Poincaré or all the other names often associated with this topic as follows:
We postulate something. We want to describe how the world behaves, how certain particles behave and therefore we need to make sure that it will always be on the same starting point. So all forces that are fundamental need to be symmetric all over space(time) they need to apply in every case and must not change or otherwise it wouldn't make sense to use them to describe anything. And on the other hand we expect the particles to behave symmetrically, too. Like if we want to describe the properties of an electron we need to make sure that the electron on Mars has the same physics as one on earth (considering that there is nothing special acting on one of them might change it's properties).
And as far as I have read the electron for example needs to be symmetric under let's say translation but also under rotation. And this rotational invariance is what tells us that we need a spin.
- Rotational Invariance and particle spins: So I remember a text about quaternions and how they behave and it said something like for a vector it's important if I apply a symmetry operation on it in what order I do this. The operators do not commute ! If I turn a vector around the z-axis for a certain degree first and then around the x-axis I get to a different direction as if I did it the other way around. And I think for quaternions this isn't the case making them quite useful but that's not important here.
So what I understood is that in a symmetrical world (is this SO(3) ???) we wouldn't have this problem we could commute all rotational operators but we life in a Hilbert Space (???) and there we have this problem in our 3D world. So in order to conserve the rotational invariance we need to add something like a commutator which is the Spin.
So I know quite much text and I'm really sorry but all explanations use mathematics and they all start with "for a ... to be ...". Well it doesn't help me if I can proof the existance with a long calculation. I would have to understand all those symmetry groups and matrix calculations. What I need to know is what exactly does one theory require (like SO(3)) for this so called "rotational invariance" and why don't we have it and how can the spin solve this. Doesn't need to be that precise. Just like I explained it. Something is wanted for some reason it doesn't happen and spin is the solution.
So would be really cool if someone could give me some hints and tell me how wrong my view on the whole topic acutally is.
Thank you in advance for all ideas, suggestions and replies and I'm sorry for my English, too but it's not my first language.