Science Forums 2

[Bo]

your questions are covered by a mathematical branche called 'group theory', which is the study of symmetries. and indeed this is a very important subject for (advanced) quantum mechanics. But group theory is quite complicated.

A Lie group is a group whereof the product of group elements is differentiable. The nice property of this is that one can define a set of generators for the symmetries of the group.
An example in physics: a spin 1/2 particle. (no calculations, just the results, still this is not a course in group theory, but i assume sanctus has some knowledge).

spin, is a form angular momentum and satisfies a symmetry called SU(2).
spin 1/2 is a doublet representation of this groups, with elements: (+1/2, -1/2) (that is spin up and down)
now SU(2) is a Lie group, so we can calculate the generators of this group. and the nice thing is that for the doublet case these are exactly the pauli matrices (which you have probably encountered). so the Lie algebra of SU(2) is also the same as that of the pauli matrices. [Sx , Sy] = Sz
So the generators of the symmetry group are the same as the operators of the quantum mechanical property. This is of course extremely powerfull. We dont have to do QM, or explicitly calculate Schrodingers equation or whatever; just knowing the symmetry group contains all information.

I hope this clarifies something, but if you have some specific questions, please ask

Bo