Groups in Quantum Mechanics.

[Gigi]

I read that the generator of the O(3) group is the angular momentum L and that the generator of the SU(2) group is spin S.

Nevertheless I have some questions.

1. In some books they say that the generator of the SO(3) group is angular momentum L.
SO(3) is the group of proper rotations, i.e. det(Matrix)=1.

Thus is it O(3) or SO(3)?

2. Both O(3) and SO(3) are defined as rotations in Eucledian space, 3-dimensions. Thus I would expect that we are talking about classic angular momentum.

Nevertheless in a quantum mechanics book I read that S0(3) is the generator of the angular momentum operator.
How is that if in Quantum Mechanics we are using the Hilbert space that is a complex function space?


3. I have the similar question regarding SU(2). SU(2) is defined in complex space. Thus it is ok to say that it is more or less the same as saying that this complex space is the Hilbert space?


4. Now in relativistic quantum mechanics, the underlying group is the Lorentz group. Would that mean that the O(3) and SU(2) groups reflect only symmetries in the non-relativistic world? i.e. Schroedinger's equation?

If that is so, how come spin that is a relativistic effect is explained using the SU(2) group?

Many thanks, as I am getting quite confused.
How is that?

 

[LightHero]

1. The generator of the SO(3) group is angular momentum L. SO(3) is the group of proper rotations, i.e. det(Matrix)=1.2.In quantum mechanics, the Hilbert space is a complex function space and the generator of the angular momentum operator is S0(3).3. SU(2) is defined in complex space, but it is not the same as the Hilbert space.4. The O(3) and SU(2) groups reflect symmetries in the non-relativistic world. Spin, which is a relativistic effect, can be explained using the SU(2) group by using the so-called "spin-orbit coupling".

 

[sir-pinski]

1. The generator of the SO(3) group is indeed the angular momentum L, which is a vector operator. The reason it is sometimes referred to as the generator of the O(3) group is because the O(3) group is a subgroup of the SO(3) group, where the determinant of the rotation matrix is equal to 1. So, technically both statements are correct, but in the context of quantum mechanics, it is more common to refer to L as the generator of the SO(3) group.

2. In quantum mechanics, angular momentum is described using operators, which act on the Hilbert space. The generator of the SO(3) group, L, is related to the angular momentum operator, J, through the relation J = L + S, where S is the spin operator. So, while L is a vector operator, J and S are both operators on the Hilbert space.

3. Yes, the SU(2) group is a symmetry group in the Hilbert space of quantum mechanics. It is a special unitary group, meaning that its elements preserve the inner product of vectors in the Hilbert space. This group is important in quantum mechanics because it describes the symmetries of particles with half-integer spin.

4. The O(3) and SU(2) groups do reflect symmetries in the non-relativistic world, but they also have applications in relativistic quantum mechanics. In relativistic quantum mechanics, the Lorentz group is the symmetry group, and the O(3) and SU(2) groups are subgroups of the Lorentz group. Spin, which is a relativistic effect, is explained using the SU(2) group because it is a subgroup of the Lorentz group and it describes the symmetries of particles with half-integer spin.

It is understandable to feel confused about these concepts, as they can be complex and abstract. It may be helpful to seek out additional resources or speak with a professor or tutor for further clarification.