Are O(3) and SO(3) Both Generators of Angular Momentum in Quantum Mechanics?

In summary, the generator of the O(3) group is angular momentum L and the generator of the SU(2) group is spin S. The SO(3) and SU(2) groups are defined in complex spaces and reflect symmetries in the non-relativistic world. However, spin, a relativistic effect, can still be explained using the SU(2) group through spin-orbit coupling.
  • #1
Gigi
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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?
 
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  • #2
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".
 
  • #3


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.
 

Related to Are O(3) and SO(3) Both Generators of Angular Momentum in Quantum Mechanics?

1. What is a group in quantum mechanics?

A group in quantum mechanics refers to a set of physical quantities or operators that obey certain mathematical rules or relationships. These rules are essential for understanding the behavior and interactions of particles at the quantum level.

2. How are groups used in quantum mechanics?

Groups are used in quantum mechanics to describe symmetries in physical systems and to classify particles based on their properties. They also help in solving complex problems and predicting the behavior of quantum systems.

3. What are the different types of groups in quantum mechanics?

There are several types of groups in quantum mechanics, such as symmetry groups, unitary groups, and permutation groups. These groups have different properties and are used for different purposes in quantum mechanics.

4. How are groups related to quantum entanglement?

Groups play a crucial role in understanding quantum entanglement, which is a phenomenon where two or more particles become connected in such a way that the state of one particle affects the state of the other. Groups help in describing the symmetries and interactions between entangled particles.

5. Can groups in quantum mechanics be applied in other fields?

Yes, the concept of groups in quantum mechanics has applications in various fields, such as particle physics, chemistry, and even cryptography. They provide a powerful mathematical framework for understanding the behavior of particles and systems at the quantum level, which has implications in many areas of science and technology.

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