# Role of Angular Momentum in Defining Vector Operator ##\mathbf{V}##

• blue_leaf77
In summary, a vector operator ##\mathbf{V}## is defined by the property ##[V_i,J_j] = i\hbar \epsilon_{ijk}V_k## where ##\mathbf{J}## is an angular momentum operator. The role of ##\mathbf{J}## is to represent the total angular momentum in the system, including both orbital and spin angular momenta. This means that in the absence of spin, ##\mathbf{J}## is equal to the orbital angular momentum operator ##\mathbf{L}##, but in the presence of spin, it is equal to the sum of the orbital and spin angular momentum operators, ##\mathbf{L}+\
blue_leaf77
Homework Helper
A vector operator ##\mathbf{V}## is defined as one satisfying the following property:

## [V_i,J_j] = i\hbar \epsilon_{ijk}V_k##

where ##\mathbf{J}## is an angular momentum operator. My question is what is the role of ##\mathbf{J}##, does it have to be the total angular momentum from all angular momenta appearing in the system, i.e. if we don't consider spin then ##\mathbf{J}=\mathbf{L}##, if we consider it then ##\mathbf{J} = \mathbf{L}+\mathbf{S}##?

Yes, exactly.

But using L instead of the total angular momentum J even in the presence of spin doesn't prevent the relation ##[x_i,L_j] = i\hbar \epsilon_{ijk}x_k## to prevail.

Of course, but if you take V=S or V=J, you will need J=L+S to rotate also the spin part.

## 1. What is angular momentum?

Angular momentum is a physical quantity that describes the rotational motion of a system. It is defined as the product of an object's moment of inertia and its angular velocity.

## 2. How is angular momentum related to vector operators?

In quantum mechanics, angular momentum is represented by vector operators, denoted by ##\mathbf{L}## or ##\mathbf{J}##. These operators act on a quantum state to produce another state with a different angular momentum value.

## 3. What is the role of angular momentum in defining vector operator ##\mathbf{V}##?

The vector operator ##\mathbf{V}## represents the total angular momentum of a system. It is defined as the sum of the orbital angular momentum operator ##\mathbf{L}## and the spin operator ##\mathbf{S}##, which describes the intrinsic angular momentum of a particle.

## 4. How does angular momentum affect the behavior of particles?

Angular momentum is a conserved quantity, meaning it remains constant unless an external torque is applied. This has important implications for the behavior of particles, as it determines their motion and interactions within a system.

## 5. What are some real-world examples of the role of angular momentum in defining vector operator ##\mathbf{V}##?

One example is the behavior of spinning tops or gyroscopes, where the angular momentum of the object determines its stability and precession around a fixed axis. Another example is the quantization of angular momentum in atoms, which is essential for understanding atomic spectra and the behavior of electrons in orbitals.

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