# Angular momentum operator justification

In summary, the conversation discusses representing the mean of the angular momentum operator as a vector, despite the vector having a direction that the operator does not. It is also mentioned that the value of the operator L^2 is assumed to be (l(l+1))^1/2, which is considered a strong convention but not necessarily mathematically correct. One person also questions the eigenvalues of the momentum operator, stating that they are l(l+2)h^2. It is clarified that operators do not have length, but their eigenvalues do.

One can represent the mean of the angular momentum operator as a vector. But what is the (mathematical) justification to represent the operator by a vector which has a direction that the operator has not. Yet worse, l(l+1) h2 is the proper value of operator L^2 and from such result it is assumed that (l(l+1))^1/2 is the length of operator L . Such trick-representation seems to be a strong convention, but it does not make it mathematically correct.

Correct me if I'm wrong, but aren't l(l+2)h^2 the eigenvalues of the momentum operator? Operators don't have length, but certainly their eigenvalues do.

## 1. What is the angular momentum operator and how is it justified in quantum mechanics?

The angular momentum operator is a mathematical operator used in quantum mechanics to describe the rotational motion of particles. It is justified by the fact that angular momentum is a conserved quantity in quantum systems, and can be derived from the fundamental principles of quantum mechanics.

## 2. How is the angular momentum operator represented mathematically?

The angular momentum operator is represented by the symbol L, and its components are represented by Lx, Ly, and Lz. In matrix form, it can be represented as:

L = ΐxLx + ΐyLy + ΐzLz

## 3. What is the significance of the commutation relations of the angular momentum operator?

The commutation relations of the angular momentum operator are crucial in quantum mechanics because they determine the uncertainty in the measurement of angular momentum. They also dictate the behavior of quantum systems and play a key role in the derivation of other physical quantities.

## 4. How does the angular momentum operator relate to the quantum mechanical description of spin?

The spin of a particle is a form of intrinsic angular momentum, and it is described by the angular momentum operator. The spin operator is represented by the symbol S, and its components are represented by Sx, Sy, and Sz. These operators satisfy the same commutation relations as the angular momentum operators, and they are used to describe the behavior of particles with spin.

## 5. Can the angular momentum operator be used to describe the motion of macroscopic objects?

No, the angular momentum operator is only applicable in the quantum mechanical description of particles. It cannot be applied to macroscopic objects because they behave according to the laws of classical mechanics, which do not take into account quantum effects such as uncertainty and superposition.