# Angular momentum theory problem probably wrong sign

• lgnr
The correct statement would be \left\langle lm | \vec{\hat{r}} \times \vec{\hat{p}} | lm\right \rangle = -\left\langle lm | \vec{\hat{p}} \times \vec{\hat{r}} | lm \right \rangle. In summary, the conversation discusses the need to prove the equality between two expressions involving the angular momentum operator and its components. The speaker explains their attempts to solve the problem and suspects that there may be a wrong sign in the problem statement. They request the listener's opinion and thank them in advance for any advice.
lgnr
I have to prove that $\left\langle lm | \vec{\hat{r}} \times \vec{\hat{p}} | lm\right \rangle = \left\langle lm | \vec{\hat{p}} \times \vec{\hat{r}} | lm \right \rangle$, where $| lm \rangle$ are eigenkets of angular momentum operator $\hat{L}^2$

And I can't figure out a way to do this correctly. I wrote the angular momentum operator-vector in terms of its components, $\hat{L_x}$, $\hat{L_y}$ and $\hat{L_z}$, and only the $\hat{k}$ component survives the bra-ket operation, because I can write $\vec{L_x}$ y $\vec{L_y}$ in terms of ladder operators, and after lowering and rising the eigenstates, the corresponding eigenkets (except for the $\hat{k}$ component) cancel with $\langle lm |$ because of the orthogonality property of these eigenkets. Probably there is a wrong sign in the problem statement. Looks trivial in that case, but I want to know your opinion.

Undoubtedly the wrong sign.

## 1. What is Angular Momentum Theory?

Angular Momentum Theory is a concept in physics that describes the rotational motion of an object. It is based on the principle of conservation of angular momentum, which states that the total angular momentum of a system remains constant unless acted upon by an external torque.

## 2. What is the problem with the Angular Momentum Theory?

The problem with the Angular Momentum Theory is that it often yields incorrect results due to the sign convention used to define angular momentum. This sign convention can be confusing and lead to incorrect interpretations of the theory.

## 3. How can the sign convention affect the results of the Angular Momentum Theory?

The sign convention used in the Angular Momentum Theory can affect the direction of the angular momentum vector, which in turn can affect the calculated values for angular momentum. This can result in incorrect predictions or explanations of rotational motion.

## 4. How can we avoid issues with the sign convention in the Angular Momentum Theory?

To avoid issues with the sign convention, it is important to carefully define the direction of angular momentum and consistently use the same sign convention throughout calculations and interpretations. It may also be helpful to double-check calculations and ask for clarification if unsure.

## 5. Are there any alternative theories to the Angular Momentum Theory?

Yes, there are alternative theories to the Angular Momentum Theory, such as the Lagrangian and Hamiltonian formulations of classical mechanics. These theories use different mathematical approaches and may provide a better understanding of certain rotational systems.

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