# Commutation relation of angular momentum

• p.p
In summary, the conversation discusses the commutation relations for the angular momentum operator of a system defined as L = R x P. The conversation concludes that the commutator of L and the scalar product of P and R is zero, as expected from the definition of a vector operator.
p.p
Just wondering if any1 would by any chance know how to calculate the commutation relations when L = R x P is the angular momentum operator of a system?
[Li, P.R]=?

P

Is it the dot product?
You can write it down as:
[Li,PiRi] (using Einstein summation notation).
and Li=Rj Pk ejki, where ejik is levi-civita tensor.
and using the basic relation for [R,P]=1 (units of hbar).

$$[L_i,p_ir_i]=p_i[L_i,r_i]+[L_i,p_i]r_i$$

Inserting:
$$L_i=\epsilon_{ijk}r_j p_k$$
Gives:

$$[L_i,p_ir_i]=\epsilon_{ijk}p_i[r_j p_k,r_i]+\epsilon_{ijk}[r_j p_k,p_i]r_i$$
$$=\epsilon_{ijk}p_i r_j[p_k,r_i]+\epsilon_{ijk}[r_j,p_i]r_i p_k$$
$$=-i\hbar\epsilon_{ijk}p_i r_j\delta_{ik}+i\hbar\epsilon_{ijk}\delta_{ij} r_i p_k$$
$$=0$$

Last edited:
The fact that the commutator of the angular momentum and the scalar product of p and r is zero follows naturally. The definition of a vector operator applies for both p and r individually, thus their scalar product should be a scalar operator, therefore the desired commutator is zero.

## What is the commutation relation of angular momentum?

The commutation relation of angular momentum is a mathematical relationship that describes how two operators representing different measurements of angular momentum behave when applied to the same quantum state. In quantum mechanics, operators do not always commute, meaning the order in which they are applied can affect the outcome. The commutation relation of angular momentum helps us understand how the operators for angular momentum behave in this context.

## Why is the commutation relation of angular momentum important?

The commutation relation of angular momentum is important because it helps us understand the fundamental properties of angular momentum in quantum systems. It also allows us to make predictions about the behavior of quantum systems and calculate the uncertainties in measurements of angular momentum.

## How is the commutation relation of angular momentum derived?

The commutation relation of angular momentum is derived from the canonical commutation relation, which describes the behavior of operators in quantum mechanics. It involves the cross product of the operators for angular momentum and their commutator, which is a mathematical operation that tells us how much the order of operators affects the outcome.

## What does the commutation relation of angular momentum tell us about the uncertainty principle?

The commutation relation of angular momentum is related to the uncertainty principle, which states that certain properties of particles, such as position and momentum, cannot be known simultaneously with precision. The commutation relation for angular momentum tells us that the more precisely we measure one component of angular momentum, the less precisely we can know another component, due to the non-commutative nature of the operators.

## How does the commutation relation of angular momentum affect the behavior of quantum systems?

The commutation relation of angular momentum affects the behavior of quantum systems by limiting our ability to know certain properties with precision. It also affects the way angular momentum is conserved in quantum systems, as the non-commutative nature of the operators means that angular momentum can only be measured in discrete values rather than continuously. Understanding the commutation relation is crucial for accurately describing and predicting the behavior of quantum systems.

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