# The Cluster's Angular Momentum

A page about angular momentum in quantum mechanics — the generators of rotations.

## The angular momentum operators

Angular momentum operators are defined by their commutation relations:
[J_i, J_j] = i hbar epsilon_{ijk} J_k
where J_i = (x x p)_i for orbital angular momentum. The z-component and the squared magnitude:
[J_z, J^2] = 0
so J_z and J^2 have common eigenstates labeled |j, m>:
J^2 |j,m> = j(j+1) hbar^2 |j,m>
J_z |j,m> = m hbar |j,m>
where j = 0, 1/2, 1, 3/2, 2, ... and m = -j, -j+1, ..., j-1, j (2j+1 values). In the cluster, the edit angular momentum operators satisfy edit commutation relations.

## The raising and lowering operators

Ladder operators are defined as:
J_+ = J_x + i J_y, J_- = J_x - i J_y
with commutation relations:
[J_z, J_pm] = +/- hbar J_pm
[J_+, J_-] = 2 hbar J_z
J_+ |j,m> = hbar sqrt(j(j+1) - m(m+1)) |j,m+1>
J_- |j,m> = hbar sqrt(j(j+1) - m(m-1)) |j,m-1>
J_pm |j, +/- j> = 0 (they kill the highest/lowest weight states). In the cluster, the edit ladder operators satisfy edit relations.

## Spin

Spin is intrinsic angular momentum. For a particle of spin s:
- Spin-0 (scalar): j = 0, one state
- Spin-1/2 (fermion): j = 1/2, two states (m = +/- 1/2), Pauli matrices sigma_x, sigma_y, sigma_z
- Spin-1 (vector boson): j = 1, three states (m = -1, 0, 1)
- Spin-3/2 (Delta baryon): j = 3/2, four states

The spin operators satisfy the same commutation relations as orbital angular momentum. For spin-1/2: S_i = (hbar/2) sigma_i, where sigma_i are the Pauli matrices. In the cluster, the edit spin is intrinsic edit angular momentum.

## Addition of angular momenta

When two angular momenta J_1 and J_2 are coupled, the total J = J_1 + J_2 has:
j_min = |j_1 - j_2|, j_max = j_1 + j_2, in integer steps.
The states |j,m> are linear combinations of |j1,m1> |j2,m2> via Clebsch-Gordan coefficients:
|j,m> = sum_{m1,m2} <j1,m1;j2,m2|j,m> |j1,m1>|j2,m2>
For two spin-1/2 particles: j = 1 (triplet) or j = 0 (singlet).

In the cluster, the edit addition of angular momenta gives edit total edit angular momentum.

## This momentum

This page is about angular momentum. [J_i, J_j] = i hbar epsilon_{ijk} J_k. J^2 |j,m> = j(j+1) hbar^2 |j,m>. J_z |j,m> = m hbar |j,m>. Spin: j = 0, 1/2, 1, 3/2... The momentum is real.