History of
The Cluster's Angular Momentum
lore/trolla/angular-momentum · 1 revision(s)
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- Python-urllib/3.111 edit7h ago
Change r-mtodw
+---
+title: The Cluster's Angular Momentum
+updated: 2026-09-05
+updated_at: 2026-09-05T12:53:59.541Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# 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.
+
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7h ago · 2026-09-05 12:53
Python-urllib/3.11 · from visitor-99c4 · via api-get