Angular momentum operator

The operator $J_{-}=J_x-i{J}_y$ acts by $J_{-}|j,m⟩=\hslash \sqrt{(j+m)(j-m+1)}|j,m-1⟩$. It generates lower magnetic states and determines Clebsch-Gordan coefficients.

In an irreducible spin representation, the angular momentum lowering operator has squared norm factor $(I+m)(I-m+1)$. Multiplying these factors from the highest-weight vector gives the factorial normalization displayed above. All factorial arguments are integers even when $I$ is half-integral. This produces a consistent positive-coefficient phase convention for all weight states and fixes signs in subsequent isospin rotations.

A joint eigenstate $|j,m⟩$ of $J^2$ and $J_z$, with eigenvalues ${\hslash }^{2}j(j+1)$ and $\hslash m$. Its allowed labels satisfy $j\in \{0,1/2,1,\dots \}$ and $m=-j,-j+1,\dots ,j$.

Every angular momentum operator satisfies

For two subsystems, the total operator is $\mathbf{J}={\mathbf{J}}_1\otimes I+I\otimes {\mathbf{J}}_2$, and $J_{\pm }=J_{1\pm }+J_{2\pm }$. The coupled basis diagonalizes ${\mathbf{J}}^2$ and $J_z$.

For a composite quantum system, the total angular momentum operator is the sum of the constituent angular momenta, with each summand acting on its own tensor factor. Its simultaneous eigenstates with $J_z$ organize the state space into total-spin sectors.

A total-spin-$J$ sector is an irreducible subspace on which ${\mathbf{J}}^2$ has eigenvalue $J(J+1){\hslash }^2$. For two spin-one particles, Clebsch-Gordan decomposition gives sectors $J=0,1,2$ of dimensions one, three, and five.

Coupling three distinguishable spin-one systems giveswhere the labels denote total spin and coefficients denote multiplicities. The dimensions are respectively $1$, $3\cdot 3$, $2\cdot 5$, and $7$, summing to $27$.

The completely symmetric subspace of three spin-one systems has dimensionand decomposes into one total-spin-$3$ multiplet and one total-spin-$1$ multiplet:The highest-weight product state generates the seven-dimensional spin-$3$ multiplet; its three-dimensional invariant complement is spin $1$.

The spin-one Cartesian basis $|x⟩,|y⟩,|z⟩$ transforms as a three-dimensional vector under rotations. In its two-site tensor square, the scalar, antisymmetric, and symmetric traceless subspaces are respectively the total-spin $0$, $1$, and $2$ sectors.

For two spins of magnitude $s$, ${\mathbf{S}}_1\cdot {\mathbf{S}}_2=\frac{1}{2}({\mathbf{J}}^2-{\mathbf{S}}_1^2-{\mathbf{S}}_2^2)$. For two spin-one particles its eigenvalues in total-spin sectors $J=0,1,2$ are $-2,-1,1$ in units of ${\hslash }^2$.

The tensor product of spin-$j_1$ and spin-$j_2$ irreducible representations decomposes into one copy of each spinThe change-of-basis entries between uncoupled and coupled states are Clebsch--Gordan coefficients.

With $J=j_1+j_2$, the maximal state is the product $|J,J⟩=|j_1,j_1⟩|j_2,j_2⟩$. Lowering it once givesThe orthogonal combination is the highest-weight state $|J-1,J-1⟩$.

Two equal spins $j$ have the unique total-spin-zero stateIt is invariant under joint rotations.

For two spin-one-half particles, the singlet isEvery component of the total spin annihilates it, and ${\mathbf{S}}^2$ has eigenvalue zero.

The symmetric two-particle statesform the spin-one triplet. On this subspace, ${\mathbf{S}}^2$ has eigenvalue $1(1+1){\hslash }^2=2{\hslash }^2$.

Spin operators satisfy $[S_i,S_j]=i\hslash {\epsilon }_{ijk}{S}_k$ and are Hermitian traceless generators of rotations in finite-dimensional irreducible representations.

The Jordan–Wigner transformation represents a one-dimensional spin one-half chain by fermions with a parity string preceding each site. With $P_n={\prod }_{m<n}(1-2n_m)$, one has $S_n^{+}=P_n{c}_n$ and $S_n^z=1/2-n_n$. The string converts the off-site canonical anticommutation relations into commuting spin operators. Adjacent exchange terms become quadratic fermion expressions, while a periodic end bond depends on global fermion parity.

In units $\hslash =1$, spin one-half is the two-dimensional irreducible spin angular momentum representation with components $S_i={\sigma }_i/2$. The Pauli matrices give eigenvalues $\pm 1/2$ along every unit axis. The normalized $x$-axis states are $(|{\uparrow }_z⟩\pm |{\downarrow }_z⟩)/\sqrt{2}$.

For $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, the normalized eigenstates of $\mathbf{n}\cdot \mathbf{S}$ arewith eigenvalues $+\hslash /2$ and $-\hslash /2$.

The Holstein–Primakoff transformation represents a spin-$S$ operator algebra with a bosonic creation operator and annihilation operator:The order of the square root and oscillator operator matters. On the physical Fock states $|n⟩$, $0\le n\le 2S$, the raising and lowering matrix elements are $\sqrt{n(2S-n+1)}$ and $\sqrt{(n+1)(2S-n)}$. Their squared difference gives $[S^{+},S^{-}]|n⟩=2(S-n)|n⟩$, proving the spin commutation relations. Expanding the square root yields the linear spin-wave approximation and its interaction corrections.

A spin-$S$ Hilbert space has dimension $2S+1$. Its Holstein–Primakoff transformation therefore uses only bosonic occupation numbers $0,1,\dots ,2S$. The square root annihilates the upper endpoint. A truncated linear spin-wave approximation formally enlarges this space; it is self-consistent only when boson depletion is small compared with $S$.

The components of spin angular momentum satisfy $[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$. In units $\hslash =1$, the spin ladder operators obey $[S^{+},S^{-}]=2S^z$ and $[S^z,S^{\pm }]=\pm S^{\pm }$.

The spin-$s$ irreducible representation has dimension $2s+1$, with basis $|s,m⟩$ for $m=-s,-s+1,\dots ,s$ and Casimir eigenvalue ${\hslash }^{2}s(s+1)$.

In the ordered spin-one basis $(m=1,0,-1)$, the second angular momentum operator satisfies $J_2^3=J_2$. Therefore its matrix exponential reduces to $I-i\sin \theta J_2+(\cos \theta -1)J_2^2$. A half-turn exchanges the two outer weights and negates the middle weight. Applied to the pion isospin triplet, this fixes the rotation factor in G parity.

The operators $S_{\pm }=S_x\pm i{S}_y$ satisfy

The spin raising operator $S_{+}=S_x+i{S}_y$ increases the magnetic quantum number by one and annihilates the maximal-weight state.

The spin lowering operator $S_{-}=S_x-i{S}_y$ decreases the magnetic quantum number by one and annihilates the minimal-weight state.

In descending $S_z$ order, the spin-$3/2$ raising operator has superdiagonal entries $\hslash \sqrt{3},2\hslash ,\hslash \sqrt{3}$; $S_x$ and $S_y$ are its Hermitian real and imaginary combinations.

A spin coherent state is obtained by rotating the maximal-weight state $|s,s⟩$ so that it is an eigenstate of spin along a chosen direction with eigenvalue $s\hslash$.

A spin coherent-state path integral inserts group-averaged resolutions of the identity between short time steps. Its kinetic term is the Spin coherent-state Berry phase, equal to $S$ times the oriented solid angle swept on the sphere.

At azimuth $\phi$, the normalized spin-$3/2$ coherent-state components are proportional to $1,\sqrt{3}{e}^{i\phi },\sqrt{3}{e}^{2i\phi },e^{3i\phi }$.

Under $H=-\gamma B{S}_z$, a coherent spin direction precesses around the $z$-axis with angular velocity $-\gamma B$.

The propagator $e^{i\gamma Bt{S}_z/\hslash }$ multiplies each magnetic component $|s,m⟩$ by $e^{im\gamma Bt}$.

The Heisenberg model couples neighboring or more general pairs of spins through ${\mathbf{\sigma }}_i\cdot {\mathbf{\sigma }}_j$. For all-to-all equal coupling, the Hamiltonian can be written in terms of the square of the total spin.

A Heisenberg antiferromagnet has positive exchange coupling in $H=J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, so neighboring spins energetically prefer antiparallel alignment.

A bipartite spin rotation rotates the spin on one sublattice by $\pi$, converting an alternating classical Néel state to an all-up reference state. Rotation about $x$ sends $S^z\mapsto -S^z$ and $S^{\pm }\mapsto S^{\mp }$ on that sublattice. For an isotropic nearest-neighbor Heisenberg antiferromagnet, a bond becomes $-S_i^z{S}_j^z+(S_i^{+}{S}_j^{+}+S_i^{-}{S}_j^{-})/2$. The transformation is unitary and preserves the spin commutation relations; it changes the reference frame, not the model's spectrum.

A Heisenberg ferromagnet has negative exchange coupling in $H=J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, so neighboring spins energetically prefer parallel alignment.

A connected isotropic spin-$S$ ferromagnet with positive exchanges has a fully polarized ground state and a maximal-total-spin multiplet. For $N$ sites the total spin is $NS$, with $2NS+1$ magnetic states generated by total-spin lowering. Global rotations span the same degenerate space. Slowly varying reorientations produce gapless ferromagnetic magnons; finite-volume degeneracy must be distinguished from a thermodynamic statement about spontaneous order.

For two spin-one-half particles,has two eigenvalues $\alpha {\hslash }^2/4$ and two further eigenvaluesFor positive $\alpha$, the zero-field limit has a unique spin-one-half singlet state ground state and a triply degenerate spin-one-half triplet state excited level.

For $N$ spin-one-half systems and one term per unordered pair,Its minimum lies in the smallest available total-spin sector.

For a bipartite spin-one-half Heisenberg antiferromagnet with coordination $z\to \infty$, the two-sublattice mean-field state becomes exact. Antiparallel pure Bloch vectors give bond energy $-J/4$ for the convention $J{\mathbf{S}}_i\cdot {\mathbf{S}}_j$.