Angular momentum operator

The chosen direction for specifying an angular-momentum projection quantum number. An applied field can provide a physically distinguished axis; polarization labels and magnetic-sublevel selection rules are defined relative to it.

A common eigenstate of $J_x$ and $J_y$ must satisfy $J_z\psi =0$ by their commutator. Applying the other angular-momentum commutators then forces both remaining eigenvalues to vanish. Thus the state lies in a spin-zero sector. Noncommuting components do not possess a general common eigenbasis, but spin-zero states are an important exception to the claim that they can never share any eigenstate.

The axis and angle of a product of rotations are properties of the group elements, independent of which spin representation acts on a state. Higher-spin generators do not obey the two-term Pauli exponential, but representations preserve the resulting rotation product. The central $2\pi$ rotation acts as $(-1{)}^{2S}I$.

The normalized periodic state $\psi =\sqrt{8/(3\pi )}{\sin }^2\varphi$ on $0<\varphi <\pi$, zero on the remaining semicircle, has $⟨L_z⟩=0$ and $\mathrm{Var}(L_z)=4{\hslash }^2/3$. Projection on $e^{i\varphi }/\sqrt{2\pi }$ gives probability $64/(27{\pi }^2)$ for $L_z=\hslash$. Its first derivative is continuous at the joins, so integration by parts has no boundary contribution.

For a normalized periodic angular wavefunction, $⟨L_z⟩=-i\hslash {\int }_0^{2\pi }{\psi }^{\ast }{\partial }_{\varphi }\psi d\varphi$. A real periodic wavefunction with matching boundary values has zero mean, even though nonzero angular momenta can have positive measurement probabilities.

A rank-$k$ spherical tensor operator has components $T_q^{(k)}$, $-k\le q\le k$, that transform under rotation like an angular momentum $k$. Equivalently, their commutators with the angular momentum operator satisfyFor a vector operator, the rank is one and a conventional choice is $T_0=V_z$, $T_{\pm 1}=\mp (V_x\pm i{V}_y)/\sqrt{2}$. The Wigner-Eckart theorem separates the matrix elements of a spherical tensor operator into Clebsch-Gordan coefficients and a reduced matrix element.

A distribution of quantum state populations over atomic magnetic sublevels with a nonzero expectation value of the angular momentum operator vector.

Each orbital angular momentum operator annihilates a radial factor and $r^2$. Direct differentiation gives $L^2{x}_3^2={\hslash }^2(6x_3^2-2r^2)$ and $L_3{x}_3^2=0$. Hence the displayed trace-free quadratic polynomial, times any admissible radial factor, has orbital labels $l=2,m=0$. Its angular dependence is proportional to the spherical harmonic $Y_{20}$; subtracting $r^2/3$ removes the rotational scalar part.

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 dimensionless angular momentum operators and a Hamiltonian $H=a\mathbf{I}\cdot \mathbf{J}$ at fixed $I,J$, addition of angular momentum gives $E_F=a[F(F+1)-I(I+1)-J(J+1)]/2$. Subtracting neighbouring energy levels proves the displayed rule. For fine structure substitute $\mathbf{L},\mathbf{S}$ and total $J$; for magnetic-dipole hyperfine structure use electronic $J$, nuclear spin $I$ and total $F$. Electric-quadrupole interactions and mixing with nearby energy levels can violate the rule.

Consecutive positive gaps obey ${\nu }_F=|A|F/h$ for an isolated hyperfine multiplet governed by the magnetic-dipole hyperfine interaction. Their common difference estimates $|A|/h$, and the first gap divided by that scale identifies the first higher F. The endpoints of the resulting F ladder constrain I and J. Gap magnitudes alone do not fix the sign of A or an absolute nuclear g-factor.

Let $d_M$ count states of fixed angular momentum projection $M$ in a finite rotational representation, and $n_J$ its spin-$J$ multiplicity. Every spin-$J$ irreducible representation contributes one state at each $M=-J,\dots ,J$. Thus $d_M={\sum }_{J\ge M}{n}_J$ for nonnegative $M$, and subtraction gives $n_J=d_J-d_{J+1}$. Counting occupation states first incorporates Bose-Einstein statistics before decomposing identical-boson angular states.

Exchanging the factors in the Clebsch-Gordan decomposition of two equal spins gives the phase $(-1{)}^{2s-j}$. This follows from the exchange symmetry of the Clebsch-Gordan coefficients. The total exchange parity includes the spatial state and any other internal factors: the spin parity alone does not impose the full symmetrization postulate. For two half-integer spins, even $j$ is spin-antisymmetric and odd $j$ spin-symmetric.

For $N$ identical fermions restricted to one angular-momentum multiplet $j$, with all other one-particle labels fixed, the Pauli exclusion principle permits distinct angular momentum projections $j,j-1,\dots ,j-N+1$. Their sum is $Nj-N(N-1)/2$, provided $N\le 2j+1$. The resulting Slater determinant is killed by the total raising angular momentum ladder operator because raising any occupied state either exceeds $j$ or duplicates an occupied state. It is therefore a highest-weight vector of precisely this total angular momentum. The formula is invariant under replacing particles by holes in the full shell.

Coupling orbital angular momentum $L>0$ to spin one-half produces $J=L\pm 1/2$. The dimensions add as $(2L+2)+2L=2(2L+1)$. A change between coupled and product bases changes labels, not the dimension of the Hilbert space.

For three distinguishable spin one-half systems, the Clebsch-Gordan decomposition first combines two spins into $0\oplus 1$, then couples the third: $0\otimes 1/2=1/2$ and $1\otimes 1/2=1/2\oplus 3/2$. Thus the tensor product has one quartet and two doublets. A Hamiltonian operator depending only on the total $S^2$ and $S_z$ acts identically on the two doublet copies. In particular $H={ϵ}_t{S}^2/{\hslash }^2+{ϵ}_z{S}_z/\hslash$ has energy eigenvalues ${ϵ}_{t}S(S+1)+{ϵ}_{z}M$, generically once for each quartet projection and twice for each doublet projection. Equal numerical energies have their multiplicities added.

An atomic addition of angular momentum scheme in which the individual orbital angular momentum vectors first combine into $\mathbf{L}$ and the individual spins into $\mathbf{S}$, then these combine into total angular momentum $\mathbf{J}$. When spin-orbit coupling is weak compared with the interactions between electrons described by electrostatics, $L$ and $S$ remain useful approximate quantum numbers. The Landé g-factor describes the resulting weak-field magnetic splitting.

For electrons in distinct radial orbitals, angular coupling permits all $L$ from $|l_1-l_2|$ to $l_1+l_2$ and both total spins $S=0,1$. A p-s pair therefore gives singlet and triplet P terms. A nonequivalent p-p pair gives singlet and triplet S, P and D terms. The equivalent-subshell exclusion restrictions do not remove these nonequivalent terms. Counting $(2S+1)(2L+1)$ gives 12 p-s and 36 p-p microstates, matching products of one-electron state counts.

Two equivalent $l=1$ electrons require an antisymmetric complete state. Orbital exchange has parity $(-1{)}^L$ and the spin singlet is antisymmetric, giving allowed $L+S$ even. The terms are ${}^3{P}_{0,1,2}$, ${}^1{D}_2$ and ${}^1{S}_0$, with $9+5+1=15$ states, equal to $\left(\genfrac{}{}{0ex}{}{6}{2}\right)$. Hund ordering puts ${}^3{P}_0$ lowest for the less-than-half-filled configuration.

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 a spin-one Hamiltonian $H=A{S}_z^2+B(S_x^2-S_y^2)$, Cartesian basis energies are ${\hslash }^2(A-B)$, ${\hslash }^2(A+B)$ and zero along the $x,y,z$ basis labels. The $S_x=\hslash$ state combines the latter two basis vectors equally. Their relative phase produces the displayed spin-reversal probability. Complete reversal occurs at odd multiples of $\pi /(|A+B|\hslash )$ if $A+B\ne 0$; when it vanishes those two components remain degenerate. Individual matrix signs depend on basis conventions, while the commutators and physical probability must remain consistent.

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 ${\hslash }^2$.

For $s_1=s_2=3/2$, angular momentum addition gives one copy of $j=0,1,2,3$. The identity $2{\mathbf{S}}_1\cdot {\mathbf{S}}_2={\mathbf{J}}^2-{\mathbf{S}}_1^2-{\mathbf{S}}_2^2$ gives the four eigenvalues in the title, with state degeneracy $1,3,5,7$. These exhaust the $4\times 4$ spin space. If the particles are identical fermions, an appropriate spatial exchange parity accommodates each sector; only a specified spatial restriction can remove sectors.

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.

Put $s=j_1+j_2$, with both $j_i>0$. At magnetic quantum number $-s+1$, let $|a⟩$ raise only the first factor of the lowest product state and $|b⟩$ raise only the second. Applying total $J_{+}$ to the lowest product gives the normalized $J=s$ combination $\sqrt{j_1/s}|a⟩+\sqrt{j_2/s}|b⟩$. Its orthogonal combination is annihilated by total $J_{-}$ and therefore has $J=s-1$. This obtains a useful Clebsch-Gordan coefficients without a table.

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.

A singlet state is the one-dimensional rotation representation; this construction supplies the angular-momentum-zero case for two equal spins.

In the product basis $|m_1,m_2⟩$ for two spin-one particles, total magnetic quantum number zero allows only $|1,-1⟩$, $|0,0⟩$ and $|-1,1⟩$. Applying the total ladder operators to $\alpha |1,-1⟩+\beta |0,0⟩+\gamma |-1,1⟩$ givesA singlet state is annihilated by $S_{+}$ and $S_{-}$, so $\alpha =\gamma =-\beta$. Wavefunction normalization gives the displayed state, up to an overall phase. Here the ket on the left of the title formula has total-spin labels, while its right-hand kets have individual magnetic labels. The state is symmetric under exchanging the particles, although no exchange constraint is needed for distinguishable particles.

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 uncertainty of a spin projection is measured by its variance. In a maximally polarized spin coherent state, the longitudinal variance is zero and each transverse variance is $S/2$ in units $\hslash =1$. The transverse fluctuations do not imply finite magnon occupation in an isotropic ferromagnet's polarized vacuum.

A classical spin is a fixed-length vector representing the orientation of a local magnetic moment. A classical limit of a large quantum spin replaces its components by this vector and minimizes the resulting energy. Its length is conventionally $S$ at leading order, whereas the exact quantum squared-spin eigenvalue is $S(S+1)$ in units $\hslash =1$.

The total spin is the vector sum of subsystem spin operators and generates their simultaneous rotation. For a rotationally invariant Heisenberg model, both its squared magnitude and any chosen component commute with the Hamiltonian. An anisotropic exchange or field can preserve only a component.

A spin rotation acts on a quantum state through the unitary representation generated by its angular momentum operators. With $\hslash =1$, rotation by angle $\theta$ about axis $\hat{\mathbf{n}}$ is $e^{-i\theta \hat{\mathbf{n}}\cdot \mathbf{S}}$. Simultaneously rotating every site is a global symmetry of an isotropic Heisenberg model.

If the Hamiltonian commutes with total $S^z$, simultaneous spin rotations $e^{-i\varphi S_{\mathrm{tot}}^z}$ form a circle group symmetry. In a state of definite total $S^z$, the transverse one-point averages vanish because $S^x\pm i{S}^y$ change that eigenvalue. A chosen classical orientation or symmetry-breaking limit requires separate care when constructing a linear spin-wave approximation.

A spin operator describes intrinsic angular momentum. Its components obey angular-momentum commutators; for spin one half, ${\hat{S}}_j=\hslash {\sigma }_j/2$ in terms of the Pauli matrices.

For a spin-half particle with $H=\alpha \mathbf{B}\cdot \mathbf{S}$ and $\mathbf{S}=\hslash \mathbf{\sigma }/2$, exponentiating the Pauli matrices gives a rotation about the field axis. A field along $y$ evolves initial $z$-up spin to $(\cos (\alpha Bt/2),\sin (\alpha Bt/2){)}^T$. The sign of $\alpha$ sets the precession direction. spin parallel to the field is conserved even while transverse components vary.

For a state initially $z$-up and precessing about $y$, measurement along $\mathbf{n}=(0,\sin \theta ,\cos \theta )$ has the displayed probabilities. eigenvectors can be chosen as $(\cos (\theta /2),i\sin (\theta /2){)}^T$ and $(i\sin (\theta /2),\cos (\theta /2){)}^T$. Their phases must be retained when taking inner products. Full $2\pi$ rotation changes their signs but leaves isolated-state probabilities unchanged.

For spin one-half, the Pauli representation makes every diagonal quadratic component a multiple of the identity. Thus $A{S}_z^2+B(S_x^2-S_y^2)=A{\hslash }^{2}I/4$ cannot split the doublet or cause spin reversal. This conclusion is special to this quadratic Hamiltonian and does not exclude linear Zeeman dynamics.

A wavefunction intended for a specified total-spin state contains components of other total spins. For an unrestricted determinant with one alpha and one beta orbital, $⟨S^2⟩=1-|⟨{\varphi }_{\alpha }|{\varphi }_{\beta }⟩{|}^2$ in units of ${\hslash }^2$.

A state with total spin angular momentum quantum number $S$ has $2S+1$ magnetic projections. A singlet state has $S=0$, a triplet state has $S=1$, and the $S=3$ chromium ground term is a septet.

The two-site quantum version of an XY model couples the transverse components of two spin one-half degrees of freedom. In the ordered basis $|\uparrow \uparrow ⟩,|\uparrow \downarrow ⟩,|\downarrow \uparrow ⟩,|\downarrow \downarrow ⟩$, its only off-diagonal entries are $H_{23}=H_{32}=-2J$. The parallel states have energies $-2h$ and $2h$; the symmetric and antisymmetric opposite-spin combinations have energies $-2J$ and $2J$ respectively. The latter is the singlet state. The formulas remain valid at accidental degeneracies.

A spin eigenstate along a specified axis is a simultaneous eigenstate of the squared spin operator $S^2$ and its component $S_{\hat{n}}$. Its eigenvalues are ${\hslash }^{2}s(s+1)$ and $\hslash m$, where $m=-s,-s+1,\dots ,s$. The axis must be specified because components along different axes generally do not commute. For spin one-half, the two field-axis eigenstates have projections $\pm \hslash /2$ and form the branches of a Stern-Gerlach experiment.

A spin-spin interaction couples two spin angular momenta. In an isotropic model it is proportional to their dot product, whose expectation in a total-pair-spin state follows from $2{\mathbf{S}}_i\cdot {\mathbf{S}}_j=S_{ij}^2-S_i^2-S_j^2$. Magnetic interactions generate examples in atoms and effective colour-magnetic interactions in hadron models.

For a spin-one-half ensemble, spin polarization measures unequal occupation or coherent alignment of spin states. A spin-dependent energy splitting does not alone determine the ensemble's response: occupation, temperature and spin relaxation matter. A spin polarization is distinct from a net change of charge density and from the optical polarization of waves.

For two levels separated by $\Delta E$ and with the plus label assigned to the upper level, thermal equilibrium gives $P=-\tanh (\Delta E/(2k_{B}T))$. The weak-splitting limit is $P\simeq -\Delta E/(2k_{B}T)$. Labels based on the lower-energy aligned moment reverse the reported sign without changing the population imbalance.

A beam's unequal population of spin projections along a specified axis. For spin-one-half particles the displayed longitudinal polarization uses populations parallel and antiparallel to the momentum. An unpolarized beam has equal populations and requires an initial-spin average in a cross section; a polarized beam weights the corresponding helicity amplitudes differently. This differs from photon linear polarization.

A spin texture describes how a spin expectation value or spin polarization varies with position or wavevector. It specifies a vector field, including its magnitude when that matters; a direction field alone is undefined wherever that magnitude vanishes. A Rashba spin texture is an example in which spin expectation values wind around circles in the plane of wavevectors.

The spin-rotation coupling contribution to a rotating-frame Hamiltonian couples angular velocity to angular momentum. When discussing spin alone, identify which part of the total angular momentum enters the effective model.

A transverse magnetic field rotates a nonequilibrium spin polarization while diffusion and spin relaxation redistribute it. The field dependence of a detector signal can constrain spin lifetime and transport parameters. A distribution of diffusion times and rotation of the injecting/detecting electrodes must be included when relevant.

Loss of transported spin polarization at an interface, distinct from bulk spin relaxation in an adjacent layer.

The distance over which a nonequilibrium spin imbalance decays appreciably in a diffusive conductor. In a simple diffusion-relaxation model it is ${\ell }_s=\sqrt{D{\tau }_s}$; interfaces can contribute additional spin-memory loss.

Exchange of energy between magnetic spin degrees of freedom and the surrounding lattice, allowing their temperatures to approach thermal equilibrium. It permits adiabatic demagnetization to cool a solid and its attached load.

A model using exponentially correlated magnetic fluctuations. For homonuclear dipolar relaxation, $T_1^{-1}=C[{\tau }_c/(1+{\omega }_0^2{\tau }_c^2)+4{\tau }_c/(1+4{\omega }_0^2{\tau }_c^2)]$. Its rate maximum occurs near ${\omega }_0{\tau }_c=0.616$; a single-frequency Lorentzian instead peaks at one.

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.

For $S_n^{+}=P_n{c}_n$ with $P_n={\prod }_{m<n}(1-2c_m^{†}{c}_m)$, neighbouring Jordan–Wigner transformation strings differ by only the local parity factor. The canonical anticommutation relations give $(c_n+c_n^{†})(1-2c_n^{†}{c}_n)=c_n^{†}-c_n$, proving the displayed local spin identity. The closing bond of a periodic chain instead retains total fermion parity; cancellation on bulk bonds does not justify discarding that global sector.

For $S_n^{+}={\prod }_{m<n}(1-2n_m)c_n$ and periodic spins, the end exchange term carries the global fermion parity $P=(-1{)}^{N_f}$. Replacing the end bond by a uniform hopping expression requires $c_{N+1}=-P{c}_1$: even parity gives antiperiodic fermions, odd parity periodic fermions. Each resulting quadratic spectrum must be restricted to states of the parity used in its boundary condition.

With occupied fermion sites representing up-spins, use $s_1^{-}=f_1$, $s_2^{-}=P_1{f}_2$ and $P_1=(-1{)}^{n_1}=1-2n_1$. The canonical anticommutation relations imply $P_1^2=I$, $P_1{f}_1=-f_1{P}_1$ and $[P_1,f_2]=0$. Consequently the two sites commute as spin operators, while $[s_j^{-},s_j^{+}]=1-2n_j=-{\sigma }_j^z$. This normalization is essential: the unscaled combinations ${\sigma }_j^x\pm i{\sigma }_j^y$ are twice $s_j^{\pm }$.

The two-spin XY Hamiltonian becomesIn particular, $f_1^{†}{P}_1=f_1^{†}$ and $f_1{P}_1=-f_1$ fix the hopping signs. The modes $b=(f_1+f_2)/\sqrt{2}$ and $c=(f_1-f_2)/\sqrt{2}$ preserve the canonical anticommutation relations and give $H=2h+(-2h-2J)b^{†}b+(-2h+2J)c^{†}c$.

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 $(|{\uparrow }_z⟩\pm |{\downarrow }_z⟩)/\sqrt{2}$.

An electron has spin one-half. Its intrinsic spin angular momentum has two possible components, $\pm \hslash /2$, along a specified unit vector $\mathbf{n}$. These are the two eigenvalues of the corresponding spin measurement; the spin is distinct from the electron's orbital angular momentum.

For a spin one-half system, an ideal projective measurement along a unit vector $\mathbf{n}$ measures ${\sigma }_{\mathbf{n}}=\mathbf{n}\cdot \mathbf{\sigma }$. Its eigenvalues are $\pm 1$, corresponding to physical spin angular momentum components $\pm \hslash /2$, and its orthogonal projections are ${\Pi }_{\pm }=(I\pm {\sigma }_{\mathbf{n}})/2$. For density matrix $\rho =(I+\mathbf{r}\cdot \mathbf{\sigma })/2$, the Born rule and the Pauli matrix multiplication law give the displayed probabilities. Thus a spin measurement along one direction tests only that component of the Bloch vector; measurements along three independent directions determine the entire density matrix.

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

A Stern-Gerlach experiment separates atomic beams according to a component of their magnetic moment in a nonuniform magnetic field. For an appropriate two-state spin system, calibrated outputs distinguish the two spin eigenvalues; Stern-Gerlach expectation estimation explains how repeated fresh preparations turn channel counts into an expectation value.

The decisive contrast is between two separated components of an atomic beam and the continuous spread predicted for randomly oriented classical magnetic moments. Neither component need be an infinitely narrow spot: velocity spread and the resolving power of the scientific instrument affect the spatial image. The apparatus converts an internal magnetic moment into a displacement large enough to inspect, connecting angular momentum eigenstates to macroscopic evidence.

The ideal projective measurement of a spin-one-half component along a unit axis $\mathbf{n}$. The two outcomes have Born rule probabilities $⟨\psi |P_{\pm }|\psi ⟩$. For an input with Bloch vector $\mathbf{r}$, they are $(1\pm \mathbf{n}\cdot \mathbf{r})/2$. This model distinguishes a mathematical spin measurement from engineering a literal free-electron beam splitter.

An electrically neutral neutron still has a magnetic moment $\mathbf{\mu }={\gamma }_n\mathbf{S}$. A collimated beam passing through an inhomogeneous magnetic field experiences the force derived from $H=-\mathbf{\mu }\cdot \mathbf{B}$. In a region whose field axis is approximately fixed, a spin one-half neutron has two field-axis spin eigenstates, producing two allowed transverse forces and two separated beam components after a drift. A homogeneous field can cause Larmor precession but cannot separate positions through this force.

The neutron's negative gyromagnetic ratio interchanges which spin projection moves toward increasing field; it does not change the number of branches. An unpolarized beam gives equal ideal populations. A beam prepared in one field-axis eigenstate populates only one branch, so splitting into two observed spots presupposes both components and sufficient spatial resolution.

In 1922, Otto Stern and Walther Gerlach used an electrically heated oven to emit neutral silver chemical atoms. Two platinum apertures, a pinhole and a slit $0.03$–$0.04\mathrm{mm}$ wide, supplied collimation. The beam passed along a $3.5\mathrm{cm}$ knife-edge pole in a nonuniform magnetic field. At its end, a glass plate collected the condensed silver. Continuous pumping maintained vacuum; the thin deposit required development even after eight hours.^[ref]

The field-off image established the undeflected slit width. A comparison put field-on and field-off deposits on the same plate. Microscopy revealed two deflected bands, no detectable central component, and broadening attributed to the velocity distribution. Alignment and distance from the knife-edge affected the pattern; precise magnetic field gradient measurements were still pending.^[ref]

The apertures act as a collimator: they constrain the initial transverse positions and velocities, so a magnetic displacement can be distinguished from an initially broad beam. Vacuum reduces collisions that would destroy this preparation. Deposition integrates arrivals over the exposure; a darkened band is not a time-resolved record of individual chemical atoms. The field-off comparison tests whether the magnet changes the spatial distribution, while alignment checks address a geometric source of apparent asymmetry. Finite sensitivity qualifies the absence of a central deposit: it is an experimental upper limit, not a proof that exactly zero atoms landed there.

Let the atomic beam travel along $x$, with velocity $v>0$, and let $z$ measure displacement along the dominant magnetic field gradient. For a fixed effective magnetic moment component ${\mu }_z$, the magnetic dipole potential energy is $U\simeq -{\mu }_z{B}_z$. Taking its negative partial derivative givesA spatially uniform magnetic field can produce torque and Larmor precession, but cannot produce this translational separation. In a quantum state following a local internal energy eigenvalue $E_n(B)$, the more general expression is $F_z=-{\partial }_z{E}_n$, with effective magnetic moment $-\partial E_n/\partial B$. Treating that moment as constant is an approximation with a physical regime, not an extra measured fact. Assume the local magnetic field direction stays nearly fixed across the sampled beam, and that internal states follow it without transitions, as in the adiabatic theorem. The scalar gradient model describes the beam's neighborhood; it omits the full magnetic field's transverse components and fringe regions.

Assume a constant $G$ over magnet length $\ell$, followed by a field-free drift $D$. Neglect transverse motion at entry and changes in longitudinal velocity. Newton's second law gives transverse acceleration $a_z={\mu }_{z}G/M$ for whole-atom mass $M$. The magnet transit time is $t_m=\ell /v$, and the drift time is $t_d=D/v$. The magnet supplies both a displacement and an exit transverse velocity:The $\ell D$ contribution is motion with the acquired transverse velocity, not continued acceleration outside the magnet. For the plate-at-exit geometry, set $D=0$. For a nonzero entrance transverse velocity $w$, add $w(\ell +D)/v$. These terms show why initial collimation and flight geometry matter when inferring a magnetic moment from a deposit.

Choose hypothetical parameters $M=1.79\times 10^{-25}\mathrm{kg}$, $G=300\mathrm{T}{\mathrm{m}}^{-1}$, $\ell =0.035\mathrm{m}$, $D=0.100\mathrm{m}$, and $v=500\mathrm{m}{\mathrm{s}}^{-1}$. Approximate the two effective magnetic moments by ${\mu }_z=\pm {\mu }_B$, where the Bohr magneton is ${\mu }_B\simeq 9.274\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$. Then $|a_z|\simeq 1.55\times 10^4\mathrm{m}{\mathrm{s}}^{-2}$, $t_m=70\mu \mathrm{s}$ and $t_d=200\mu \mathrm{s}$. The positive displacement is about $0.0381\mathrm{mm}$ inside the magnet plus $0.2176\mathrm{mm}$ during drift: $d=0.2557\mathrm{mm}$, with full separation $2d=0.5114\mathrm{mm}$. These chosen values illustrate the model and are not reconstructed 1922 measurements.