Spin (physics)

Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion. Spin is described mathematically as a vector for some particles such as photons, and as a spinor for other particles such as electrons. Spinors behave similarly to vectors: they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number. The SI units of spin are the same as classical angular momentum (i.e., N·m·s, J·s, or kg·m 2 ·s −1 ).…

Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion. Spin is described mathematically as a vector for some particles such as photons, and as a spinor for other particles such as electrons. Spinors behave similarly to vectors: they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number. The SI units of spin are the same as classical angular momentum (i.e., N·m·s, J·s, or kg·m 2 ·s −1 ). In quantum mechanics, angular momentum and spin angular momentum take discrete values proportional to the Planck constant. In practice, spin is usually given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant . Often, the "spin quantum number" is simply called "spin". S = \hbar \, \sqrt{s(s + 1)} = \frac{h}{2\pi} \, \sqrt{\frac{n}{2}\frac{(n + 2)}{2}} = \frac{h}{4\pi} \, \sqrt{n(n + 2)}, where is the Planck constant, and \hbar = \frac{h}{2\pi} is the reduced Planck constant. In contrast, orbital angular momentum can only take on integer values of ; i.e., even-numbered values of . \boldsymbol{\mu} = \frac{g_\text{s} q}{2m} \mathbf{S}, where the dimensionless quantity is called the spin -factor. For exclusively orbital rotations, it would be 1 (assuming that the mass and the charge occupy spheres of equal radius). The electron, being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron -factor, which has been experimentally determined to have the value , with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation. The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties; and the deviation from arises from the electron's interaction with the surrounding quantum fields, including its own electromagnetic field and virtual particles. Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions. Neutrinos are both elementary and electrically neutral. The minimally extended Standard Model that takes into account non-zero neutrino masses predicts neutrino magnetic moments of: \mu_\nu \approx 3 \times 10^{-19} \mu_\text{B} \frac{m_\nu c^2}{\text{eV}}, where the are the neutrino magnetic moments, are the neutrino masses, and is the Bohr magneton. New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. It can be shown in a model-independent way that neutrino magnetic moments larger than about 10 −14   are "unnatural" because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses are known to be at most about , fine-tuning would be necessary in order to prevent large contributions to the neutrino mass via radiative corrections. The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than  times the electron's magnetic moment. On the other hand, elementary particles with spin but without electric charge, such as the photon and Z boson, do not have a magnetic moment. S_i = \hbar s_i, \quad s_i \in \{ -s, -(s - 1), \dots, s - 1, s \}, where is the spin component along the -th axis (either , , or ), is the spin projection quantum number along the -th axis, and is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the  axis: S_z = \hbar s_z, \quad s_z \in \{ -s, -(s - 1), \dots, s - 1, s \}, where is the spin component along the  axis, is the spin projection quantum number along the  axis. One can see that there are possible values of . The number " " is the multiplicity of the spin system. For example, there are only two possible values for a spin- particle: and . These correspond to quantum states in which the spin component is pointing in the +z or −z directions respectively, and are often referred to as "spin up" and "spin down". For a spin- particle, like a delta baryon, the possible values are + , + , − , − . For a given quantum state, one could think of a spin vector \lang S \rang whose components are the expectation values of the spin components along each axis, i.e., \lang S \rang = [\lang S_x \rang, \lang S_y \rang, \lang S_z \rang] . This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it cannot be measured directly: , and cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern–Gerlach apparatus, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin- particles, this probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%. As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum-mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "torque" on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment—see the following section). The result is that the spin vector undergoes precession, just like a classical gyroscope. This phenomenon is known as electron spin resonance (ESR). The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance (NMR) spectroscopy and imaging. Mathematically, quantum-mechanical spin states are described by vector-like objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin- particle by 360° does not bring it back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a 720° rotation. (The plate trick and Möbius strip give non-quantum analogies.) A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through. \left[\hat S_j, \hat S_k\right] = i \hbar \varepsilon_{jkl} \hat S_l, where is the Levi-Civita symbol. It follows (as with angular momentum) that the eigenvectors of \hat S^2 and \hat S_z (expressed as kets in the total basis) are \begin{align} \hat S^2 |s, m_s\rangle &= \hbar^2 s(s + 1) |s, m_s\rangle, \\ \hat S_z |s, m_s\rangle &= \hbar m_s |s, m_s\rangle. \end{align} The spin raising and lowering operators acting on these eigenvectors give \hat S_\pm |s, m_s\rangle = \hbar \sqrt{s(s + 1) - m_s(m_s \pm 1)} |s, m_s \pm 1\rangle, where \hat S_\pm = \hat S_x \pm i \hat S_y . But unlike orbital angular momentum, the eigenvectors are not spherical harmonics. They are not functions of and . There is also no reason to exclude half-integer values of and . All quantum-mechanical particles possess an intrinsic spin s (though this value may be equal to zero). The projection of the spin s on any axis is quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not \psi = \psi(\mathbf r) , but \psi = \psi(\mathbf r,s_z) , where s_z can take only the values of the following discrete set: s_z \in \{-s\hbar, -(s - 1)\hbar, \dots, +(s - 1)\hbar, +s\hbar\}. One distinguishes bosons (integer spin) and fermions (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin. \hat{\mathbf{S}} = \frac{\hbar}{2} \boldsymbol{\sigma}, where in Cartesian components S_x = \frac{\hbar}{2} \sigma_x, \quad S_y = \frac{\hbar}{2} \sigma_y, \quad S_z = \frac{\hbar}{2} \sigma_z. For the special case of spin- particles, , and are the three Pauli matrices: \sigma_x = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}. \psi(\dots, \mathbf r_i, \sigma_i, \dots, \mathbf r_j, \sigma_j, \dots ) = (-1)^{2s} \psi(\dots, \mathbf r_j, \sigma_j, \dots, \mathbf r_i, \sigma_i, \dots). Thus, for bosons the prefactor will reduce to +1, for fermions to −1. This permutation postulate for -particle state functions has most important consequences in daily life, e.g. the periodic table of the chemical elements. |a_{+1/2}|^2 + |a_{-1/2}|^2 = 1. For a generic particle with spin , we would need such parameters. Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices: \sum_{m=-j}^j a_m^* b_m = \sum_{m=-j}^j \left(\sum_{n=-j}^j U_{nm} a_n\right)^* \left(\sum_{k=-j}^j U_{km} b_k\right), \sum_{n=-j}^j \sum_{k=-j}^j U_{np}^* U_{kq} = \delta_{pq}. Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2). There is one -dimensional irreducible representation of SU(2) for each dimension, though this representation is -dimensional real for odd and -dimensional complex for even (hence of real dimension ). For a rotation by angle in the plane with normal vector \hat{\boldsymbol{\theta}} , U = e^{-\frac{i}{\hbar} \boldsymbol{\theta} \cdot \mathbf{S}}, where \boldsymbol{\theta} = \theta \hat{\boldsymbol{\theta}} , and is the vector of spin operators. A generic rotation in 3-dimensional space can be built by compounding operators of this type using Euler angles: \mathcal{R}(\alpha, \beta, \gamma) = e^{-i\alpha S_x} e^{-i\beta S_y} e^{-i\gamma S_z}. An irreducible representation of this group of operators is furnished by the Wigner D-matrix: D^s_{m'm}(\alpha, \beta, \gamma) \equiv \langle sm' | \mathcal{R}(\alpha, \beta, \gamma) | sm \rangle = e^{-im'\alpha} d^s_{m'm}(\beta)e^{-i m\gamma}, where d^s_{m'm}(\beta) = \langle sm' | e^{-i\beta s_y} | sm \rangle is Wigner's small d-matrix. Note that for and ; i.e., a full rotation about the  axis, the Wigner D-matrix elements become D^s_{m'm}(0, 0, 2\pi) = d^s_{m'm}(0) e^{-i m 2 \pi} = \delta_{m'm} (-1)^{2m}. Recalling that a generic spin state can be written as a superposition of states with definite , we see that if is an integer, the values of are all integers, and this matrix corresponds to the identity operator. However, if is a half-integer, the values of are also all half-integers, giving for all , and hence upon rotation by 2 the state picks up a minus sign. This fact is a crucial element of the proof of the spin–statistics theorem. \psi' = \exp{\left(\tfrac{1}{8} \omega_{\mu\nu} [\gamma_{\mu}, \gamma_{\nu}]\right)} \psi, where are gamma matrices, and is an antisymmetric 4 × 4 matrix parametrizing the transformation. It can be shown that the scalar product \langle\psi|\phi\rangle = \bar{\psi}\phi = \psi^\dagger \gamma_0 \phi is preserved. It is not, however, positive-definite, so the representation is not unitary. \begin{array}{lclc} \psi_{x+} = \left|\frac{1}{2}, \frac{+1}{2}\right\rangle_x = \displaystyle\frac{1}{\sqrt{2}} \!\!\!\!\! & \begin{pmatrix}{1}\\{1}\end{pmatrix}, & \psi_{x-} = \left|\frac{1}{2}, \frac{-1}{2}\right\rangle_x = \displaystyle\frac{1}{\sqrt{2}} \!\!\!\!\! & \begin{pmatrix}{1}\\{-1}\end{pmatrix}, \\ \psi_{y+} = \left|\frac{1}{2}, \frac{+1}{2}\right\rangle_y = \displaystyle\frac{1}{\sqrt{2}} \!\!\!\!\! & \begin{pmatrix}{1}\\{i}\end{pmatrix}, & \psi_{y-} = \left|\frac{1}{2}, \frac{-1}{2}\right\rangle_y = \displaystyle\frac{1}{\sqrt{2}} \!\!\!\!\! & \begin{pmatrix}{1}\\{-i}\end{pmatrix}, \\ \psi_{z+} = \left|\frac{1}{2}, \frac{+1}{2}\right\rangle_z = & \begin{pmatrix} 1 \\ 0 \end{pmatrix}, & \psi_{z-} = \left|\frac{1}{2}, \frac{-1}{2}\right\rangle_z = & \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \end{array} (Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as SymPy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.) By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the , , or  axis can only yield an eigenvalue of the corresponding spin operator ( , or ) on that axis, i.e. or . The quantum state of a particle (with respect to spin), can be represented by a two-component spinor: \psi = \begin{pmatrix} a + bi \\ c + di \end{pmatrix}. When the spin of this particle is measured with respect to a given axis (in this example, the  axis), the probability that its spin will be measured as is just \big|\langle \psi_{x+}|\psi\rangle\big|^2 . Correspondingly, the probability that its spin will be measured as is just \big|\langle\psi_{x-}|\psi\rangle\big|^2 . Following the measurement, the spin state of the particle collapses into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (since \big|\langle\psi_{x+} | \psi_{x+}\rangle\big|^2 = 1 , etc.), provided that no measurements of the spin are made along other axes. S_u = \frac{\hbar}{2} (u_x \sigma_x + u_y \sigma_y + u_z \sigma_z). The operator has eigenvalues of , just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three -, -, -axis directions. A normalized spinor for spin- in the direction (which works for all spin states except spin down, where it will give ) is \frac{1}{\sqrt{2 + 2u_z}} \begin{pmatrix} 1 + u_z \\ u_x + iu_y \end{pmatrix}. The above spinor is obtained in the usual way by diagonalizing the matrix and finding the eigenstates corresponding to the eigenvalues. In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity. \begin{align} S_x &= \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, & \left|1, +1\right\rangle_x &= \frac{1}{2} \begin{pmatrix} 1 \\{\sqrt{2}}\\ 1 \end{pmatrix}, & \left|1, 0\right\rangle_x &= \frac{1}{\sqrt{2}} \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, & \left|1, -1\right\rangle_x &= \frac{1}{2} \begin{pmatrix} 1 \\{-\sqrt{2}}\\ 1 \end{pmatrix} \\ S_y &= \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}, & \left|1, +1\right\rangle_y &= \frac{1}{2} \begin{pmatrix} -1 \\ -i\sqrt{2} \\ 1 \end{pmatrix}, & \left|1, 0\right\rangle_y &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, & \left|1, -1\right\rangle_y &= \frac{1}{2} \begin{pmatrix} -1 \\ i\sqrt{2} \\ 1 \end{pmatrix} \\ S_z &= \hbar \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}, & \left|1, +1\right\rangle_z &= \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, & \left|1, 0\right\rangle_z &= \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, & \left|1, -1\right\rangle_z &= \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \\ \end{align} | For spin they are \begin{array}{lclc} S_x = \frac\hbar2 \begin{pmatrix} 0 &\sqrt{3} &0 &0\\ \sqrt{3} &0 &2 &0\\ 0 &2 &0 &\sqrt{3}\\ 0 &0 &\sqrt{3} &0 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{+3}{2}\right\rangle_x =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix} 1 \\{\sqrt{3}}\\{\sqrt{3}}\\ 1 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{+1}{2}\right\rangle_x =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix}{-\sqrt{3}}\\ -1 \\ 1 \\{\sqrt{3}}\end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{-1}{2}\right\rangle_x =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix}{\sqrt{3}}\\ -1 \\ -1 \\{\sqrt{3}}\end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{-3}{2}\right\rangle_x =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix} -1 \\{\sqrt{3}}\\{-\sqrt{3}}\\ 1 \end{pmatrix} \\ S_y = \frac\hbar2 \begin{pmatrix} 0 &-i\sqrt{3} &0 &0\\ i\sqrt{3} &0 &-2i &0\\ 0 &2i &0 &-i\sqrt{3}\\ 0 &0 &i\sqrt{3} &0 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{+3}{2}\right\rangle_y =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix}{i}\\{-\sqrt{3}}\\{-i\sqrt{3}}\\ 1 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{+1}{2}\right\rangle_y =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix}{-i\sqrt{3}}\\ 1 \\{-i}\\{\sqrt{3}}\end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{-1}{2}\right\rangle_y =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix}{i\sqrt{3}}\\ 1 \\{i}\\{\sqrt{3}}\end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{-3}{2}\right\rangle_y =\!\!\! & \frac{1}{2\sqrt{2}} \begin{pmatrix}{-i}\\{-\sqrt{3}}\\{i\sqrt{3}}\\ 1 \end{pmatrix} \\ S_z = \frac\hbar2 \begin{pmatrix} 3 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &-1 &0\\ 0 &0 &0 &-3 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{+3}{2}\right\rangle_z =\!\!\! & \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{+1}{2}\right\rangle_z =\!\!\! & \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{-1}{2}\right\rangle_z =\!\!\! & \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \!\!\! & \left|\frac{3}{2}, \frac{-3}{2}\right\rangle_z =\!\!\! & \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \\ \end{array} |For spin they are \begin{align} \boldsymbol{S}_x &= \frac{\hbar}{2} \begin{pmatrix} 0 &\sqrt{5} &0 &0 &0 &0 \\ \sqrt{5} &0 &2\sqrt{2} &0 &0 &0 \\ 0 &2\sqrt{2} &0 &3 &0 &0 \\ 0 &0 &3 &0 &2\sqrt{2} &0 \\ 0 &0 &0 &2\sqrt{2} &0 &\sqrt{5} \\ 0 &0 &0 &0 &\sqrt{5} &0 \end{pmatrix}, \\ \boldsymbol{S}_y &= \frac{\hbar}{2} \begin{pmatrix} 0 &-i\sqrt{5} &0 &0 &0 &0 \\ i\sqrt{5} &0 &-2i\sqrt{2} &0 &0 &0 \\ 0 &2i\sqrt{2} &0 &-3i &0 &0 \\ 0 &0 &3i &0 &-2i\sqrt{2} &0 \\ 0 &0 &0 &2i\sqrt{2} &0 &-i\sqrt{5} \\ 0 &0 &0 &0 &i\sqrt{5} &0 \end{pmatrix}, \\ \boldsymbol{S}_z &= \frac{\hbar}{2} \begin{pmatrix} 5 &0 &0 &0 &0 &0 \\ 0 &3 &0 &0 &0 &0 \\ 0 &0 &1 &0 &0 &0 \\ 0 &0 &0 &-1 &0 &0 \\ 0 &0 &0 &0 &-3 &0 \\ 0 &0 &0 &0 &0 &-5 \end{pmatrix}. \end{align} | The generalization of these matrices for arbitrary spin is \begin{align} \left(S_x\right)_{ab} & = \frac{\hbar}{2} \left(\delta_{a,b+1} + \delta_{a+1,b}\right) \sqrt{(s + 1)(a + b - 1) - ab}, \\ \left(S_y\right)_{ab} & = \frac{i\hbar}{2} \left(\delta_{a,b+1} - \delta_{a+1,b}\right) \sqrt{(s + 1)(a + b - 1) - ab}, \\ \left(S_z\right)_{ab} & = \hbar (s + 1 - a) \delta_{a,b} = \hbar (s + 1 - b) \delta_{a,b}, \end{align} where indices a, b are integer numbers such that 1 \le a \le 2s + 1, \quad 1 \le b \le 2s + 1. }} Also useful in the quantum mechanics of multiparticle systems, the general Pauli group is defined to consist of all -fold tensor products of Pauli matrices. The analog formula of Euler's formula in terms of the Pauli matrices \hat{R}(\theta, \hat{\mathbf{n}}) = e^{i \frac{\theta}{2} \hat{\mathbf{n}} \cdot \boldsymbol{\sigma}} = I \cos \frac{\theta}{2} + i \left(\hat{\mathbf{n}} \cdot \boldsymbol{\sigma}\right) \sin \frac{\theta}{2} for higher spins is tractable, but less simple. p + p \rightarrow \pi^+ +d The known spin values for protons and deuterium allows analysis of the collision cross-section to show that \pi^+ has spin s_\pi = 0 . A different approach is needed for neutral pions. In that case the decay produced two gamma ray photons with spin one: \pi^0 \rightarrow 2\gamma This result supplemented with additional analysis leads to the conclusion that the neutral pion also has spin zero.

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