In quantum mechanics, spin is an intrinsic property of all elementary particles. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made Fermions, the particles that constitute ordinary matter, have half-integer spin. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Mathematics, a half-integer is a Number of the form n + 1/2 where n is an Integer. Spin-½ particles constitute an important subset of such fermions. All known elementary particles that are fermions have spin ½.
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Overview
Particles having spin ½ include the electron, proton, neutron, neutrino, and quarks. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. Neutrinos are Elementary particles that travel close to the Speed of light, lack an Electric charge, are able to pass through ordinary matter almost In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. The dynamics of spin-½ objects cannot be accurately described using classical physics; they are among the simplest systems which require quantum mechanics to describe them. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons As such, the study of the behavior of spin-½ systems forms a central part of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons
General properties
Spin-½ objects are all fermions (a fact explained by the spin statistics theorem) and satisfy the Pauli exclusion principle. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 Spin-½ particles can have a permanent magnetic moment along the direction of their spin, and this magnetic moment gives rise to electromagnetic interactions that depend on the spin. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of One such effect that was important in the discovery of spin is the Zeeman effect. The Zeeman effect (ˈzeɪmɑːn is the splitting of a Spectral line into several components in the presence of a static Magnetic field.
Unlike in more complicated quantum mechanical systems, the spin of a spin-½ particle can be expressed as a linear combination of just two eigenstates, or eigenspinors. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Quantum mechanics, eigenspinors are thought of as Basis vectors representing the general spin state of a particle These are traditionally labeled spin up and spin down. Because of this the quantum mechanical spin operators can be represented as simple 2 × 2 matrices, as opposed to the infinite dimensional matrices commonly needed to represent operators like energy or position. In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained These matrices are called the Pauli matrices. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.
Creation and annihilation operators can be constructed for spin-½ objects; these obey the same commutation relations as other angular momentum operators. In Physics, an annihilation operator is an Operator that lowers the number of particles in a given state by one In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Quantum mechanics, the angular momentum operator is an Operator analogous to classical Angular momentum.
Connection to the uncertainty principle
One consequence of the generalized uncertainty principle is that the spin projection operators (which measure the spin along a given direction like x, y, or z), cannot be measured simultaneously. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Physically, this means that it is ill defined what axis a particle is spinning about. A measurement of the z-component of spin destroys any information about the x and y components that might previously have been obtained.
Stern–Gerlach experiment
When a spin-½ particle with non-zero magnetic moment like an electron is placed in an inhomogenous magnetic field, it experiences a force. In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current For other uses see Homogeneous. In Physics, homogeneous mixtures are mixtures that have definite consistent composition and properties This acts to separate out particles in the spin up state from particles in the spin down state. This is the idea behind the Stern–Gerlach experiment. In Quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is an important 1922 experiment on the Deflection
Symmetry
Mathematically, quantum mechanical spin is not described by a vector as in classical angular momentum. It is described using a family of objects known as spinors. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the There are subtle differences between the behavior of spinors and vectors under coordinate rotations. In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed Rotating a spin-1/2 particle by 360 degrees 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. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern To return the particle to its exact original state, one needs a 720 degree rotation!
Mathematical description
The quantum state of the spin of a spin-½ particle can be described by a complex-valued vector with two components called a two-component spinor. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the When spinors are used to describe the quantum states, quantum mechanical operators are represented by 2 × 2, complex-valued Hermitian matrices. In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint
For example, the spin projection operator Sz effects a measurement of the spin in the z direction.
Sz operator has two eigenvalues, of 
, which correspond to the eigenvectors
These vectors form a complete basis for the Hilbert space describing the spin-½ particle. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes This article assumes some familiarity with Analytic geometry and the concept of a limit. Thus, linear combinations of these two states can represent all possible states of the spin.
See also
References
Griffiths, David J. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. (2005) Introduction to Quantum Mechanics (2nd ed. ). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-111892-7.
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