Spin

The spin is a quantum property Intrinsèque associated with each particle, which is characteristic of the nature of the particle, as well as its Masse and its electric Charge. It makes it possible to characterize the behavior of the particle under the effect of the Symétrie of rotation of the space.

Simplified definition

The concept of spin makes it possible mathematically to classify the way in which the objects under the effect of rotations of space with three change Dimension S. In a general way, an object has a spin $s\; \backslash ,$ if it is invariant under a rotation of angle $\backslash \; frac\; \{2\; \backslash \; pi\}\; \{S\}\; \backslash ,$. For example:

• an object without particular symmetry, for example a Chart to play representing one three of clover, has a spin 1 because it is necessary to carry out a rotation of $2\; \backslash \; pi\; \backslash ,$ (a full rotation) so that it is found in its starting position.
• an object having a little more symmetry, like a queen of spades for example, has a spin 2 because so that it returns to its starting position, one can be satisfied to make him carry out a rotation of $\backslash \; pi=\; \backslash \; frac\; \{2\; \backslash \; pi\}\; \{2\}$ (a half turn).
• a star with five branches has a spin 5 because it is sufficient to make him make a rotation of $\backslash \; frac\; \{2\; \backslash \; pi\}\; \{5\}\; \backslash ,$.
• a completely symmetrical object, like a sphere for example, is invariant by rotation of any angle. The simplified definition that one gave is difficult to apply in this case but mathematically it is natural to say that such an object has an infinite spin.

Let us notice that usually, since a rotation of angle $2\; \backslash \; pi\; \backslash ,$ is equal to the identity, it would seem that any object is of whole spin because in the worst of the cases an object should always be identical to itself under a rotation of angle $2\; \backslash \; pi\; \backslash ,$. However the rigorous mathematical analysis of the Groupe of rotations shows a subtle structure which makes it possible certain objects to have a spin half-entirety. For such objects, to make a full rotation on themselves is not sufficient to make them return to their starting position but it is necessary to carry out a rotation of angle $4\; \backslash \; pi\; \backslash ,$. One does not meet such objects on our scale but in the microscopic world they are current. One calls them Fermion S, whose well-known example is the electron, which has precisely a spin $\backslash \; frac\; \{1\}\; \{2\}\; \backslash ,$.

In a more rigorous way, as one will see it low, the analysis of the behavior of the objects under the effect of rotations requires to take into account the mathematical structure of Groupe formed by those. With an object changing under rotations a representation of group is then associated. Two objects having similar properties of symmetry will thus be associated with equivalent representations of the group of rotations. From this point of view, the spin is anything else only one number which allows to classify the various representations inéquivalentes group of rotations (one calls that the irreducible representations). Thus one can say that a particle of spin 2 such as the Graviton (see below) has the same symmetry from the point of view of rotations as a queen of spades because both change in equivalent representations.

Spin of the usual particles

The spin of a particle is a Integer or positive half-entirety, noted $s\; \backslash ,$.

Although related to the phenomena of quantification of the Angular momentum, the spin is indeed an intrinsic property of the particles. In particular, it does not correspond to any hypothetical rotation movement of these particles.

The particles having a spin half-entirety are called Fermion S, those having a whole spin are called Bosons. More specifically:

• Spin 0: the Boson of Higgs, hypothetical particle, not yet in experiments discovered.
• Spin 1/2: the electron, the Positron, the Proton, the Neutron, the Neutrino S, the Quark S, etc
• Spin 1: the Photon, the W^± bosons and Z⁰ vectors of the weak Interaction.
• Spin 2: the Graviton, hypothetical particle vector of the Gravitation.

The spin of composed particles, like the Proton, the Neutron, the Atomic nucleus or the Atom, is consisted of the spins of the particles which compose them to which the angular momentum of the elementary particles one compared to the other is added.

History

The concept of spin was introduced by Pauli in December 1924. for the electron in order to explain an experimental result which remained incomprehensible within the framework incipient from the quantum Mécanique not relativist: the Effect abnormal Zeeman. The approach developed by Pauli consisted in introducing on an ad-hoc basis the spin by adding a additional postulate to the other postulates of the quantum Mécanique not relativist (equation of Schrödinger, etc).

The introduction of the spin makes it possible to also include/understand other experimental effects, like the doublets of the spectra of the alkaline metals, or the result of the Expérience of Stern and Gerlach.

In 1928, Paul Dirac built a quantum and relativistic version of the equation of Schrödinger, called today equation of Dirac, which makes it possible to describe the fermions of spin 1/2. The spin seems there like a property derived from its equation, and not an additional postulate to add on an ad-hoc basis.

Lastly, it is in Quantum theory of the fields that the spin shows its most fundamental character. The analysis of the group of Poincaré carried out by Wigner in 1939 showed indeed that a particle is associated with a quantum field, operator who changes like an irreducible representation of the group of Poincaré. These irreducible representations are classified by two positive real numbers: mass and the spin.

“Clean Rotation”

Historically, the spin was initially interpreted by Uhlenbeck and Goudsmit in September 1925, as being a kinetic Moment intrinsic, i.e. as if the particle “rotated”. This traditional vision of a “clean rotation” of the particle is in fact too naive; indeed:

• if the particle is specific , the clean concept of rotation around its axis is quite simply stripped of physical direction.

• if the particle is not specific, then the concept has a direction, but one encounters in this case another difficulty. Let us suppose for example that the particle is an electron, modelled as being a spherical body of ray $a$. One obtains an estimate of the ray $a$ by writing that the energy of mass of the electron is of about size of its electrostatic potential energy, that is to say:

the numerical value of this “traditional ray” of the electron is: $a\; \backslash \; \backslash \; simeq\; \backslash \; 10^\; \{-\; 15\}$ Mr. If one allots then to this electron one kinetic moment equal to $\backslash \; hbar\; /2$, one obtains for a point of the checking equator a speed $v$:

the numerical value would be worth: $v\; \backslash \; \backslash \; simeq\; \backslash \; 6\; \backslash \; 10^\; \{+10\}$ m/s, therefore speed would be higher than speed of light in the vacuum, which poses problems with the theory of the restricted Relativité.

This calculation neglects the relativistic effects on the mass.

Relativistic calculation

MacGregor, in its book, The Enigmatic Electron, watch which should be taken account of the relativistic variation of the mass with the number of revolutions. It makes the assumption that the equatorial speed of the electron at rest is equal at the speed C of the light. Let us calculate the mass m observed of a sphere full (a ball) with ray R in relativistic rotation at the speed $v=\; \backslash \; Omega\; r$. The number of revolutions angular is supposed to be constant, i.e. the electron is a rigid solid and not a fluid. That is to say $\backslash \; rho\; (R)$ mass specific to the distance R of the axis of rotation and $\backslash \; rho\_0$ on the axis of rotation where the matter is at rest. Volume being unchanged, the relativistic specific mass is: The height of the elementary cylinder is h. the linear velocity of rotation, $v=\; \backslash \; Omega\; r$, is equal to C on the equator, which gives the angular velocity $\backslash \; omega=c/R$. One thus has, by taking account of the relativistic variation of the mass:

2 \ pi \ rho_0 R^2 \ int_0^R \ frac {2 \ sqrt {R^2-r^2}} {\ sqrt {1 \ frac {(\ Omega R) ^2} {c^2}}} D \ frac {r^2} {2R ^2}

2 \ pi \ rho_0 R^3 \ int_0^R \ frac {\ sqrt {1 \ frac {r^2} {R^2}}} {\ sqrt {1 \ frac {c^ 2 r ^2} {R^2 c^2}}} D \ frac {r^2} {R^2}

2 \ pi \ rho_0 R^3 \ int_0^Rd \ frac {r^2} {R^2} 2 \ pi \ rho_0 R^3

|} One calculates in the same way the relativistic moment of inertia of the sphere, moreover rather not very different from the traditional moment $\backslash \; frac\; \{3\}\; \{5\}\; mR^2$:

2 \ pi \ rho_0 R^3 \ int_0^R \ frac {2 \ sqrt {1 \ frac {r^2} {R^2}}} {\ sqrt {1 \ frac {r^2} {R^2}}} D \ frac {r^4} {4R ^4}

\ pi \ rho_0 R^3 \ int_0^R D \ frac {r^4} {R^4} \ pi \ rho_0 R^5 \ frac {1} {2} mR^2

|} Simplification is crucial in this calculation. It is practically inextricable in the general case. The relation of Einstein-Planck gives the Eigen frequency $\backslash \; nu$ of the electron. By identifying it at its number of angular revolutions $\backslash \; omega$, one a: One obtains for the electron a ray different from the ray " classique"

$R\_C\; \backslash$ is the ray of Compton, that one even which appears in the equation of Klein-Gordon and, of course, in the Compton effect. MacGregor thus obtains the intrinsic kinetic moment of the electron:

in agreement with the observation. That gives a spin of 1/2 in units of $\backslash \; hbar$.

This calculation was made for an electron in rotation but with its axis at rest. It would be to re-examine for an electron moving relativistic unspecified. The model of the spinning top for the spin of the electron seems in agreement at the same time with the theories relativist and quantum. The difficulty announced to the preceding paragraph is solved. It has remained however to measure directly the still unknown ray of this elementary particle however known for one century. It is currently considered that its ray is null what would require that its mass be it too. That could the being with the rigor in the absence of rotation. One can evacuate the problem while striking " an elementary particle does not have size. The spin is a " objet" purely quantique."

Physical interpretation of the spin

Operator spin

In Mechanical quantum, the spin is a vectorial Opérateur Hermitien comprising three components, usually noted $\backslash \; hat\; \{S\}\; \_x,\; \backslash ,\; \backslash \; hat\; \{S\}\; \_y$ and $\backslash \; hat\; \{S\}\; \_z$ by reference to the three axes of Cartesian coordinates of physical space. These components check the relations of commutations:

$\backslash \; left\; \backslash ,\; \backslash \; hat\; \{S\}\; \_i\; \backslash ,\; \backslash \; \backslash \; hat\; \{S\}\; \_j\; \backslash ,\; \backslash \; right\; \backslash \; =\; \backslash \; I\; \backslash \; \backslash \; hbar\; \backslash \; \backslash \; epsilon\_\; \{ijk\}\; \backslash \; \backslash \; hat\; \{S\}\; \_k$

where $\backslash \; epsilon\_\; \{ijk\}$ is the Symbole of Levi-Civita. These relations of commutations are similar to those discovered in November 1925 by Born, Heisenberg and Jordan for the components of the orbital kinetic Moment: $\backslash \; left\; \backslash ,\; \backslash \; hat\; \{L\}\; \_i\; \backslash ,\; \backslash \; \backslash \; hat\; \{L\}\; \_j\; \backslash ,\; \backslash \; right\; \backslash \; =\; \backslash \; I\; \backslash \; \backslash \; hbar\; \backslash \; \backslash \; epsilon\_\; \{ijk\}\; \backslash \; \backslash \; hat\; \{L\}\; \_k$

By analogy with the results obtained for the moment kinetic orbital (or more generally for a quantum kinetic Moment), there exists for the operator spin a base of noted clean vectors $|\; S,\; m\_s\; >$, where $s$ is whole or half-entirety, and $m\_s$ are an entirety or half-entirety taking one of the $2s\; +\; 1$ values $-\; \backslash ,\; S\; \backslash \; the\; m\_s\; \backslash \; the\; s$, such as:

$\backslash \; hat\; \{S\}\; ^2\; \backslash \; |\; S,\; m\_s\; \backslash \; rangle\; \backslash \; =\; \backslash \; S\; (s+1)\; \backslash ,\; \backslash \; hbar^2\; \backslash \; |\; S,\; m\_s\; \backslash \; rangle$

$\backslash \; hat\; \{S\}\; \_z\; \backslash \; |\; S,\; m\_s\; \backslash \; rangle\; \backslash \; =\; \backslash \; m\_s\; \backslash ,\; \backslash \; hbar\; \backslash \; |\; S,\; m\_s\; \backslash \; rangle$

Spin 1/2 - matrices of Pauli

See also: Matrices of Pauli

For a particle of spin 1/2 like the electron, there is $s\; =\; 1/2$, therefore $2s\; +\; 1\; =\; 2$: there exists only two distinct states of spin, characterized by $m\_s\; =\; \backslash \; pm\; 1/2$.

One often notes the two clean states corresponding: $|+\; \backslash \; rangle$ and $|-\; \backslash \; rangle$, or symbolically: $|\backslash \; uparrow\; \backslash \; rangle$ and $|\backslash downarrow\backslash rangle$.

Pauli introduced three matrix S 2 X 2, noted $\backslash \; hat\; \{\backslash \; sigma\}\; \_i,\; \backslash \; I\; =\; 1,2,3$ such as the operator of spin is written:

$\backslash \; hat\; \{S\}\; \_i\; \backslash \; =\; \backslash \; \backslash \; frac\; \{\backslash \; hbar\}\; \{2\}\; \backslash \; \backslash \; hat\; \{\backslash \; sigma\}\; \_i$

These three matrices of Pauli are written explicitly:

They satisfy the relations of commutation:

$\backslash \; left\; \backslash ,\; \backslash \; hat\; \{\backslash \; sigma\}\; \_i\; \backslash ,\; \backslash \; \backslash \; hat\; \{\backslash \; sigma\}\; \_j\; \backslash ,\; \backslash \; right\; \backslash \; =\; \backslash \; 2\; \backslash \; I\; \backslash \; \backslash \; epsilon\_\; \{ijk\}\; \backslash \; \backslash \; hat\; \{\backslash \; sigma\}\; \_k$

Geometrical representation of the spin by a Sphere of Riemann

A quantum state unspecified of a particle of spin 1/2 can be expressed in the general form:

$|\backslash \; nearrow\; \backslash \; rangle\; =\; has\; |\backslash \; uparrow\; \backslash \; rangle\; +\; B\; |\backslash downarrow\; \backslash rangle$

(has and B being two complex numbers). This formula expresses a superposition of the two clean states.

Depending on the rules of quantum mechanics, the quantum state represented by $|\backslash \; psi\; \backslash \; rangle$ and $\backslash \; alpha\; |\backslash \; psi\; \backslash \; rangle$ is physically rigorously the same ones. Consequently, one can also express the general state of a particle of spin 1/2 by:

$|\backslash \; nearrow\; \backslash \; rangle\; =\; |\backslash \; uparrow\; \backslash \; rangle\; +\; \backslash \; frac\; Ba\; |\backslash downarrow\; \backslash rangle$

The state of spin 1/2 is thus entirely characterized by a complex number $U\; =\; \backslash \; frac\; ba$. This report/ratio which can be infinite when = 0 (pure state of spin " has; down"), it is necessary to use a Sphère of Riemann to represent this report/ratio, the sphere of Riemann being an extension of the body of the complexes with the infinite one.

According to this representation, any state of spin 1/2 finds a representation geometrical (see figure opposite). The vector passing by the origin and pointing on the projection of the complex U on the sphere of Riemann gives a geometrical visualization of the state of spin 1/2 as being a direction in space .

Although seeming a priori purely mathematical, this representation of the state of spin as being a direction in space has a certain relevance. In particular, one can simply find using this geometrical representation the probability of obtaining the state $|\backslash \; uparrow\; \backslash \; rangle$ and $|\backslash \; downarrow\; \backslash \; rangle$ during a measurement of the state $|\backslash \; nearrow\; \backslash \; rangle$ (one should not lose sight of the fact that the state measured of a state of spin 1/2 will be always is $|\backslash \; uparrow\; \backslash \; rangle$ is $|\backslash \; downarrow\; \backslash \; rangle$).

Magnetic moment of spin

See also: magnetic Moment of spin

Definition. Factor of Moor

At the orbital kinetic time of a particle of load $q$ and mass $m$ is associated a magnetic Moment orbital:

$\backslash \; vec\; \{\backslash \; driven\}\; \_L\; \backslash \; =\; \backslash \; \backslash \; frac\; \{Q\}\; \{2\; m\}\; \backslash \; \backslash \; vec\; \{L\}$

The factor $q/2m$ is called gyromagnetic Rapport. In the same way, one associates with a particle of load $q$, mass $m$, and spin given a magnetic Moment of spin :

$\backslash \; vec\; \{\backslash \; driven\}\; \_S\; \backslash \; =\; \backslash \; G\; \backslash \; \backslash \; frac\; \{Q\}\; \{2\; m\}\; \backslash \; \backslash \; vec\; \{S\}$

where $g$ is a Nombre without dimension, called Facteur of Moor (1921). This number varies according to the nature of the particle: there is roughly $g=2$ for the electron, $g=5,586$ for the Proton, and $g=-\; \backslash ,\; 3,826$ for the Neutron.

Magneton of Bohr

For the electron, one has the following values: $s=\; \backslash \; hbar\; /2$ and $g=2,002$; one then introduces the “magnetic quantum” following, called magneton of Bohr : $\backslash \; mu\_\; \{B\}\; =\; \backslash \; frac\; \{E\; \backslash \; hbar\}\; \{2\; m\_e\}$

Abnormal magnetic moment of the electron

See also: abnormal magnetic Moment

The equation of Dirac predicted for the electron a factor of Moor exactly equal to: $g=2$. However, the allowed experimental value in 2005 is worth:

$g\; \backslash \; \backslash simeq\; \backslash \; 2,002\; \backslash \; 319\; \backslash \; 304\; \backslash \; 373\; \backslash \; 7$

There thus exists a variation, detected for the first time in 1947 in the hyperfine structure of hydrogen and deuterium: one then speaks about the abnormal magnetic Moment of the electron. The Quantum theory of the fields of the standard Modèle makes it possible to give an account of this anomaly with a very high degree of accuracy.

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