What does no spin mean
The Spin (from Engl. spin, Rotation, twist) is a quantum mechanical property of elementary particles. Though the classic idea of a rotating particles is wrong from today's and logical point of view, this picture can help with understanding. The spin behaves physically (e.g. with rotations of space) like the angular momentum. In addition, the conservation law for the total angular momentum only applies to the sum of the (classic) orbital angular momentum and the spin of a system. Therefore, in contrast to isospin, spin is not just a property that is (mathematically) analogous to angular momentum, but actually a type of angular momentum.
Relativistically (i.e. within four-dimensional space-time) spin is described with the aid of spinors.
Spin, spin quantum number and spin eigenstate
In quantum mechanics, an object given by its wave function (i.e. the state vector) Ψ is thrown into an eigenstate with an associated eigenvalue during an (ideal) measurement; the eigenvalues are the possible measured values. The spin eigenstates and eigenvalues result - in analogy to the quantum mechanical angular momentum - from the solutions of the following eigenvalue equations:
Are there and S.z the spin operators and s and sz the spin quantum number and the magnetic spin quantum number. To simplify matters, one also says that the particle has the spin s or be it a spinsParticles. An important property of spin is that only discrete values are possible, in contrast to angular momentum, but also half-integer values: A particle can have a spin of 0, 1/2, 1 (and so on, in steps of 1/2) .
The spin quantum number s of an elementary particle is fixed and cannot change. The possible szValues then result in
That means that a spin-0 particle only one Eigenvalue concerning. S.z owns. A spin 1/2 particle has two eigenvalues and in general a spin-s-particle has 2s+1 eigenvalues wrt S.z.
The states of the spin are given by 2s+ 1-component spinors shown. Instead of calculating in spinor space, however, it is mostly calculated in spin space. A spinor can be developed according to the basis vectors of the spin space:
In spin space, the spin operators are represented by matrices and the states by vectors.
Since this property, called spin, can only assume a very limited number of states, it cannot logically be a matter of an intrinsic rotation of the particles, as in the original idea. There would be no explanation for certain rotational speeds and identical orientation of the axes of rotation of all particles of the same kind.
Spin as a maintenance factor
The spin quantum number s of an elementary particle is unchangeable, but the spin orientation is not. The total spin of a system made up of several particles is also not a conserved quantity, but its total angular momentum. So if reactions are observed in atomic physics, for example, then the sum of all angular momentum eigenvalues before and after the reaction is the same.
The symmetry responsible for the conserved quantity "spin" is described by the actual Lorentz group SO (3,1). Seen in this way, the spin is a conserved quantity of relativistic theories. The compact part of the special orthochronous Lorentz group is isomorphic to SU (2) x SU (2). This fact justifies the systematics used in physics of the representations used for relativistic wave functions, which then deal with the two eigenvalues j1 and J2 let the SU (2) address:
|(1/2, 1/2)||Four-vectors, the defining transformation|
|(1/2, 0) x (0, 1/2)||Dirac representation|
|(1, 0) x (0, 1)||the antisymmetric tensor Fij|
Spin and Magnetic Moment
The spin of an elementary particle can be measured using the magnetic moment associated with it (Einstein-de-Haas effect, Stern-Gerlach experiment). Via this magnetic moment, the spin interacts with magnetic fields, so that a particle contains different amounts of energy depending on the orientation of its spin in a magnetic field. In this way, interactions occur in the atom between electron and atomic nucleus or between different electrons. This interaction is used technically in nuclear magnetic resonance (e.g. in magnetic resonance imaging) or in electron spin resonance.
Spin and Statistics
According to the standard model of physics, elementary particles are grouped according to their spin into bosons (integer spin) and fermions (half-integer spin). The allocation of "half" or "whole number spin" to the respective type of particle is justified by the spin statistics theorem. Bosons and fermions have a different symmetry behavior under rotations: The wave function of a boson turns into itself under a rotation of 360 degrees. In the case of a fermion, however, a rotation through 360 degrees does not produce the identical wave function, but - Ψ. A rotation of 720 degrees only results in Ψ.
This is the ultimate reason that the Pauli principle applies to fermions. If you swap two fermions, the sign of the total wave function of the system is negated, while the swap of two bosons leaves the wave function unaffected. The result is that two fermions can never be in the same state, whereas two bosons can. According to the spin statistics theorem, all fermions obey the Fermi-Dirac statistics, and all bosons obey the Bose-Einstein statistics.
Due to these properties and the indistinguishability of elementary particles, only two fermions can occupy one energy level, namely one with spin-up and one with spin-down. This law is known as the Pauli principle. In contrast, any number of bosons can occupy an energy state (Bose-Einstein condensate).
Spin, practical meaning
As shown in the previous section, all elementary particles are either fermions or bosons - depending on their spin. In particular, electrons are fermions. Therefore only one electron can occupy a state in an atom (see Pauli principle, the electrons must differ in an orbital at least by the spin (+1/2 or -1/2)). This is how the matter known to us is formed. If electrons were bosons, they would all occupy the lowest energy level in the atom. The matter known to us and in particular the connection of atoms to molecules would not exist. Therefore, spin is one of the most important properties of matter, along with electrical charge and mass.
A direct practical application of the spin of atomic nuclei is magnetic resonance imaging.
The spin was first noticed in connection with the measurement of emission spectra of alkali metals, namely through the splitting of spectral lines into two adjacent partial lines. In 1924, Wolfgang Pauli proposed a quantum mechanical degree of freedom that can assume two values for the electron. This enabled him to explain the splitting of the lines and to justify the fact that exactly two electrons share an atomic orbital (see also atomic model).
Ralph Kronig, an assistant to Alfred Landé, suggested in 1925 that this unknown degree of freedom was caused by the electron's own rotation. Due to Pauli's criticism of this idea, Kronig's proposal remained unpublished. Also in 1925, Samuel Abraham Goudsmit and George Eugene Uhlenbeck postulated the electron spin to explain the line splitting in the spectra of hydrogen-like atoms (e.g. the sodium D line) and the anomalous Zeeman effect.
In 1927 Pauli formulated a quantum theory of the spin for the electron. With the help of the Pauli matrices he was able to represent electron wave functions as 2-component spinors.
In 1928 Paul Dirac set up a relativistic equation of motion for the electron. The Dirac equation describes half-integer spin and also predicted an antiparticle of the electron, the later detected positron.
Category: quantum physics
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