- Spin quantum number
In

atomic physics , the**spin quantum number**is aquantum number that parameterizes the intrinsicangular momentum (or spin angular momentum, or simply spin) of a given particle. The**spin quantum number**is the fourth of a set ofquantum number s (theprincipal quantum number , theazimuthal quantum number , themagnetic quantum number , and the spin quantum number) which describe the uniquequantum state of an electron and is designated by the letter "s".**Derivation**As a quantized angular momentum, (see

angular momentum quantum number ) it holds that :$Vert\; mathbf\{s\}\; Vert\; =\; sqrt\{s\; ,\; (s+1)\}\; ,\; hbar$ where : $mathbf\{s\}$ is the quantized spin vector,: $Vert\; mathbf\{s\}Vert$ is the norm of the spin vector,:**$s$**is the spin quantum number associated with the spin angular momentum,: $hbar$ is Planck's reduced constant (Dirac's constant ).Given an arbitrary direction "z" (usually determined by an external magnetic field) the spin "z"-projection is given by :$s\_z\; =\; m\_s\; ,\; hbar$

where "m

_{s}" is the**secondary spin quantum number**, ranging from −"s" to +"s" in steps of one. This generates 2"s"+1 different values of "m_{s}".The allowed values for "s" are non-negative

integer s orhalf-integer s.Fermion s (such as theelectron ,proton orneutron ) have half-integer values, whereasboson s (e.g.photon ,meson s) have integer spin values.**Algebra**The algebraic theory of spin is a carbon copy of the Angular momentum in quantum mechanics theory.First of all, spin satisfies the fundamental commutation relation::$[S\_i,\; S\_j\; ]\; =\; i\; hbar\; epsilon\_\{ijk\}\; S\_k$, $left\; [S\_i,\; S^2\; ight]\; =\; 0$where ε

_{lmn}is the (antisymmetric)Levi-Civita symbol . This means that is impossible to know two coordinates of the spin at the same time because of the restriction of theUncertainty principle .Next, the eigenvectors of $S^2$ and $S\_z$ satisfy::: $S^2\; |\; s,\; m\_s\; ang\; =\; \{hbar\}^2\; s(s+1)\; |\; s,\; m\_s\; ang$:: $S\_z\; |\; s,\; m\_s\; ang\; =\; hbar\; m\_s\; |\; s,\; m\_s\; ang$:: $S\_pm\; |\; s,\; m\_s\; ang\; =\; hbar\; sqrt\{s(s+1)-m\_s(m\_s\; pm\; 1)\}\; |\; s,\; m\_s\; pm\; 1\; ang$where $S\_pm\; =\; S\_x\; pm\; mathrm\{i\}\; S\_y$ are the creation and annihilation (or "raising" and "lowering" or "up" and "down") operators.

**Electron spin**Early attempts to explain the behavior of electrons in atoms focused on solving the Schrödinger wave equation for the hydrogen atom, the simplest possible case, with a single electron bound to the

atomic nucleus . This was successful in explaining many features of atomic spectra.The solutions required each possible state of the electron to be described by three "quantum numbers", "n", "l", and "m". These were identified as, respectively, the electron "shell" number, "n", the "orbital" number, "l", and the "orbital angular momentum" number "m".

Angular momentum is a so-called "classical" concept measuring themomentum of a mass in circular motion about a point. The shell numbers start at 1 and increase indefinitely. Each shell of number "n" contains "n²" orbitals. Each orbital is characterized by its number "l", where "l" takes integer values from "0" to "n-1", and its angular momentum number "m", where m takes integer values from "+l" to "-l". By means of a variety of approximations and extensions, physicists were able to extend their work on hydrogen to more complex atoms containing many electrons.Atomic spectra measure radiation absorbed or emitted by electrons "jumping" from one "state" to another, where a state is represented by values of "n", "l", and "m". So called "

Transition rule " limit what "jumps" are possible. Generally a jump or "transition" is only allowed if all three numbers change in the process. This is because a transition will only be able to cause the emission or absorption of electromagnetic radiation if it involves a change in the electromagneticdipole of the atom.However, it was recognized in the early years of quantum mechanics that atomic spectra measured in an external magnetic field (see

Zeeman effect ) cannot be predicted with just "n", "l", and "m". A solution to this problem was suggested in early 1925 byGeorge Uhlenbeck andSamuel Goudsmit , students ofPaul Ehrenfest (who rejected the idea), and independently byRalph Kronig , one of Landé's assistants. Uhlenbeck, Goudsmit, and Kronig introduced the idea of the self-rotation of the electron, which would naturally give rise to an angular momentum vector in addition to the one associated with orbital motion (quantum numbers "l" and "m").The spin angular momentum, is characterized by a quantum number;

**s = 1/2**specifically for electrons. In the an analogous way of other quantized angular momenta, L, it is possible to obtain an expression for the total spin angular momentum::$mathbf\{S\}\; =\; hbarsqrt\{frac\{1\}\{2\}(frac\{1\}\{2\}+1)\}\; =\; frac\{sqrt\{3\{2\}hbar$

where: $hbar$ is Planck's reduced constant (

Dirac's constant ).The hydrogen spectra fine structure is observed as a doublet corresponding to two possibilities for the z-component of the angular momentum, where for any given direction z:

:$mathbf\{S\_z\}\; =\; pm\; frac\{1\}\{2\}hbar$

which solution has only two possible "z" components for the electron. In the electron, the two different spin orientations are sometimes called "spin-up" or "spin-down".

The spin property of an electron would classically give rise to

magnetic moment which was a requisite for the fourth quantum number. The electron spin magnetic moment is given by the formula::$mathbf\{mu\_s\}\; =\; -frac\{e\}\{2m\}gS$

where:e is the charge of the electron:g is the

Lande g-factor and by the equation:

:$mathbf\{mu\_z\}\; =\; pm\; frac\{1\}\{2\}g\{mu\_B\}$

where:g is the Lande g-factor:$mu\_B$ is the

Bohr magneton When atoms have even numbers of electrons the spin of each electron in each orbital has opposing orientation in different directions. However, many atoms have an odd number of electrons or an arrangement of electrons in which the number of "spin-up" and "spin-down" orientations are not the same. These atoms or electrons are said to have unpaired spins which are detected in

electron spin resonance .**Detection of spin**When the spectral lines of the hydrogen spectrum are examined at very high resolution, they are found to be closely-spaced doublets. This splitting is called fine structure and was one of the first experimental evidences for electron spin. The direct observation of the electron's intrinsic angular momentum was achieved in the

Stern-Gerlach experiment .**Dirac equation solves spin**When the idea of electron spin was first introduced in 1925, even Wolfgang Pauli had trouble accepting Ralph Kronig's model. The problem was not that a rotating charged particle would have given rise to a magnetic field, but that the electron was so small that the equatorial speed of the electron would have to be greater than the speed of light for the magnetic moment to be of the observed strength.

In 1930,

Paul Dirac developed a new version of the Schrödinger Wave Equation which was relativistically invariant, and predicted the magnetic moment correctly, and at the same time treated the electron as a point particle. In theDirac equation all four quantum numbers including the additional quantum number "s" arose naturally during its solution.**ee also***

Quantum number

**Azimuthal quantum number

**Magnetic quantum number

**Principal quantum number

**Total angular momentum quantum number

*Basic quantum mechanics

*Dirac equation

*Ralph Kronig

*Schrödinger equation

*Spin (physics)

*Quantum state **External references*** [

*http://math.ucr.edu/home/baez/spin/spin.html Full treatment of Spin--including origins, evolution of Spin Theory, and details of the Spin equations*]

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