Spin quantum number - Citizendia

In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle. Atomic physics (or atom physics) is the field of Physics that studies atoms as an isolated system of Electrons and an atomic nucleus. Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position 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 The spin quantum number is the fourth of a set of quantum numbers (the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number) which describe the unique quantum state of an electron and is designated by the letter s. Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. In Atomic physics, the principal quantum number symbolized as n is the first of a set of Quantum numbers (which includes the principal quantum The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number In Atomic physics, the magnetic quantum number is the third of a set of Quantum numbers (the Principal quantum number, the Azimuthal quantum number In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system.

1 Derivation
2 Algebra
3 Electron spin
4 Detection of spin
5 Dirac equation solves spin
6 See also
7 External references

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). The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta.

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 2s+1 different values of m_s.

The allowed values for s are non-negative integers or half-integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a half-integer is a Number of the form n + 1/2 where n is an Integer. Fermions (such as the electron, proton or neutron) have half-integer values, whereas bosons (e. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. 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. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein g. photon, mesons) have integer spin values. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin.

Algebra

The algebraic theory of spin is a carbon copy of the Angular momentum in quantum mechanics theory. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position First of all, spin satisfies the fundamental commutation relation:

$[S_i, S_j ] = i \hbar \epsilon_{ijk} S_k$ , $\left[S_i, S^2 \right] = 0$

where ε_lmn is the (antisymmetric) Levi-Civita symbol. The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor This means that is impossible to know two coordinates of the spin at the same time because of the restriction of the Uncertainty principle. 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

Next, the eigenvectors of $S 2$ and $S z$ satisfy:

$S^2 | s, m_s \rang = {\hbar}^2 s(s+1) | s, m_s \rang$

$S_z | s, m_s \rang = \hbar m_s | s, m_s \rang$

$S_\pm | s, m_s \rang = \hbar \sqrt{s(s+1)-m_s(m_s \pm 1)} | s, m_s \pm 1 \rang$

where $S_\pm = S_x \pm \mathrm{i} S_y$ are the creation and annihilation (or "raising" and "lowering" or "up" and "down") operators. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

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. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom This was successful in explaining many features of atomic spectra. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ

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 the momentum of a mass in circular motion about a point. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product 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. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ So called "selection rules" 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 electromagnetic dipole of the atom. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a

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. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ The Zeeman effect (ˈzeɪmɑːn is the splitting of a Spectral line into several components in the presence of a static Magnetic field. A solution to this problem was suggested in early 1925 by George Uhlenbeck and Samuel Goudsmit, students of Paul Ehrenfest (who rejected the idea), and independently by Ralph Kronig, one of Landé's assistants. George Eugene Uhlenbeck ( December 6 1900, Batavia Dutch East Indies &ndash October 31 1988, Boulder Colorado) Samuel Abraham Goudsmit (born July 11, 1902 Den Haag, The Netherlands, died December 4, 1978 in Reno Nevada Paul Ehrenfest ( January 18, 1880 – September 25, 1933) was an Austrian Physicist and Mathematician, who Ralph Kronig was a German-American Physicist ( March 10, 1904 — November 16, 1995) Alfred Landé ( 13 December, 1888 &ndash 30 October 1976) was a German-American physicist known for his contributions to quantum theory 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).

A spin angular momentum, characterized by a quantum number s = 1/2, is an intrinsic property of electrons. In the pattern of other quantized angular momenta, it gives a total spin angular momentum:

$\mathbf{S} = \hbar\sqrt{\frac{1}{2}(\frac{1}{2}+1)} = \frac{\sqrt{3}}{2}\hbar$

where

$\hbar$ is Planck's reduced constant (Dirac's constant). The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta.

The energy of any wave is the frequency multiplied by Planck's constant. When the electron was being described by wavefunctions in Dirac's equation, it was found that the spin property of all fundamental particles is a multiple $\hbar$ . The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides If this multiple is even the particle is a boson and if it is odd the particle is a fermion. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi.

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. In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current 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

μ 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. In Physics, the Landé g-factor is a particular example of a G-factor, namely for an Electron with both spin and Orbital angular In Atomic physics, the Bohr magneton (symbol \mu_\mathrm{B} is named after the Physicist Niels Bohr. 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. Electron paramagnetic resonance (EPR or electron spin resonance (ESR Spectroscopy is a technique for studying Chemical species that have one or more unpaired

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. In Quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is an important 1922 experiment on the Deflection

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. Ralph Kronig was a German-American Physicist ( March 10, 1904 — November 16, 1995) 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 the Dirac equation all four quantum numbers including the additional quantum number s arose naturally during its solution. In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides

External references

Full treatment of Spin--including origins, evolution of Spin Theory, and details of the Spin equations

Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. The Azimuthal quantum number (or orbital angular momentum quantum number, second quantum number) symbolized as l (lower-case L is a Quantum number In Atomic physics, the magnetic quantum number is the third of a set of Quantum numbers (the Principal quantum number, the Azimuthal quantum number In Atomic physics, the principal quantum number symbolized as n is the first of a set of Quantum numbers (which includes the principal quantum See also Azimuthal quantum number#Addition of quantized angular momenta In Quantum mechanics, the total angular quantum momentum numbers parameterize the total Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides Ralph Kronig was a German-American Physicist ( March 10, 1904 — November 16, 1995) In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system.

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Contents

Derivation

Algebra

Electron spin

Detection of spin

Dirac equation solves spin

See also

External references