Spin quantum number

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In atomic physics, the spin quantum number is a quantum number that parametrizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle. 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.

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

[edit] 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 2s+1 different values of m_s.

The allowed values for s are non-negative integers or half-integers. Fermions (such as the electron, proton or neutron) have half-integer values, whereas bosons (e.g. photon, mesons) have integer spin values.

[edit] 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:

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

This means that is impossible to know two coordinates of the spin at the same time because of the restriction of the Uncertainty principle.

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

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

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

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

where $S_\pm = S_x \pm \mathrm{i} S_y$ are the up and down operators.

[edit] Electron spin

There are a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, l, m, and s specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The wavefunction of the Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. However, line emission spectra of some atoms when measured in an external magnetic field turned out to be more complicated than predicted by the first three quantum numbers. There needed to be a fourth quantum number that could properly predict spectra that matched the complexity found in nature so that this new quantum number had to behave as if it were also derived from the algebra of angular momentum vectors. 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, by introducing the idea of the self-rotation of the electron which would naturally be an angular momentum vector.

An electron spin s = 1/2 is an intrinsic property of electrons. Electrons have intrinsic angular momentum characterized by quantum number 1/2. In the pattern of other quantized angular momenta, this gives total angular momentum:

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

where

$\hbar$ is Planck's reduced constant (Dirac's constant).

The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels. The reduced Planck's constant is used because in a wave, a cycle is defined by the return from a certain position to the same position such as from the top of one crest to the next crest. This actually is equivalent to a circle both having 360 degrees. There are 2 pi radians per cycle in a wave. Therefore, dividing h by 2π describes a constant that when multiplied by the frequency of a wave gives the energy of one cycle. When the subatomic particle the electron was being described by wavefunctions in Dirac's equation, it was found that the property of spin of all particles is a multiple of h-bar denoted by $\hbar$ , that is, h (Planck's constant) divided by 2π. H-bar or $\hbar$ has an even multiple for bosons and an odd multiple for fermions.

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:

$\mathbf{S_z} = \pm 1/2\hbar$

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

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. 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.

[edit] 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.

[edit] Dirac equation solves spin

When the idea of electron spin was first introduced in 1925, even Wolfgang Pauli had trouble accepting Ralph Kronigs 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 the Dirac equation all four quantum numbers including the additional quantum number s arose naturally during its solution.