Spin quantum number

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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. 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.

Derivation[edit]

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 the reduced Planck 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 ms is the secondary spin quantum number, ranging from −s to +s in steps of one. This generates 2 s + 1 different values of ms.

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.

Algebra[edit]

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

where εlmn is the (antisymmetric) Levi-Civita symbol. This means that it 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 \rangle= {\hbar}^2 s(s+1) | s, m_s \rangle
 S_z | s, m_s \rangle = \hbar m_s | s, m_s \rangle
 S_\pm | s, m_s \rangle = \hbar \sqrt{s(s+1)-m_s(m_s \pm 1)} | s, m_s \pm 1 \rangle

where  S_\pm = S_x \pm i S_y are the creation and annihilation (or "raising" and "lowering" or "up" and "down") operators.

Electron spin[edit]

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". 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[citation needed] 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. The so-called "Transition rule" limits what "jumps" are possible. In general, a jump or "transition" is allowed only if all three numbers change in the process. This is because a transition will be able to cause the emission or absorption of electromagnetic radiation only if it involves a change in the electromagnetic dipole 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 by George Uhlenbeck and Samuel Goudsmit, students of Paul Ehrenfest (who rejected the idea), and independently by Ralph 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 a way analogous to other quantized angular momenta, L, it is possible to obtain an expression for the total spin angular momentum:

S = \hbar\sqrt{\frac{1}{2}\left( \frac{1}{2}+1 \right) } = \frac{\sqrt{3}}{2}\hbar

where

\hbar is the reduced Planck 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

whose 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 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 Landé g-factor

and by the equation:

\mathbf{\mu_z} = \pm \frac{1}{2}g{\mu_B}

where \mu_B is the Bohr magneton.

When atoms have even numbers of electrons the spin of each electron in each orbital has opposing orientation to that of its immediate neighbor(s). However, many atoms have an odd number of electrons or an arrangement of electrons in which there is an unequal number of "spin-up" and "spin-down" orientations. These atoms or electrons are said to have unpaired spins that are detected in electron spin resonance.

Detection of spin[edit]

When 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.

The Stern–Gerlach experiment[edit]

The theory of spatial quantization of the spin moment of the momentum of electrons of atoms situated in the magnetic field needed to be proved experimentally. In 1920 (two years before the theoretical description of the spin was created) Otto Stern and Walter Gerlach observed it in the experiment they conducted.

Dirac equation solves spin[edit]

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 Wave Equation which was relativistically invariant (unlike Schrödinger's one), 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.

See also[edit]

External references[edit]