math lessons - Spin quantum number

The spin quantum number is a quantum number that parametrize the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle.

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

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, while bosons (e.g. photon, mesons) have integer values.

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