Derivation
As a quantized angular momentum, (see angular momentum quantum number) it holds that
where
- is the quantized spin vector
- is the norm of the spin vector
- is the spin quantum number associated with the spin angular momentum
- is the reduced Planck constant.
Given an arbitrary direction z (usually determined by an external magnetic field) the spin z-projection is given by
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.
Read more about this topic: Spin Quantum Number