Spin Angular Momentum Calculator

Spin is an intrinsic form of angular momentum carried by every fundamental and composite particle, quite separate from any orbital motion. It is quantized, characterized by a spin quantum number s that can be a non-negative multiple of 1/2 (0, 1/2, 1, 3/2, ...), and it behaves mathematically like angular momentum even though particles are not literally spinning objects.

The magnitude of the total spin angular momentum is S = ħ√(s(s+1)), where s is the spin quantum number and ħ is the reduced Planck constant. The component along a chosen axis (usually z) is Sz = msħ, where ms ranges from −s to +s in integer steps, giving 2s+1 possible values.

Even in the state with the largest possible z-component, ms = s, the total magnitude S = ħ√(s(s+1)) is still larger than Sz = sħ, because s(s+1) is always greater than s². This reflects the fact that the spin vector can never point exactly along one axis, quantum uncertainty always leaves some spin 'hidden' in the other two components.

Electrons, protons, and neutrons all have spin s = 1/2 (fermions), photons have spin s = 1 (a boson), and the Higgs boson has spin s = 0. Composite particles combine the spins of their constituents, for example a helium-4 nucleus has total spin 0 even though it is built from spin-1/2 protons and neutrons.

For s = 1/2, ms can only be +1/2 or −1/2, giving exactly two spin states, often called 'spin up' and 'spin down'. This two-state structure is the basis of the qubit in quantum computing and explains why electron shells hold at most 2 electrons per orbital (the Pauli exclusion principle).

A particle's spin angular momentum gives it an intrinsic magnetic moment, roughly proportional to Sz, which is why particles with spin interact with magnetic fields (as seen in the Stern-Gerlach experiment and MRI). The proportionality constant, the gyromagnetic ratio, differs from particle to particle and even includes quantum corrections for the electron.

No. Spin is a purely quantum mechanical property with no exact classical counterpart, a point-like particle such as an electron has no physical structure that could be 'spinning' in the everyday sense. The name comes from an early, now-superseded classical picture, but the mathematics of spin angular momentum is entirely quantum.

This follows from the general theory of angular momentum in quantum mechanics: for any allowed s, the z-component quantum number ms takes exactly 2s+1 evenly-spaced values from −s to +s, each one unit of ħ apart. This gives, for example, 2 states for spin 1/2, 3 states for spin 1, and 4 states for spin 3/2.

Orbital angular momentum comes from a particle's motion through space around some point, quantized by the orbital quantum number l, and it can be zero. Spin angular momentum is intrinsic, present even for a particle at rest, quantized by s, and every fundamental particle has a fixed, unchangeable spin value (an electron is always spin 1/2, never anything else).

The spin quantum number s is fixed for a given type of particle (always 1/2 for an electron, for example), but ms can take any of the 2s+1 allowed values depending on the particle's actual spin state or how it was measured. Choosing s selects the particle type, while ms describes which specific spin orientation is being considered.