Spin Angular Momentum Calculator
Calculates the magnitude of a particle's spin angular momentum from its spin quantum number s. Used in quantum mechanics courses and magnetic resonance physics to quantify intrinsic particle spin.
About this calculator
In quantum mechanics, every particle possesses an intrinsic angular momentum called spin, which has no classical analogue. The magnitude of the spin angular momentum vector is given by: |S| = ℏ √(s(s + 1)), where ℏ = 1.055×10⁻³⁴ J·s is the reduced Planck constant and s is the spin quantum number. For electrons, s = ½, giving |S| = ℏ√(3/4) ≈ 9.133×10⁻³⁵ J·s. Protons and neutrons also have s = ½, while photons have spin s = 1 and delta baryons have s = 3/2. Importantly, |S| is not the same as the z-component of spin, which is quantised as m_s = −s, −s+1, …, +s and takes values ±ℏ/2 for an electron. Spin angular momentum is responsible for the Pauli exclusion principle, NMR spectroscopy, and the fine structure of atomic spectra.
How to use
Calculate the spin angular momentum of an electron (s = ½). Step 1 — compute s(s + 1) = 0.5 × 1.5 = 0.75. Step 2 — take the square root: √0.75 ≈ 0.8660. Step 3 — multiply by ℏ: |S| = 1.055×10⁻³⁴ × 0.8660 ≈ 9.136×10⁻³⁵ J·s. For comparison, try a spin-1 particle such as a photon: s(s+1) = 1 × 2 = 2, √2 ≈ 1.4142, |S| = 1.055×10⁻³⁴ × 1.4142 ≈ 1.492×10⁻³⁴ J·s — noticeably larger than for a spin-½ particle.
Frequently asked questions
What values can the spin quantum number s take for different particles?
The spin quantum number s is a fixed property of each particle type and can be any non-negative integer or half-integer. Fermions (electrons, protons, neutrons, quarks) have half-integer spin: s = ½, 3/2, etc. Bosons (photons, W/Z bosons, Higgs boson) have integer spin: s = 0, 1, 2. The spin determines the particle's statistics — fermions obey Fermi–Dirac statistics and the Pauli exclusion principle, while bosons obey Bose–Einstein statistics and can occupy the same quantum state. The graviton, if it exists, is predicted to have s = 2.
Why is the magnitude of spin angular momentum larger than its maximum z-component?
The magnitude |S| = ℏ√(s(s+1)) is always greater than the maximum z-projection S_z = ℏs, because quantum mechanics forbids perfect alignment of the spin vector along any axis. If |S| equalled S_z, the x and y components would both be exactly zero, violating the uncertainty relations [S_x, S_y] = iℏS_z. This means the spin vector precesses around the measurement axis rather than pointing exactly along it — a purely quantum mechanical effect with no classical counterpart.
How is spin angular momentum relevant to magnetic resonance imaging (MRI)?
Protons in the human body have spin s = ½ and possess a magnetic moment proportional to their spin angular momentum. In a strong external magnetic field, spin energy levels split into two states (spin-up and spin-down) separated by ΔE = 2μ_p B, where μ_p is the proton magnetic moment. Radiofrequency pulses tuned to the Larmor frequency flip the spins; when they relax back, they emit detectable RF signals. Spatial variation of the magnetic field encodes position information, allowing MRI scanners to reconstruct three-dimensional images of soft tissue non-invasively.