|S| = √(s(s+1))ℏ for integer or half-integer s: multiplicity 2s+1, the mₛ ladder, S_z and polar angle for a chosen projection, and why |S| > s always holds.
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Key facts
Category
Science & Education
Input types
number
Output type
text
Sample coverage
4
API ready
Yes
Overview
The Spin Angular Momentum Calculator computes the magnitude of spin angular momentum, total multiplicity, allowed spin projections, and precession polar angles for any integer or half-integer spin system from s = 0 to 9.5.
When to use
Analyzing quantum mechanical spin states and multiplicity in atomic physics or spectroscopy problems.
Finding the magnitude |S| and z-axis projection S_z for electrons, nucleons, or composite particles.
Determining the polar cone angle θ between the total spin vector and the quantization axis.
How it works
1Enter a valid integer or half-integer spin quantum number s between 0 and 9.5.
2Optionally provide a spin projection value mₛ within the allowed range from −s to +s in unit steps.
3The tool calculates the total magnitude |S| = √(s(s+1))ℏ in both units of ℏ and Joules-seconds (J·s).
4It outputs the multiplicity (2s + 1), the ladder of allowed projections, and the polar angle θ = arccos(mₛ / √(s(s+1))).
Use cases
Quantum physics homework verifying spin-1/2 electron states and the 54.74° magic angle orientation.
Nuclear magnetic resonance (NMR) analysis of quadrupolar nuclei with half-integer spins such as s = 3/2.
Particle physics calculations of vector bosons (s = 1) or higher-spin resonance states like the Δ baryon (s = 3/2).
Examples
1. Electron Spin-1/2 Vector and Magic Angle
Physics Undergraduate
Background
A student is solving an introductory quantum mechanics problem on intrinsic electron spin and Zeeman splitting.
Problem
Calculate the magnitude of electron spin angular momentum and the polar angle of the spin-up state.
How to use
Set the spin quantum number s to 0.5 and enter mProjection as 0.5.
s = 0.5, mProjection = 0.5
Outcome
Yields |S| = 0.866025 ℏ (9.133e-35 J·s), multiplicity 2, and a polar angle θ = 54.735610° from the +z axis.
2. Δ Baryon Spin-3/2 State Analysis
Nuclear Physics Researcher
Background
A researcher is modeling spin projections and angular distributions for a spin-3/2 resonance particle.
Problem
Determine all allowed projection states and the orientation angle for the maximally down projection mₛ = -1.5.
How to use
Input 1.5 into the spin quantum number field and -1.5 into the spin projection field.
FAQ
What values of spin quantum number s are supported?
The calculator supports integer and half-integer values from 0 up to 9.5 in steps of 0.5.
Why is the spin magnitude |S| always larger than the maximum projection s?
Quantum mechanics dictates |S| = √(s(s+1))ℏ, which strictly exceeds s·ℏ for any s > 0 due to non-zero transverse uncertainty.
What is the physical meaning of multiplicity (2s + 1)?
Multiplicity represents the total number of distinct allowed orientations (projections mₛ) along the quantization axis.
How is the polar angle θ calculated?
The polar angle is evaluated using θ = arccos(mₛ / √(s(s+1))), describing the tilt of the spin vector relative to the +z axis.
What happens if I leave the mₛ projection field empty?
The calculator displays the total magnitude, multiplicity, and the complete ladder of allowed mₛ values without computing a single projection angle.