Quantum Spin and Angular Momentum Calculator
Calculates the magnitude, z-component, magnetic moment, and multiplicity of quantum spin or orbital angular momentum. Use it when solving atomic physics problems or interpreting spectral fine structure.
About this calculator
In quantum mechanics, angular momentum is quantised. For a quantum number ℓ (orbital) or s (spin), the magnitude of the angular momentum vector is |L| = ℏ√(ℓ(ℓ+1)), where ℏ = 1.055×10⁻³⁴ J·s is the reduced Planck constant. The z-component along an applied field is Lz = ℏ·mₗ, where mₗ is the magnetic quantum number ranging from −ℓ to +ℓ. The number of allowed orientations is the multiplicity 2ℓ+1. The magnetic moment associated with orbital motion is μ = μ_B√(ℓ(ℓ+1)), where μ_B = 9.274×10⁻²⁴ J/T is the Bohr magneton. These quantities determine how atoms interact with magnetic fields (Zeeman effect), the selection rules for optical transitions, and the structure of the periodic table. Spin follows identical mathematics with s = 1/2 for electrons.
How to use
Calculate the magnitude and z-component of orbital angular momentum for an electron in a p-orbital (ℓ = 1) with magnetic quantum number mₗ = −1. Magnitude: |L| = ℏ√(1×2) = 1.055×10⁻³⁴ × √2 = 1.055×10⁻³⁴ × 1.4142 ≈ 1.492×10⁻³⁴ J·s. Z-component: Lz = ℏ·mₗ = 1.055×10⁻³⁴ × (−1) = −1.055×10⁻³⁴ J·s. Multiplicity = 2×1+1 = 3 (mₗ can be −1, 0, or +1). Enter quantum number = 1, magnetic number = −1, and select the desired property to read each result.
Frequently asked questions
What is the difference between spin angular momentum and orbital angular momentum in quantum mechanics?
Orbital angular momentum arises from an electron's motion around the nucleus and is described by the azimuthal quantum number ℓ = 0, 1, 2, … (s, p, d, f orbitals). Spin angular momentum is an intrinsic property of particles with no classical analogue; for electrons s = 1/2, giving only two orientations (spin-up and spin-down). Both obey the same quantisation rules |J| = ℏ√(j(j+1)) and Jz = ℏ·mⱼ. They can couple together via spin-orbit interaction to produce fine-structure splitting in atomic spectra.
Why can the angular momentum vector not point exactly along the z-axis in quantum mechanics?
The Heisenberg uncertainty principle prevents simultaneously knowing all three components of angular momentum with arbitrary precision. Only the magnitude |L|² and one component (conventionally Lz) can be simultaneously well-defined. Because |L| = ℏ√(ℓ(ℓ+1)) is always larger than the maximum Lz = ℏ·ℓ (since √(ℓ(ℓ+1)) > ℓ), the vector can never be perfectly aligned with the z-axis — it always has indeterminate x and y components. This is visualised by the vector precessing around the z-axis on a cone.
How does the magnetic quantum number determine the energy splitting in a magnetic field?
In an external magnetic field B, each orientation of the angular momentum corresponds to a slightly different energy given by ΔE = mₗ·μ_B·B for orbital angular momentum (or g·mₛ·μ_B·B for spin). Because mₗ takes 2ℓ+1 distinct integer values, a single spectral line splits into multiple lines — the normal or anomalous Zeeman effect. This splitting is directly proportional to B and to mₗ, so measuring it allows physicists to determine quantum numbers experimentally and to map magnetic field strengths in stellar atmospheres via spectropolarimetry.