Spin Precession
Explore signed Larmor precession, physical frequencies and a normalized spin expectation vector.
About this tool
The valid vector appears immediately at t=0. Field, moment and tilt update directly and begin a fresh view; no Calculate action is needed. Entering t/T or adding T/4 inspects exact model time. Optional Play/Pause retains that time; Reset returns to t=0 with the same parameters. The component plot shows nₓ, nᵧ and n_z over one period, with markers at the same vector as the sphere. The plot uses (t/T) modulo1; full model time remains in the readout. Static axial or zero-field cases need no timeline or component plot. Options contains custom γ, screen speed, signed ω and γ. Screen speed preserves the physical state. Only the four raw parameter strings and moment choice are saved, including blank, invalid, comma and inactive custom-γ drafts. Legacy numeric inputs migrate. Model times,cycles,vectors and playback are never saved; reload begins at t=0. Hidden tabs stop playback; reduced motion retains manual controls.
Convention: μ=γS, H=−γBSz and B along the signed z direction. The Bloch expectation vector has x=sinθ cosφ, y=sinθ sinφ, z=cosθ, with φ=−γBt. Thus ω=φ̇=−γB, f=|γB|/(2π) and T=2π/|γB|. Positive azimuth runs from x towards y. Electron γ is negative and proton γ positive; reversing the field reverses rotation. For Bγ=0 there is no precession; an exactly axial vector also stays fixed.
Manual time is a positive number of periods t/T (0–1000); azimuth displays the signed remainder within one turn. Playback deliberately maps one physical period to 2.5/speed screen seconds, independently of B. Model time is the actual t=(t/T)T in SI seconds. Changing field magnitude changes f and T, not the illustrative screen rate. Playback stops at1000 periods. At zero frequency no period exists and the stationary state uses t=0.
The blue vector is 2⟨S⟩/ħ for a pure spin-½ state; it is not a spinning ball or a particle trajectory. Its length and z component stay fixed. The magnetic moment is antiparallel to spin when γ is negative. The diagram is an orthographic projection with a fixed camera; the dashed curve is the exact orbit, not a history dependent on frame rate. No relaxation, RF pulses, decoherence, mixed states or spinor interference are modelled; a Bloch vector repeats after2π although a spinor can change global sign.
Presets use 2022 CODATA magnitudes: |γe|=1.76085962784×10¹¹ and γp=2.6752218708×10⁸ rad/(s·T). The electron sign is attached by the μ=γS convention; NIST lists its magnitude. A custom signed γ, including zero, describes an ideal moment.