Larmor Equation - QM Derivation

The direct relationship between the magnetic moment (μ) and the spin angular momentum vector (S) is quantized thus:

μ = γS = γhm_s

where γ is the gyromagnetic ratio, m_s is the azimuthal spin quantum number (m_s = ±½ for ¹H) and h is the Planck constant over 2π. The formula for the energy E associated with a magnetic moment μ in an external magnetic field B is

E = -μ · B (the dot product)

Thus, discrete energy levels may be defined by

E = -γhm_sB₀ (the cosθ term of the dot product = ±1 since the values of m_s correspond to spin polarisation with and against the field; θ is 0° or 180°).

This equation corresponds to the Zeeman energy eigenstates. The quantum of energy which can be absorbed or released by the spin system is, therefore,

ΔE = E_higher - E_lower
= E(m_s=-½) - E(m_s=+½)
= ½γhB₀ - (-½γhB₀)
= γhB₀

for a constant field B=B₀. According to the de Broglie relation E=hω we may write

ΔE = γhB₀ = hω₀

from which the Larmor equation results:

ω₀ = γB₀.