# Larmor Equation - QM Derivation

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

μ = γS = γ~~h~~m_{s}

where γ is the gyromagnetic ratio, m_{s} is the azimuthal spin quantum number (m_{s} = ±½ for ^{1}H) 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 = -γ~~h~~m_{s}B_{0} (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)

= ½γ~~h~~B_{0} - (-½γ~~h~~B_{0})

= γ~~h~~B_{0}

for a constant field B=B_{0}. According to the Planck-Einstein relation E=~~h~~ω we may write

ΔE = γ~~h~~B_{0} = ~~h~~ω_{0}

from which the Larmor equation results:

**ω _{0} = γB_{0}**.

Further reading on this topic:

Books: MRI: Physical Principles & Sequence Design p71, MRI From Picture to Proton p138