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» Basic Physics

Where does the MRI signal come from? This section explores the basic physics of magnetic resonance imaging.

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Larmor Equation - QM Derivation

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Derive the Larmor equation from a quantum mechanical perspective.

This derivation is shorter than the classical derivation. The de Broglie relation E=hω is required.

Reveal the QM derivation.

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

μ = γS = γhms

where γ is the gyromagnetic ratio, ms is the azimuthal spin quantum number (ms = ±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 = -γhmsB0 (the cosθ term of the dot product = ±1 since the values of ms 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 = Ehigher - Elower
= E(ms=-½) - E(ms=+½)
= ½γhB0 - (-½γhB0)
= γhB0

for a constant field B=B0. According to the Planck-Einstein relation E=hω we may write

ΔE = γhB0 = hω0

from which the Larmor equation results:

ω0 = γB0.

Further reading on this topic:
Books: MRI: Physical Principles & Sequence Design p71, MRI From Picture to Proton p138

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