“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

Two-Level Systems by Hand — Every MRI magnet has a frequency printed on it

Exercises

Introductory Quantum Mechanics › Unit 3 · Two-level systems & spin-1/2 › Two-Level Systems by Hand › all problems

Practice Problem 2 of 9

Every MRI magnet has a frequency printed on it

Problem

A proton has gyromagnetic ratio

γ = 2.675 \times 1 0^{8}

rad·s

^{- 1}

·T

^{- 1}

. Find the Larmor frequency at 1 T and at 3 T. Why does an MRI facility quote its scanner in tesla and its radio hardware in MHz interchangeably?

Solution

Predict before reading on. The magnetic field plays the role ammonia’s tunneling played: it sets the level splitting. Which matrix element of the worked example corresponds to $γ B$ here, and what happened to the coupling $V$ ?

f_{L} = \frac{γ B}{2 π} = 42.58 MHz at 1 T, 127.7 MHz at 3 T

For a spin in a static field the 2×2 is already diagonal ( $V = 0$ ): splitting $ℏ γ B$ , and any transverse state precesses at exactly $f_{L}$ . Field strength and resonance frequency are the same dial in different units — 42.58 MHz/T is the proton’s conversion factor.

Result

f_{L} = 42.58 MHz (1 T), 127.7 MHz (3 T)