0.1. POTENTIAL ENERGY OF A MAGNETIC DIPOLE 221
randomsuperpositions of those states, ones that are as likely to be eigenstates ofSxorSy(or the
spin resolved relative to some randomz-axis directionSz′) as in pureSzeigenstates.
We are thus forced into using astatistical description of the state of the protons. Basically,
their spin angular momentum (and magnetic moments) are equally likelyto point in all possible
directions. This is not saying that each proton doesn’thavea particular spin direction at any given
time, only that we don’t know what it is.
If we put this collection of more or less isolated spins into a strong external magnetic field,
however, the situation changes. Now the field itselfdefinesa suitablez-direction. Furthermore,
we know that theenergyof the protons changes. The potential energy of each proton in astrong
external magnetic fieldB~=B 0 zˆis given by:
U=−~mp·B~=−mzB 0 (483)This energy is minimized when the spin isalignedwith the external field, and is maximized when
it isantialignedwith the external field. At any finite temperature, the lower energy of spins when
they are aligned makes this statemore probablethan spins aligned the other way. In equilibrium one
then expects to havemorespins aligned with the field than anti-aligned and the system develops
a finite total magnetic moment aligned with the field (where in the absence of the field, its total
magnetic moment is within random fluctuations of zero).
This state of affairs is not realized instantly, however. If one starts with no field and then “sud-
denly” turns on a strong field, the system will relax towards the equilibrium state (approximately)
exponentially, with a characteristic relaxation time that we will callT 1. During this relaxation, the
systemloses electromagnetic energy via radiation and spin-lattice relaxation processes but these
processes are largely incoherent and do not produce a detectablesignal.