# Zeeman Eigenstates (QM)

Zeeman eigenstates associated with m

_{s}

CORRECT. A spin-half particle (such as the hydrogen nucleus) is not restricted to either state, but is in a superposition of the two energy eigenstates along the direction of the external magnetic field, B_{0}. The eigenvalues of the two states are +½ (*polarised* along the z-axis) and -½ (*polarised* along the minus-z-axis).

With reference to these spin polarisations, we can *imply* that a number of particles are in the higher energy state as "spin down" or "antiparallel" with respect to the external magnetic field B_{0}, though this is not strictly true. Similarly, we can imply that particles are in the lower energy state, referred to as "spin up" or "parallel" with respect to B_{0}. In fact, assigning spins a parallel or antiparallel direction is only a way of describing the *observable population difference* between the two energy *eigenstates*, and any notion of alignment of spins with B_{0} is only a *physical interpretation* of the relevant quantum mechanical equations (not given here).

INCORRECT. The *spin up* and *spin down* spin states are sometimes compared with the Zeeman energy levels. But this is a poor comparison, and may lead to confusion about what is classical mechanics (CM) and what is quantum mechanics (QM). Even in the classical description, spins are not restricted to align with (parallel or spin up) or against (antiparallel or spin down) the external magnetic field direction. However, a simplification of the classical description of MRI does describe these particular spin states, assigning *equivalent* numbers of spins to the two states to produce the same net magnetisation vector. So this simplification can be helpful.

It is probably clearer to consider the Zeeman energy levels only as part of the QM description of MRI. They are named eigenstates for a reason: that these energy levels only describe the states which are observable. Spins are not restricted to either state but exist in linear combinations of both states. This is certainly not a CM concept.

Try again.

Further reading on this topic:

Books: Spin Dynamics p241, Q&A in MRI p22

Online: Wikipedia