Application to MRI scans
In nuclear magnetic resonance
(NMR), which is the techno-
logical basis for medical MRI
scans, a very large DC mag-
netic field,∼ 3 T, is applied
to the sample using a super-
conducting magnet. Protons
in hydrogen atoms have their
spin states split in energy by by
∆E= 2ε=k~. After∼ 1 s,
the protons reach a new ther-
mal equilibrium state in which
the probability of| ↓〉and| ↑〉
differ by∼ 10 −^5.
A brief radio-frequency pulse is
then applied at the frequency
ωsuch that ∆E=~ω, so that
a radio photon has the correct
energy to cause a transition
between the two spin states.
Since there is a large num-
ber of protons, and they inter-
act with one another, their re-
sponse can be described semi-
classically. The magnetiza-
tion vector of the sample pre-
cesses in a complicated man-
ner, which can be affected by
the polarization and duration
of the pulse.
After the radio pulse has
stopped, the protons return
to equilibrium again, and this
changing magnetic field causes
induced electric fields in a coil,
which picks up a signal at the
frequencyω. Spatial resolution
for imaging is accomplished by
adding a gradient to the mag-
netic field, amounting to a few
percent over a distance of one
meter, so thatωhas different
values for different points in
space.Quantum-mechanically, the components of the magnetic field
will act like ordinary numbers (since the field is static, and we
aren’t trying to describe its dynamics quantum-mechanically), but
the components of the angular momentum are observable properties
of the proton, to be represented by operators. There is not always a
foolproof procedure for translating a classical expression into some-
thing quantum-mechanical, but in this example it seems sensible to
imagine that the classical expression for the energy can be made
into a quantum-mechanical energy operator that is obtained simply
by substituting the components of the angular momentum operator
into the expression.
What we have determined so far is that the HamiltonianHˆwill
simply be a weighted sum of ˆsx, ˆsy, and ˆsz, with the weighting
determined by the components of the magnetic field.
From our previous study of angular momentum in quantum me-
chanics, we know that a full description of our proton’s angular
momentum can be given by specifying the magnitude of the angu-
lar momentum, which is a fixed~/2, and its component along some
arbitrarily chosen axis, sayz. We have a state|↑〉which has eigen-
valuesz = +~/2, and a| ↓〉with−~/2. If the magnetic field is
parallel to thezaxis, then the action of the Hamiltonian is easy to
define in terms of these two states,
Hˆ|↑〉= ε|↑〉 and
Hˆ|↓〉=−ε|↓〉,where to keep the notation compact we writeε=k~/2, which is an
energy. The interpretation is that if there is no external magnetic
field (k= 0), then the energies of these two states are the same (and
set to zero because we choose that as an arbitrary definition), while
in the presence of aBzthe two energies become unequal. The pair
of states is “split” in energy by the field. Note that the above two
equations are sufficient to define the Hamiltonian forallstates, not
just for states in whichszhas a definite value. This follows from
the completeness principle — a state having a definite value of, say,
sxcan be written as some kind of linear combination of the form
α|↑〉+β|↓〉, and we then haveHˆ=αε|↑〉−βε|↓〉.
Now suppose that the magnetic field is not parallel to the z
axis. One way to handle this situation would be simply to redefine
the coordinate system so that thez axis was back in alignment
with the direction of the field. But suppose that’s not convenient.
Then the Hamiltonian will have a different form. But because the
Hamiltonian must be Hermitian (see p. 982), there is not much
freedom in choosing this form. It must look something like this:
Hˆ|↑〉=ε|↑〉+f|↓〉
Hˆ|↓〉=f∗|↑〉−ε|↓〉.Section 14.7 Applications to the two-state system 991