18.10.13Time Development of a Spin^12 State in a B field
Assume that we are in an arbitrary spin stateχ(t= 0) =
(
a
b)
and we have chosen the z axis tobe in the field direction. The upper component of the vector (a) is the amplitude to have spin up
along the z direction, and the lower component (b) is the amplitude tohave spin down. Because
of our choice of axes, the spin up and spin down states are also the energy eigenstates with energy
eigenvalues ofμBBand−μBBrespectively. We know that the energy eigenstates evolve with time
quite simply (recall the separation of the Schr ̈odinger equation whereT(t) =e−iEt/ ̄h). So its simple
to write down the time evolved state vector.
χ(t) =(
ae−iμBBt/ ̄h
beiμBBt/ ̄h)
=
(
ae−iωt
beiωt)
whereω=μB ̄hB.
So let’s say we start out in the state with spin up along the x axis,χ(0) =
( 1
√
2
√^1
2)
. We then have
χ(t) =( 1
√
2 e−iωt
√^1
2 eiωt)
.
〈χ(t)|Sx|χ(t)〉 =( 1
√ 2 e+iωt √^12 e−iωt
) ̄h
2(
0 1
1 0
)(√ 1
2 e−iωt
√^1
2 eiωt)
=
̄h
2( 1
√
2 e+iωt √ 1
2 e−iωt)( 1
√
2 e+iωt
√^1
2 e−iωt)
=
̄h
21
2
(
e+2iωt+e−^2 iωt)
=
̄h
2
cos(2μBBt/ ̄h)So again the spin precesses around the magnetic field. Becauseg= 2 the rate is twice as high as for
ℓ= 1.
18.10.14Nuclear Magnetic Resonance (NMR and MRI)
Nuclear Magnetic Resonanceis an important tool in chemical analysis. As the name implies,
it uses the spin magnetic moments of nuclei (particularly hydrogen)and resonant excitation.Mag-
netic Resonance Imaginguses the same principle to get an image (of the inside of the body for
example).
In basic NMR, a strong static B field is applied. A spin^12 proton in a hydrogen nucleus then has
two energy eigenstates. After some time, most of the protons fall into the lower of the two states.
We now use an electromagnetic wave (RF pulse) to excite some of theprotons back into the higher
energy state. Surprisingly, we can calculate this process already. The proton’s magnetic moment
interacts with the oscillating B field of the EM wave.