Biomedical Engineering Reference
In-Depth Information
It is understood that the sum runs over distinct energy levels and not the individual quantum
states. I have used the symbol
n
in the summation to indicate the number of available
energy levels.
7.6 Two-Level Quantum System
You should be familiar with the phenomenon of spin and know that charged particles
with spin behave essentially as small bar magnets. According to the laws of (classical)
electromagnetism, any particle with charge
Q
, mass
M
and angular momentum
l
has a
magnetic dipole moment, a vector quantity that is usually written
p
m
:
Q
2
M
l
p
m
=
(7.21)
The energy of a magnetic dipole in an external field
B
(Figure 7.9) is given by
U
=−
p
m
B
(7.22)
and according to the rules of classical electromagnetism, all orientations of the dipole in
the field are allowed but each orientation corresponds to a different energy.
p
m
N
S
Figure 7.9
Magnetic dipole in a magnetic field
The quantummechanical picture is quite different; only certain orientations are allowed,
and the energy is quantized. The Stern-Gerlach experiment demonstrated that electrons
have an 'internal'magnetic moment called
electron spin
, and only two orientations are pos-
sible. These are visualized as being (roughly) parallel to the field and (roughly) antiparallel
to the field and the energies are given by
2π
g
e
e
h
2
m
e
B
where
m
s
is the spin quantumnumber, which takes values of
ε
ms
=
m
s
±
1
/
2
for electron spin. The lower
energy state ε
l
is therefore that with
m
s
=−
1
/
2
, the higher energy state ε
u
that corresponding
to
m
s
=+
1
/
2
.
