Biomedical Engineering Reference
In-Depth Information
It can be shown that any particle whose wavefunction satisfies the Dirac equation must
be a spin -
1
/
2
particle. Not only that, the Dirac treatment gives the correct value for the
magnetic moment in that it gives
e
2
m
e
s
p
spin
=−
2
13.12 Measurement in the QuantumWorld
The process of measurement in quantum mechanics is subtler than in classical mechanics.
As discussed inAppendixA, the possible results of measurements depend on the eigenvalues
of the appropriate operators. Also, if we wish to make simultaneous measurements of two
observables (such as the linear momentum and position, or two components of an angular
momentum vector), we have to take account of Heisenberg's uncertainty principle. Certain
pairs of observables can be measured simultaneously to arbitrary precision whilst other
pairs cannot.
There is worse to come, for the word
measurement
gets used in two different ways in
quantum mechanics. Suppose we have a one-electron atom as discussed above; we know
that the energies are found from Schrödinger's equation, which I will write in Hamiltonian
form as
H
ψ
i
=
ε
i
ψ
i
If the atom is in state ψ
i
then repeated energy measurements on the same atom will always
yield the same result, ε
i
(we say that the system is in a
pure state
).
If on the other hand we pass a beam of electrons through a Stern-Gerlach apparatus, the
magnet separates the beam into two components that correspond to the two spin eigenvalues
m
s
=
+
−
1
/
2
and
m
s
=
1
/
2
. This kind of measurement is referred to as
state preparation
for
if we pass the
m
s
=
1
/
2
beam through a further Stern-Gerlach apparatus oriented in the
same way as the first, we simply observe the one beam. I have shown this schematically in
Figure 13.11.
+
m
s
=
+1/2
Beam
SG
Apparatus
SG
Apparatus
m
s
=
-1/2
Figure 13.11
Repeated Stern-Gerlach (SG) measurements
However, electrons in the incident beamare not in pure states, and their spinwavefunction
can be written as linear combinations of the two spin functions α and β:
ψ
=
a
α
+
b
β
They are said to be in
mixed states
. Here,
a
and
b
are scalar (complex) constants. The
question is how do we interpret the measurement process? For example, does a given
