Biomedical Engineering Reference
In-Depth Information
and they are degenerate (they have the same energy). In such situations, it turns out that
any linear combination of degenerate solutions is also an acceptable solution. This is easily
seen by rewriting the problem in Hamiltonian operator form
H
Ψ
1,2
=
E
Ψ
1,2
H
Ψ
2,1
=
E
Ψ
2,1
b
Ψ
2,1
In the above equation,
a
and
b
are nonzero scalars that could be complex.
We live in a world of mass production where items that come off a production line appear
to be identical. Ashopping trip to the supermarket might yield four tins of dog food, all with
the same label, the same contents and the same price, but on close examination it usually
turns out that there are differences. One tin might have a scratch, one might be slightly
dented, the label might be peeling from the third and so on. In the everyday world, objects
that seem at first sight to be identical may very well turn out to be distinguishable. Even
if the cans were exactly identical, we could label them A, B, C and D with an indelible
marker and so make them distinguishable.
Things are very different in the world of atoms and molecules. A visit to MolMart
for four hydrogen atoms would reveal that all hydrogen atoms are exactly the same; there
is no question of one atom having a scratch or a dent, and it is certainly not possible to
label a hydrogen atom with an indelible marker. We say that the four hydrogen atoms are
indistinguishable
. This simple observation has far-reaching consequences.
I mentioned in earlier sections that wavefunctions themselves do not have any phys-
ical interpretation, but that the modulus squared of a wavefunction is an experimentally
observable probability density. So if ψ(
x
) describes a single particle, then
|
H
a
Ψ
1,2
+
b
Ψ
2,1
=
E
a
Ψ
1,2
+
ψ
(
x
)
|
2
d
x
gives the chance of finding the particle between
x
and
x
+
d
x
and this can be measured in
experiments such as X-ray and neutron diffraction.
The wavefunctions Ψ
1,2
and Ψ
2,1
above involve the coordinates of two particles and in
this case the physical interpretation is that, for example,
|
2
d
x
A
d
x
B
represents the probability of simultaneously finding particle A between
x
A
and
x
A
+
Ψ (
x
A
,
x
B
)
|
d
x
A
with particle B between
x
B
and
x
B
+
d
x
B
. If we now add the condition that the particles are
truly indistinguishable then we should get exactly the same probability on renaming the
particles B and A. So if we consider (Ψ
1,2
)
2
Ψ
1,2
2
2
L
sin
1π
x
A
sin
2π
x
B
L
2
=
L
it is clear that the probability density treats the two particles on a different footing and so
implies that they are distinguishable. The acid test is to write down the probability density
and then interchange the names of the particles. The probability density should stay the
same.
The two wavefunctions Ψ
1,2
and Ψ
2,1
do not satisfy this requirement and so they are not
consistent with our ideas about indistinguishability. I can illustrate the problem by plotting
the squares of the wavefunctions as density maps, as in Figures 12.10 and 12.11.
