Quantum Mechanics for Engineers |
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© Leon van Dommelen |
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A.9 Wave function symmetries
Symmetries are very important in physics. For example, symmetries in
wave functions are often quite helpful to understand the physics
qualitatively.
As an example, the hydrogen molecular ion is mirror symmetric around
its midplane. This midplane is the plane halfway in between the two
nuclei, orthogonal to the line connecting them. To roughly understand
what the mirror symmetry around this plane means, think of the
midplane as an infinitely thin mirror. Take this mirror to be
two-sided, so that you can look in it from either side. That allows
you to see the mirror image of each side of the molecule. Simply put,
the mirror symmetry of the ion means that the mirror image looks
exactly the same as the original ion.
(If you would place the entire molecule at one side of the mirror, its
entire mirror image would be at the other side of it. But except for
this additional shift in location, everything would remain the same as
in the case assumed here.)
Under the same terms, human beings are roughly mirror symmetric around
the plane separating their left and right halves. But that symmetry
is far from perfect. For example, if you part your hair at one side,
your mirror image parts it at the other side. And your heart changes
sides too.
To describe mirror symmetry more precisely, take the line through the
nuclei to be the -axis. And take to be zero at the
mirror. Then all that the mirror does mathematically is replace
by . For example, the mirror image of the nucleus at
positive is located at the corresponding negative value.
And vice-versa.
The effect of mirroring on any molecular wave function can be
represented by a “mirror operator” . According to the above, all
this operator does is replace by :
By definition a wave function is mirror symmetric if the mirror
operator has no effect on it. Mathematically, if the mirror
operator does not do anything, then must be the same as
. So mirror symmetry requires
The final equality above shows that a mirror-symmetric wave function
is the same at positive values of as at the corresponding negative
values. Mathematicians might simply say that the wave function is
symmetric around the -plane, (i.e. the mirror). The ground
state of the molecular ion is mirror symmetric in this
sense. The big question to be addressed in this addendum is, why?
The fundamental reason why the ion is mirror symmetric is a
mathematical one. The mirror operator commutes with the
Hamiltonian . Recall from chapter 4.5.1 what
this means:
In words, it does not make a difference in which order you apply the
two operators.
That can be seen from the physics. The Hamiltonian consists of
potential energy and kinetic energy . Now it does not
make a difference whether you multiply a wave function value by the
potential before or after you flip the value over to the opposite
-position. The potential is the same at opposite values,
because the nuclei at the two sides of the mirror are the same. As
far as the kinetic energy is concerned, if it involved a first-order
-derivative, there would be a change of sign when you flip
over the sign of . But the kinetic energy has only a second
order -derivative. A second order derivative does not
change. So all together it makes no difference whether you first
mirror and then apply the Hamiltonian or vice-versa. The two
operators commute.
Also according to chapter 4.5.1, that has a consequence.
It implies that you can take energy eigenfunctions to be mirror
eigenfunctions too. And the ground state is an energy eigenfunction.
So it can be taken to be an eigenfunction of too:
Here is a constant called the eigenvalue. But what would
this eigenvalue be?
To answer that, apply twice. That multiplies the wave
function by the square eigenvalue. But if you apply twice, you
always get the original wave function back, because
. So the square eigenvalue must be 1, in order
that the wave function does not change when multiplied by it. That
means that the eigenvalue itself can be either 1 or 1. So
for the ground state wave function , either
or
If the first possibility applies, the wave function does not change
under the mirroring. So by definition it is mirror symmetric. If the
second possibility applies, the wave function changes sign under the
mirroring. Such a wave function is called “mirror
antisymmetric.” But the second possibility has wave
function values of opposite sign at opposite values of .
That is not possible, because the previous addendum showed that the
ground state wave function is everywhere positive. So it must be
possibility one. That means that the ground state must indeed be
mirror symmetric as claimed.
It may be noted that the state of second lowest energy will be
antisymmetric. You can see the same thing happening for the
eigenfunctions of the particle in a pipe. The ground state figure
3.8, (or 3.11 in three dimensions), is
symmetric around the center cross-section of the pipe. The first
excited state, at the top of figures 3.9, (or
3.12), is antisymmetric. (Note that the grey tones show
the square wave function. If the wave function is antisymmetric, the
square wave function is symmetric. But it will be zero at the
symmetry plane.)
Next consider the rotational symmetry of the hydrogen molecular ion
around the axis through the nuclei. The ground state of the molecular
ion does not change if you rotate the ion around the -axis through
the nuclei. That makes it rotationally symmetric. The big question
is again, why?
In this case, let be the operator that rotates a wave
function over an angle around the -axis. This
operator too commutes with the Hamiltonian. After all, the only
physically meaningful direction is the -axis through the nuclei.
The angular orientation of the axes system normal to it is a
completely arbitrary choice. So it should not make a difference at
what angle around the axis you apply the Hamiltonian.
Therefore the ground state must be an eigenfunction of the rotation
operator just like it is one of the mirror operator:
But now what is that eigenvalue ? First note that the
magnitude of all eigenvalues of must be 1. Otherwise
the magnitude of the wave function would change correspondingly
during the rotation. However, the magnitude of a wave function does
not change if you simply rotate it. And if the eigenvalue is a
complex number of magnitude 1, then it can always be written as
where is some real number. So the rotated
ground state is some multiple of the original ground
state. But the values of the rotated ground state are real and
positive just like that of the original ground state. That can only
be true if the multiplying factor is real and positive
too. And if you check the Euler formula (2.5), you see
that is only real and positive if it is 1. Since
multiplying by 1 does not change the wave function, the ground state
does not change when rotated. That then makes it rotationally
symmetric around the -axis through the nuclei as claimed.
You might of course wonder about the rotational changes of excited
energy states. For those a couple of additional observations apply.
First, the number must be proportional to the rotation
angle , since rotating twice is equivalent to
rotating it once over twice the angle. That means that, more
precisely, the eigenvalues are of the form
, where is a real constant independent of
. Second, rotating the ion over a full turn
puts each point back to where it came from. That should reproduce
the original wave function. So an eigenvalue for a
full turn must be 1. According to the Euler formula, that requires
to be an integer, one of ..., 2, 1, 0, 1, 2, .... For
the ground state, will have to be zero; that is the only way to
get equal to 1 for all angles .
But for excited states, can be a nonzero integer. In that case,
these states do not have rotational symmetry.
Recalling the discussion of angular momentum in chapter
4.2.2, you can see that is really the magnetic quantum
number of the state. Apparently, there is a connection between
rotations around the -axis and the angular momentum in the
-direction. That will be explored in more detail in chapter
7.3.
For the neutral hydrogen molecule discussed in chapter 5.2,
there is still another symmetry of relevance. The neutral molecule
has two electrons, instead of just one. This allows another
operation: you can swap the two electrons. That is called “particle exchange.” Mathematically, what the particle
exchange operator does with the wave function is swap the
position coordinates of electron 1 with those of electron 2.
Obviously, physically this does not do anything at all; the two
electrons are exactly the same. It does not make a difference which
of the two is where. So particle exchange commutes again with the
Hamiltonian.
The mathematics of the particle exchange is similar to that of the
mirroring discussed above. In particular, if you exchange the
particles twice, they are back to where they were originally. From
that, just like for the mirroring, it can be seen that swapping the
particle positions does nothing to the ground state. So the ground
state is symmetric under particle exchange.
It should be noted that the ground state of systems involving three or
more electrons is not symmetric under exchanging the positions
of the electrons. Wave functions for multiple electrons must satisfy
special particle-exchange requirements, chapter 5.6. In
fact they must be antisymmetric under an expanded
definition of the exchange operator. This is also true for systems
involving three or more protons or neutrons. However, for some
particle types, like three or more helium atoms, the symmetry under
particle exchange continues to apply. This is very helpful for
understanding the properties of superfluid helium,
[18, p. 321].