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

The effect of mirroring on any molecular wave function

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

The final equality above shows that a mirror-symmetric wave function is the same at positive values of

The fundamental reason why the ion is mirror symmetric is a
mathematical one. The mirror operator

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

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

Here

To answer that, apply

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

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

In this case, let

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

You might of course wonder about the rotational changes of excited
energy states. For those a couple of additional observations apply.
First, the number

Recalling the discussion of angular momentum in chapter
4.2.2, you can see that

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

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].