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The hydrogen molecule 81

takes place, and a bond is formed. ‘Something’ goes over from the proton to
the neutron which causes the change, and this ‘something’ is called a positively
charged π-meson. Thus, just as an electron holds together two protons in a
hydrogen molecular ion, in the same way a positively charged π-meson holds
together a proton and a neutron in the nucleus.

5.6 The hydrogen molecule

The hydrogen molecule differs from the hydrogen molecular ion by having one
more electron. So we may choose our states as shown in Fig. 5.8. State (1) is a b
when electron a is with proton 1 and electron b with proton 2, and state (2) is (1) + +
obtained when the electrons change places.
How do we know which electron is which? Are they not indistinguishable?
Yes, they are, but we may distinguish them by assigning opposite spins to them.
b a
We may now explain the bond of the hydrogen molecule in a manner ana-
logous to that of the hydrogen molecular ion, but instead of a single electron (2) + +
jumping to and fro, we now have two electrons changing places. Thus, we may
argue again that owing to symmetry, the energies of the two states are identical. Fig. 5.8
The coupling between the states—due to the exchange of electrons—splits The two basic states of the hydrogen
the energy levels, one becoming somewhat higher, the other somewhat lower. molecule. Each electron can be
attached to either proton leading to a
Having the chance to lower the energy results in an attractive force which is
coupling between the states.
eventually balanced by the repulsive force between the protons. And that is the
reason why the hydrogen molecule exists.
It is interesting to compare this picture with the purely intuitive one de-
scribed earlier, based on the atoms’ ‘desire’ to fill the energy shells. In the
present explanation we are saying that the bond is due to the exchange of elec-
trons; previously we said the bond was due to sharing of the electrons. Which
is it; is it sharing or swapping? It is neither. Both explanations are no more than
physical pictures to help the imagination.
We could equally well have said that the hydrogen molecule exists be-
cause it comes out mathematically from our basic premises, that is the spin
and Pauli’s principle added to Schrödinger’s equation. The problem is a purely
mathematical one, which can be solved by numerical methods. There is no
need, whatsoever, for a physical picture. This argument would hold its ground
if numerical solutions were always available. But they are not available. Com-
puters are not powerful enough, not as yet and will not be for a long time to
come. So we need mathematical approximations based on a simplified phys-
ical picture and then we must strive to build up a new, more sophisticated
physical picture from the mathematical solution obtained, and then attempt a
better mathematical approximation based on the new physical picture, and so
on, and so on. It seems a tortuous way of doing things, but that is how it is.
It is a lot easier in classical physics. Our physical picture is readily acquired
in conjunction with our other faculties. We do not need to be taught that two
bricks cannot occupy the same place: we know they cannot.
In studying phenomena concerned with extremely small things beyond the
powers of direct observation, the situation is different. The picture of an atom
with filled and unfilled energy shells is not a picture acquired through personal
experience. It has come about by solving a differential equation. But once the
solution is obtained, a physical picture starts emerging. We may visualize little