The wavefunctions seem to make sense: they tend to zero as x tends to plus or minus infinity so it is normalisable. The number of nodes increases by one with increasing energy levels etc.
However, when you take a look at $|\psi(x)|^2$ which represents the probability density of the particle to be found at that point in space (or the diatomic to be in that particular state/bond length) I struggle to see how a node can exist. This is because a diatomic will stretch and contract through the full range of extension/contraction - the quantum representation does not seem to conflict this in the sense that there is a non zero probability when the potential is equal to the total energy. However, to reach this state from, for example, $x=0$ in the $n=2$ energy level it must pass through a node. What does this mean? surely a bond can't cease to exist as it stretches from equilibrium to some extended or contracted state? What's really going on here?