Bound states, scattering states and infinite potentials

The distinction to be made here is that, for the quantum harmonic oscillator system, there are no unbound states, only bound states thus, there is no benefit to insisting the states have negative energy, no reason to 'subtract infinity' in order to zero the potential at infinity.

However, in systems that permit both bound and unbound states, it is reasonable to zero the potential at infinity for the same reason that we do this classically.

For example, in the classical central force problem, there is a state in which particle can 'escape to infinity' where it will have zero kinetic energy (more precisely, the kinetic energy of the particle asymptotically approaches zero). If we set the potential energy to be zero at infinity, then the total energy 'at infinity' is zero. Thus, the particle with zero total energy 'sits on the boundary' between those particles with not enough energy to 'reach' infinity and those that do.

But, for the classical harmonic oscillator potential, no particle can escape to infinity. The kinetic energy of the particle will periodically and instantaneously be zero. In this case, it is reasonable that the state where the total energy is always equal to the potential energy (the state where the kinetic energy is always zero) be the zero total energy state; all other states having positive total energy.

So the conclusion is that nothing really exists according to Quantum Mechanics... which can't be right, surely?

That's not remotely the correct conclusion to draw. One might conclude instead that

(1) The conception of bound state must be modified in the passage from classical mechanics to quantum mechanics and

(2) the physical (normalizable) unbound states are not eigenstates of the Hamiltonian, i.e., the physical unbound states are not states of definite energy but are, instead, a distribution of energy eigenstates, e.g., a wavepacket.

Part 1. Essentially you're right. You can think of it as subtracting infinity from the energy. A better way to view it is that the convention that the zero of energy should correspond to potential at infinity was always an arbitrary choice. Normally it is a very sensible convention, but if the potential diverges at infinity, as is the case for the harmonic oscillator, clearly another choice would be better. In practice we normally use harmonic oscillators as approximations to more complex potentials which do not diverge and this often works pretty well as we require the wavefunction to go to zero at infinity anyway.

Part 2 Bound States are defined to be those states with a lower energy than a free particle in a given potential. A particle cannot go from a bound state to a continuum state without an input of energy. If I have an isolated hydrogen atom the electron cannot spontaneously escape from the proton because this would increase its energy.

Quantum tunnelling occurs when there are two areas of low potential separated by an area of high potential, where the particle would be forbidden to enter classically. This can be a pair of potential wells or two areas of free space separated by a potential barrier (which is what Griffith was referring to) So for example if I have my hydrogen atom and bring another proton close to it the electron can tunnel from one proton to the other, even though it could not became a free particle and so classically could not leave the atom it started out bound to. Generally what happens in these situations with multiple potential wells is that the in the stationary states the particle is in a superposition of being in both wells. This is what happens when a covalent bond forms.