In quantum mechanics no less than any other aspect of physics, the Simple Harmonic Oscillator is an extremely important system. In quantum mechanics, though, it leads to the 'zero-point energy', a common source of confusion among novices (and back when quantum mechanics was new, everyone was a novice).

The potential of the SHO is Aω²x²/2 + V, where V is the minimum of the potential, ω is the angular frequency associated with the oscillator, x is the parameter that's oscillating (position is a common case), and A is a constant. Note that though V is most often set to 0, this is merely a convention, and the value put here does not change any observable aspect of the system (this property is called Gauge Symmetry).

Just as with any other potential, the set of energy eigenstates (i.e. special states with precisely defined energy) of this potential spans the space of wavefunctions (i.e. states in general). That is to say, any conceivable state of the SHO can be expressed as a linear combination of states with definite known energies.

"So what?" you ask. Well, it turns out that the energy eigenvalues are

En = V + (n + 1/2)ℏω;

where ℏ is Dirac's constant, and each eigenvalue is labelled by n, and n is a nonnegative integer. This means that the minimum energy of the SHO is not the minimum of the potential, V, but V + ℏω/2. This is the zero-point energy.

What does this mean for energy extraction? Nothing. Ab-so-lute-ly nothing. You can't extract this energy, since there is no state with lower energy. Remember what I said about 'any conceivable state' up above. It does have a variety of implications for other things, principally that the mean kinetic energy and potential energy energies are nonzero; but that doesn't mean they can be gotten out without taking apart the oscillator1.

Still, this freaked theoretical physicists out for quite a while, especially when it came time to quantize the electromagnetic field. You see, the electromagnetic field can be expressed as a sum of an infinite number of harmonic oscillators, one for each wavevector (read: frequency and direction). Having the ground state (one with no photons) have not only some energy but actually an infinite amount of energy really bothered them. Now, from a practical point of view, it made no difference, as energy in itself doesn't do anything2, it's only differences in energy. The more pragmatic and less timid theoretical physicists simply chose the arbitrarily-set minimum of the potential, V, not to be 0 but rather be -ℏω/2. Thus, the total ground state energy was 0, and they could go with their business. Eventually, when their predictions ended up correct, the furor died down a bit.

As far as the public was concerned, though, the damage was done. The concept of 'zero-point energy', some energy that was at all points in space, just sitting there with no one using it, had leaked into the collective popular science consciousness. Ever since, we have received claims of 'harnessing the zero-point energy' from people who confuse Energy with Power, or think they've found a new solution of the SHO which has less energy than V + ℏω/2.

Recent experiments with attempts to find regions of negative energy (mentioned in the previous writeup) necessarily use the zero-point energy, but even in that interpretation, nowhere was the zero-point energy of the photon field extracted; merely it was observed to have an effect.

Also note that any bound state has a zero-point energy, but the SHO of the photon field is the one that gets most of the attention. Why? For most other bound states, V is selected so that it is 0 at all points infinitely far away from the points of interest. If the state is truly bound, that means that the energy of the system is less than V at infinity: 0. For example, the zero-point energy of the hydrogen atom is -13.6 eV. A negative zero-point energy is unexciting to the average person, but it's basically the same. The only reason this isn't done for the SHO is because the potential diverges at infinity, so that convention is impossible.

Related to Zero-Point Energy is the notion that one can extract huge amounts of energy from water. This isn't the same thing exactly, but it does arises from the same problem - thinking that there's a lower-energy state that all the water on Earth keeps on not finding its way into, for no particular reason. Suffice it to say, the oceans are not critically unstable high explosives.

1 this caveat is important. You can extract the ZPE of an oscillator which you made yourself, by relaxing the oscillator in such a fashion as to absorb the energy. But this is just recovering energy you had to put in to make the oscillator in the first place...

2 except in General Relativity. This was one of the more serious problems, except in that we could readily measure that there was a non-infinite energy density at every point in space. In this sense, the 'convenience' solution became a necessity.