Grab a particle and put it in a box.

According to elementary quantum mechanics, that particle isn’t described by the classical model in which it can have any value of energy as it bounces around. Instead, the possible energy levels of that particle are described by a discrete set. When you measure the energy of that particle in the box, it will always have one of those specific energy levels and nothing else. Exactly which level it will occupy depends on the details of how you put it into the box, but whatever you do you’re constrained to having the particle in those levels or at least a superposition thereof. The possible energy levels are:

Where each n is a positive integer, L is the length of the sides of the box, and m is the mass of the particle. However, the smallest the energy can possibly be is where all three n values are equal to 1. This is the famous zero-point energy, and for this particular system the energy of the particle cannot possibly be any less than exactly

Occasionally you’ll hear some intrepid thinker propose to use this as a source of power. If the particle always has that minimum energy, why can’t we bleed off energy from the particle as nature constantly replenishes it to keep it at the zero-point value? Unfortunately it doesn’t work. As weird as quantum mechanics is, it still doesn’t give us away around thermodynamics. Imagine we put a particle into a large box with large L and thus small E, then squashed the box down to decrease L and increase E. Then we plan to open the box and extract the power from our newly-energetic particle.

Here’s the problem – changing the energy of a system by changing its volume requires doing work. Pressure by definition is the change in energy with respect to volume (with a minus sign because pressure is pushing in the opposite direction as the change in volume):

With V = L^3, of course. So as you adiabatically compress the box, you have to do work against the pressure. The work you have to do is equal to the increasing zero-point energy. This isn’t a proof – it could have been that the energy/volume relationship doesn’t hold in quantum mechanics – but it’s what thermodynamics predicts, it’s pretty easily testable, and it has in fact been experimentally demonstrated to work. For that matter, the fact that it does work underlies the description of all kinds of phenomena from the bulk modulus of metals to the pressure holding up neutron stars .

Those two equations above also give us a chance to take a look at why zero-point energy isn’t something we notice in everyday life. An electron in a 1 mm box has a zero point energy of about 6×10^{-32} J, which is pretty darn tiny even by electron standards. It’s well under a trillionth of an electron volt. The associated pressure is about 4×10 ^{-23} pascals, and atmospheric pressure is about a hundred thousand pascals. This is also pretty tiny.

Big or tiny though, it’s not going to give us a free lunch of any size.