The Real Point of Zero Point

While Kenneth Ford’s 101 Quantum Questions was generally good, there was one really regrettable bit, in Question 23: What is a “state of motion?” When giving examples of states, Ford defines the ground state as the lowest-energy state of a nucleus, then notes that its energy is not zero. He then writes:

An object brought to an absolute zero of temperature would have zero-point energy and nothing else. Because of zero-point energy, there is indeed such a thing as perpetual motion.

This is really the only objectionable content in the book, but he certainly made up in quality what it lacks in quantity. When I read that, on an airplane, it made me want to bash my head repeatedly against the seat in front of me (I didn’t, because it probably would’ve been interpreted as some sort of terrorist act, and I don’t want to be on the no-fly list).

Zero-point energy, of course, is the modern justification for all manner of perpetual motion and “free energy” schemes. Every lunatic or scam artist with a web site, it sometimes seems, proposes some scheme to extract the zero-point energy from some quantum system. While I understand what he’s trying to get at, it would be extremely difficult to come up with a worse way to phrase that.

This is another place where I think the odd ordering of the book comes back to bite him. This comes forty-odd pages before he discusses the wave nature of matter, which means that zero-point energy is something whose existence is just asserted, rather than demonstrated. In reality, zero-point energy, like the uncertainty principle is a natural and unavoidable consequence of the wave nature of matter. You can’t extract that energy from a system because doing so would require you to fundamentally change the nature of every object in the universe.

One of the key starting points for quantum physics is the realization that every object in the universe has both particle and wave properties. Light, which we normally think of as a wave, comes in discrete bundles of energy now called photons, whose existence has been experimentally verified. Electrons, which we normally think of as particles, have wave nature that we can directly observe, as do heavy organic molecules, and all other material objects.

All objects in the universe have some particle characteristics– they can be detected as discrete entities at a specific position– and some wave characteristics– they can show interference effects, and extend over some range in position– and the need to have both of those properties at once places tight restrictions on what is and is not possible. The best-known consequence of this is the uncertainty principle, but it’s also the basis for the zero-point energy.

For an object like an electron to have both wave and particle properties, it must have a wavelength associated with it. This wavelength is related to the momentum– specifically, the wavelength is Planck’s constant h divided by the momentum, so when you double the momentum of a particle (say, by doubling its speed), you necessarily cut its wavelength in half. If you cut the momentum in half, you double the wavelength.

Zero-point energy is the energy associated with a particle in the lowest possible energy state. Since momentum is related to energy (for particles moving at speeds well below that of light, the kinetic energy of a moving particle is approximately one-half the momentum squared divided by the mass), low energy means low momentum, which means long wavelength. So if we want the lowest-energy state for a particle, we’re looking for the longest wavelength possible.

So what determines the longest wavelength possible? Well, for any particle that is confined to be in some region of space– a particle trapped in a box, a mass on the end of a spring, an electron attracted to the nucleus of an atom– the longest possible wavelength is set by the “size” of the container holding the particle. The wavelength can’t be infinitely large and still fit within the container, which means that the momentum and thus the energy can’t be infinitesimally small, but must have some minimum value.

This is simplest to see in the case of a particle in a box, which has impermeable walls but otherwise allows the particle to move about freely. In that case, the wave inside the box is just a pure sine wave, but it must be zero at the walls of the box. The longest wavelength you can fit in a box under these constraints is twice the length of the box (that is, one half-wavelength of the wave fits across the box, starting at zero, going up (or down) to a maximum (or minimum) in the center, then back to zero at the far end), and that determines the lowest possible energy for the system. Any particle placed in a box that it can’t escape must have at least that much energy, because of its wave nature.

“That’s crazy,” you say. “Clearly, the lowest state for the particle is sitting perfectly still in the middle of the bottom of the box.” Which would be true, if it were a classical particle. But “sitting perfectly still” implies a momentum of zero, and a momentum of zero would give you an infinitely large wavelength. You can’t fit an infinitely large wavelength inside a box of finite size.

“So the wave’s a little big. You just put a piece of it in the box,” you say. Except it’s a wave, and that means it will interfere with itself. And if the wavelength doesn’t match one of the allowed wavelengths for a box of that size, the interference will be destructive, and a wave of that size will not be able to exist in the box.

“How can it interfere with itself?” you ask, indignantly. “It’s a particle– maybe it’s waving a little bit in the region where it is inside the box, but it wouldn’t affect the rest of the box.” It’s true that you can make a state that has its waviness confined to a smallish region of space, but such a state necessarily has a short wavelength, and thus a high energy. Which means it can’t very well be the lowest-energy state in the box.

“You know, I don’t appreciate the way you’re characterizing my perfectly reasonable questions,” you say. Sorry. I’ll try to do better with my imaginary rhetorical devices.

The end result is that the lowest-energy state of a particle confined to a box has a well-defined wavelength, which is twice the length of the box. There’s no way to get around that while preserving both the wave and the particle natures of the particle in the box. And again, we know absolutely and unambiguously that the description of particles in quantum physics must include both particle and wave aspects– we’ve got the experiments to prove it.

What does this mean, physically? That is, what is the particle “really” doing? Well, if you prepare your particle in the lowest energy state, then stick some sort of measuring apparatus inside the box and measure the momentum of the particle at some instant, then repeat the whole cycle many times (prepare the state, measure the momentum, prepare the state again, measure the momentum again…), you’ll find the particle having the same magnitude of momentum (plus or minus some uncertainty) every time, but moving in different directions. So, in a sense, it’s moving at the same minimum speed all the time (more or less), but bouncing back and forth off the walls. The average of the momentum, including the direction, will be zero, but the average magnitude of the momentum will be the value corresponding to a wavelength of twice the length of the box.

(This is what Ford means by his regrettably phrased comment about “perpetual motion.” A particle in its lowest possible energy state is effectively in constant motion at some minimum speed. On average it will be at rest, but when you look closely, it’s always jiggling around.)

The mathematical treatment is more complicated for something like a mass-on-a-spring, or an electron in a hydrogen atom, but the basic physics remains the same: because the particle is confined to some region of space, there is some maximum possible wavelength that it can have, which sets a minimum value for the energy. You can even express that energy in terms of the characteristic size– the ground state energy of hydrogen, for example, can be written in terms of the Bohr radius giving the average size of the atom. That minimum energy is the zero-point energy, and it is an unavoidable consequence of the need to have both particle and wave nature. There is no way to extract that energy while retaining the quantum properties that we know have to exist.

Lunatics, of course, regard this as a feature of models allowing the extraction of zero-point energy to do useful work– they mostly don’t like quantum physics in the first place, and are happy to discard its weirder aspects even in the face of overwhelming experimental evidence to the contrary. Scam artists don’t really care one way or the other, but they know that lots of ordinary people are uneasy about the weird features of quantum physics, and are happy to prey upon that uneasiness as a way of extracting more cash for their perpetual motion scams.

But if you are not a lunatic or a scam artist, and I know you’re not (well, most of you, anyway), the wave nature of matter explains both the origin of zero-point energy, and why zero-point energy can never be extracted to power a perpetual motion machine.

(Note: I am aware that the above explanation doesn’t really encompass the Bohmian picture, in that it asserts that particles are “really” waves, not being guided by a wave-like potential. In the Bohmian picture, the wavefunction isn’t “really” the particle, but just guides the particle; the wavelength associated with the wavefunction is still constrained by the size of the confining potential, though, so the end result is the same.)