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.)

Comments

  1. #1 Wilson
    May 11, 2011

    In a box, the diagonal distance from one corner to the opposite corner is (slightly) longer than the orthogonal measurement from one end to the other. However, it seems, from your explanation, that the wave has to travel parallel to the four sides that are half its wavelength long. Am I reading that correctly? If so, why is that the case? I hope this isn’t a stupid question. :)

  2. #2 Chad Orzel
    May 11, 2011

    In a three-dimensional rectangular box, the situation is a little more complicated, but in general, you get half a wavelength along each of the three axes of the box (left-right, up-down, back-front), with each wavelength being twice the length of the box in that direction. The size of the box is still the key factor in determining the minimum energy for the system.

  3. #3 Roberto Baginski
    May 11, 2011

    Beware of the boundary conditions!

    Casimir effect hints that zero-point energy is more “real” than expected. In fact, with the advent of dark energy, vacuum fluctuations may have the last word in the fate of the Universe.

    Of course, whether we attribute these phenomena to vacuum fluctuation or to radiation reaction is a matter of taste.

  4. #4 Kate Nepveu
    May 11, 2011

    for any particle that is confined to be in some region of space

    But what about particles unconfined in space with nothing else around??!! Why can’t they be at absolute zero? Huh, Mr Smarty-Pants??!!!

    (Just hoping to head it off.)

  5. #5 Chad Orzel
    May 11, 2011

    Casimir effect hints that zero-point energy is more “real” than expected. In fact, with the advent of dark energy, vacuum fluctuations may have the last word in the fate of the Universe.

    I’m not trying to claim that the zero-point energy isn’t real– it’s a real and unavoidable consequence of quantum mechanics. It is not, however, a source of energy that can be tapped to do useful work or drive perpetual motion machines.

    But what about particles unconfined in space with nothing else around??!! Why can’t they be at absolute zero? Huh, Mr Smarty-Pants??!!!

    In the end, the idea of a totally unconfined particle doesn’t make physical sense. You always have some boundaries within which the particle is contained, even if they’re the size of the visible universe (or whatever). That will still give you a zero-point energy– an infinitesimal energy, for an infinitely large “box,” but not exactly zero.

  6. #6 Andy Perrin
    May 11, 2011

    Why do the waves need to be zero on the sides of the box in general? I took a stat mech class one time and when we did the Planck distribution problem, we were told that since light waves are electromagnetic fields, the electric field has to be zero at the wall since the walls are “perfectly conducting.” I was never happy about that, since it seems like you should get a different answer if the walls are, say, insulators, or just ordinary conductors. Is that really what happens? You get a different result?

  7. #7 Alan
    May 11, 2011

    I think your tilting at windmills. Your quote from the book, (that I haven’t read), doesn’t say anything about extracting energy from perpetual motion. It just states that it exists, which you seem to agree with, albeit using different terminology.

  8. #8 Marion Delgado
    May 11, 2011

    I thought it was only socialist regulations that prevented almost-free perpetual zero-point energy machines from being distributed in the most efficient possible way – an Amway™ system. More people should know about the suppression of neo-technology by altruism, don’t you think?

    :)

  9. #9 quasihumanist
    May 11, 2011

    As a mathematician who is philosophically mostly nominalist, I just regard “energy” as a term found in equations which happen to make predictions consistent with experiments up to experimental error. Therefore, I must say I regard this discussion with some amusement. You’re talking about the meaning of a mathematical artifact of combining several mathematical models!

  10. #10 Remo
    May 11, 2011

    Just destroy it (I mean the electron or whatever particle you are using). All the energy comes out including the zero point energy. E = m c^2

  11. #11 Neil B
    May 12, 2011

    As I gather, there are in effect two types of “zero point energy”. One is that by the formal name, the vacuum energy issue based on energy-time uncertainty, virtual particles etc. I hear contradictory things about that, such as the expected point about particles “borrowing energy from the vacuum” and giving it back. But there’s also an expectation of an actual net energy density, one interpretation of which had it predicted to around one Planck mass per Planck volume, which would have been horrendously high – off by around 10^120 over actual dark energy values IIRC. Then compare oddities like Casimir force, etc. That’s what most “cranks” try to take advantage of.

    Then there’s the concept that a collection of atoms – stuff and not “space” – cooled “to absolute zero” really can’t be all “motionless” because their confinement requires a momentum spread ΔpΔx >= ħ/2 or so. That is equivalent to a temperature, but not “effectively” so, right? BTW Chad, you have a point but in principle an “isolated” particle (!) could have a wave function that is a very pure momentum state (very low amplitude harmonics) and thus ironically “at rest” despite being all over the place!
    PS – It’s OK to reply to me, time to move on …

  12. #12 Neil B
    May 12, 2011

    (Clarification – the pure momentum state would be at rest relative to a given reference frame, as for any other tardyon.) BTW, don’t hear much tardyon/luxon talk since with neutrinos having mass, light is all that’s left in the luxon bracket. And I haven’t heard about tachyons for ages, must have been a dud beyond just theory toy.)

  13. #13 AES
    May 13, 2011

    To the best of my knowledge of QM, there is a perfectly good way of rewriting the Hamiltonian for a simple harmonic oscillator (as just one example of a common physical system) in such a way that the zero point energy disappears — that is, acquires a predicted value of zero — while the equations of motion and the predicted experimental observations for absolutely every physically observable quantity associated with that SHO — including the predicted zero-point fluctuations — remain absolutely unchanged. In fact, you can rewrite the Hamiltonian to give the ZPE any value you want, without changing any predictions associated with any measurable quantities.

    So, why do quantum mechanicians not adopt this simple, sensible, and straightforward way of writing the quantum Hamiltonian for SHOs, and all other quantum systems? Why do they keep spouting this unending nonsense about ZPE? — a totally unmeasureable or unobservable quantity — and hence a totally meaningless quantity in QM?

  14. #14 moo
    May 14, 2011

    What if the space has no boundary? For example a circle, a infinite wavelength perfectly wraps around on a circle, would also work for a spherical universe.

  15. #15 Emeris
    May 15, 2011

    So, why do quantum mechanicians not adopt this simple, sensible, and straightforward way of writing the quantum Hamiltonian for SHOs, and all other quantum systems? Why do they keep spouting this unending nonsense about ZPE? — a totally unmeasureable or unobservable quantity — and hence a totally meaningless quantity in QM?

    Because the way you do this is a constant added to the potential. In that case we would be talking about the deeper meaning of the SHO Shift Value which depends on the details of the SHO.

  16. #16 MAM
    May 25, 2011

    ZPE was utilized in a utube blip about ufos–it is the ufo’s energy source. But ZPE cannot be an energy source. Therefore, ufos do not exist.