Built on Facts

Zero-point free lunches

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:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

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

i-88516ca505d84f969b7ab15a965c3fb3-2.png

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):

i-d18063683dcc0d42b9be45451a84d1e3-3.png

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.

Comments

  1. #1 Joseph Smidt
    June 1, 2010

    Really well put. Every once and a while I am asked why we can’t extract zero point energy and now I will have to use this as a helpful explanation.

  2. #2 Peeter Joot
    June 1, 2010

    Shouldn’t that be hbar squared?

  3. #3 Matt Springer
    June 1, 2010

    Yes, in all three cases. This is when I wish ScienceBlogs had a LaTeX editor instead of me having to redo all the equation imaged by hand…

  4. #4 NickBMorgan
    June 1, 2010

    Now I thought it was because Mother Nature had a very large dog that would bite you if you tried to steal her energy.

  5. #5 John S. Wilkins
    June 2, 2010

    I thought it was because Maxwell’s demon needed to use more energy to get the energy than he got.

  6. #6 Uncle Al
    June 2, 2010

    A collapsing Casimir etalon extracts zero point energy, F(dot)d. A 100 nm gap gives 130 dyne/cm^2. That is 33 times the weight of aluminum foil/cm^2. Aside from fabrication energy pre-debt… only 1/2 cycle obtains, and it doesn’t have much throw before Casimir becomes van der Waals. Vacuum propeller proposals, even sober ones (arxiv:0912.1031), are similarly modest. Here is what we need to do,

    1) Return to the moon!
    2) ?
    3) Energy independence!

  7. #7 rob
    June 4, 2010

    Uncle Al:

    i thought step 1 was steal underwear.

  8. #8 Nathan Myers
    June 4, 2010

    Nah, that’s only if you’re after Profit(!!!). Energy independence(!) definitely needs the moon thing.

  9. #9 Marc
    June 4, 2010

    I think the last line of this post deserves to be put into a list of great quotes!

  10. #10 ball valves
    March 27, 2011

    The unveiling truth about this post is, that it spoke to me deeply. Thank you for sharing your thoughts and concerns.

The site is undergoing maintenance presently. Commenting has been disabled. Please check back later!