Absolute Zero and the Uncertainty Principle

I make the dangerous mistake quite often of thinking I understand the concepts of quantum mechanics without actually being able to do any quantum cookery. But I am largely certain this question has a factual answer, or at least a generally accepted one.

If the title didn’t make it clear, my concern is what happens to particles at absolute zero, or indeed if they can ever reach absolute zero.

In my head, if a particle made it to absolute zero it would instantly be everywhere in the universe at the same time, making it extremely impossible to stay absolute zero for any period of time.

But then I think: hey, it is just this reason that absolute zero can’t even be theoretically reached, because eventually the uncertainty in position will become large enough that an interaction will have to happen before zero is ever reached.

The perspective I’m approaching this from is the “cold death” of the universe idea.

I vaguely remember reading a Scientific American article on Bose-Einstein Condenstates some time ago, but I do not recall at all what sort of temperatures were achieved and what the uncertainty in position implied other than the creation of a perhaps aptly named “super atom”. What kind of “size” or distance are we talking about here? Are there practical limits to approaching absolute zero because of the HUP? What sort of limits are we talking about here? Does not having a quantum theory of gravity limit realistic hypotheses here?

Sorry if this sounds way out in left field. :slight_smile:

When I first read your title, I though, “Wow, I never considered that!” Then I read your post and you were on the opposite end of the spectrum from me.

I was thinking that if you removed all energy from an electron it would just sit there and you would be able to determine position AND rotation (guess it would be none) thus violating the uncertainty rule.

I don’t understand why you think it would be everywhere only if it reached absolute zero though. From what I’ve read it seems that electrons and such are alway everywhere when acting like a wave.

As for a Bose-Einstein Condenstate, they reached a fraction of a degree I believe. And the different particles melted into one larger (but still unmeasurable I guess) particle.

Just a few of my thoughts. I’m not sure why we can’t remove all energy from an electron.

When a Bose-Einstein Condensate was observed they had a pile of atoms at only a few billionths of a degree above absolute zero.

You guessed correctly that true absolute zero is theoretically impossible. Even when the universe cools off completely it will never ever quite reach absolute zero…always approaching but never arriving.

You also have it right that when they got the condensate it ‘spread-out’ in such a fashion as to maintain the Uncertainty Principle.

AFAIK true absolute any temeperature is theoritically impossible too. This is something independent of the measurement error.

I don’t remember the reasoning - but it was similar on the lines of having an absolutely pure element or an element with absolutely known purity.

Which measurement error would that be?

Absolute zero means the system is in its lowest-energy state. This is totally divorced from position or momentum being precisely specified. In other words a system with all its particles in their ground states would have a non-zero energy and yet the system could still be at absolute zero, and therefore there would be no violation of the uncertainty principle.

However, if you could reach absolute zero then the entropy of the system would go to zero and this would violate the second law of thermodynamics which would allow perpetual motion and all kinds of good things.

I believe that the Third Law of Thermodynamics, the one nobody ever remembers, states that you can’t reach Absolute Zero in a finite number of steps. This doesn’t even take quantum into account.

I don’t know much about this stuff, but is it kinda like, if each step you take halves the distance to your destination, will you ever get there?

If every particle of matter continuously produces heat at a constant rate, no matter how negligent the amount, would not that amount of heat/energy immediately counteract the approach to absolute zero? If so, does that make absolute zero a theorectical state? Can we get so close that it doesn’t matter, or is that where quantum physics come into play?

If I am so totally off-base that it would take a book to set me straight, just tell me to go back to sleep. I have extremely little education in physics, so you’d be wasting your time. If I’m wrong but close enough that it makes the concept conceivable to a layman, I’m happy :slight_smile:

This is the Third Law of Thermodynamics.

That’s one way of saying it, but the Second Law isn’t the only one that has different ways of being expressed.


You’re going at this from the wrong direction. As you said, if a particle had exactly zero momentum, then it would be infinitely delocalized. However, absolute zero temperature does not mean the particle has exactly zero momentum. It just has zero momentum from thermal energy. It still has the Heisenberg uncertainty in its momentum, which is inversely proportional to the degree to which the atoms are localized. (In a liquid, the average localization is simply the average volume per atom.) The kinetic energy due to Heisenberg uncertainty in the momentum is known as zero-point energy.

This has a very striking example in nature: at ambient pressure, helium does not freeze at T=0 K, because of its large zero-point energy, caused by its low mass. (Kinetic energy = p[sup]2[/sup]/2m, so the same amount of localization causes a much larger kinetic energy in helium than in heavier elements.)

Sorry for the hi-jack but why would a particle with zero momentum be ‘everywhere’?

Heisenburg’s Uncertainty principle:
(uncertainty in x)*(uncertainty in p[sub]x[/sub]) >= Planck’s constant (hbar)

For a quantum particle, uncertainty in x is equivalent to how localized the particle is. If you put a particle in a 1 nm-sized bucket, then that is how localized it is, and it’s momentum can not be less than hbar/(1 nm). Conversely, if you can somehow force it to have exactly zero momentum, then the uncertainty in its position is a constant/zero = infinity.

Ah, I understand now, thanks!


The Uncertainty Principle merely states that you cannot precisely measure a pair of characteristics (in this case position and momentum) at the same time.

Urban, that is true. However, Heisenburg uncertainty is not just a rule about how well you can do experiments. It is a physical property of a system – a particle with very small uncertainty in its momentum would have very large uncertainty in its position, which means the particle’s position would not just be hard to measure – it would physically occupy a large region of space simultaneously.

What about the speed of light limitation? If a particle was so localized that it’s momentum would have to be greater than the speed of light, then what happens?

At relativistic speeds, momentum is not equal to mass*velocity, but rather mv/sqrt(1-v[sup]2[/sup]/c[sup]2[/sup]). So infinite momentum is equivalent to having velocity equal to the speed of light.

:smack: And damn it, I should have remembered that the ground state doesn’t mean “no energy” or anything like that. This is related to that affect when two plates are brought together, right? (Now I can’t remember that name)

If we measure a particle’s momentum, exactly which is not impossible, “it would physically occupy a large region of space simultaneously.”

How large a region?