Silly thermodynamics and quantum mechanics questions

[amrhima]

Hello,

I have recently studied that a quantum particle can't have a zero kinetic energy, as it would violate the uncertainty principle, so I thought of these questions that are related to the topic:

1- At very heigh energies the speed of a particle cannot be less than a certain amount, so is it possible to apply the speed of light limit and ΔP would be limited to the minimum possible speed of that energy and the speed of light, would that increase ΔX?

2- Is the zero kinetic energy not possible for the same reason that the absolute zero temperature is not allowed? I mean is that the reason; if something reaches zero Kalvin temperature then the energies of it's particles are 0, and that would violate the uncertainty principle?

3- If these are the cases, then would it make sense, given that the temperature is related to the average speed (root mean square speed), that at very low temperatures and very high temperatures a macroscopic object would have a high ΔX, or at least each of its parts? I can't imagine what would happen then.

 

[bhobba]

I have recently studied that a quantum particle can't have a zero kinetic energy

That's a misunderstanding - it can have a fixed kinetic energy (by which I assume you mean momentum). But then its position is completely unknown - its wavefunction is a plane wave.

Zero kinetic energy however is a slightly different matter. It means the particle is at rest (ie has a fixed position) and fixed momentum so that's not possible. However if you go to another frame its moving so you don't have the issue. As you move closer to a frame where it is at rest its wavefunction becomes flat - which is probably the best way of looking at a particle with zero momentum.

Not that it ever can actually be achieved in practice - but theoretically its what QM predicts.

The theory, for mathematical convenience, contains mathematical objects like Dirac Delta functions and plane waves of infinite extent that don't actually exist but make calculations easier - applied math does that sort of thing all the time.

And it is indeed true you can't achieve absolute zero which would mean the objects have a definite position and momentum - which the uncertainly principle does not allow - ie if at rest then its wavefunction is such it could be anywhere.

Combining relativity and QM leads to QFT which is an entirely new, and much more complicated ball game.

Thanks
Bill

 

[kith]

I have recently studied that a quantum particle can't have a zero kinetic energy, as it would violate the uncertainty principle

The uncertainty principle is a statement about the variance of an observable in a given quantum state, so it doesn't refer to single particles.

1- At very heigh energies the speed of a particle cannot be less than a certain amount, so is it possible to apply the speed of light limit and ΔP would be limited to the minimum possible speed of that energy and the speed of light, would that increase ΔX?

The particles in a gas never move with a speed equal to or even greater than the speed of light.

 

[bhobba]

The uncertainty principle is a statement about the variance of an observable in a given quantum state, so it doesn't refer to single particles.

Not sure what you mean.

Suppose a particle has fixed momentum - its wavefunction is a plane wave. Its a single particle state as far as I can see. Of course if you measure its position it could be anywhere.

Thanks
Bill

 

[amrhima]

Bhobba,

But is it at least true that a very cold object would have a high uncertainty in position? I'm assuming that if the average root mean square speed is v for N particles, then no particle with mass m can have a momentum more than mvN, which would be very low for temperatures near 0 K.

I just understood what you said about the relative speed, that was helpful, thank you. However in this case of a macroscopic object, the movements of particles are in all directions, so no reference frame can have much of an effect I think.

I hope it is clear that I'm asking, not arguing. I have no idea what the answer is to argue for it :)

 

[kith]

Not sure what you mean.

The HUP makes statements of the form Δa ≥ const*(Δb)⁻¹. This doesn't imply that a can't take a specific value in a single measurement. Also -as you have already written- the particle can be in an eigenstate where it has a fixed value.

 

[bhobba]

But is it at least true that a very cold object would have a high uncertainty in position?

What you are doing is intermixing quantum and classical ideas - that's fraught with danger.

If the atoms have no motion, none, zero, which is what absolute zero is, then you know its momentum - its zero. QM is very clear in that case - it could be anywhere - but we know its located in this confined gas - so that's not possible.

Actually when you take gasses to very low temperatures, not quite absolute zero, QM effects start to predominate and some very weird effect are observed eg:
http://www.scientificamerican.com/article.cfm?id=superfluid-can-climb-walls

The reason is its now reached its lowest energy state for Helium - its actually not absolute zero but a bit above it - and since it can't go lower does things like flow with no friction - if it had friction it would dissipate energy - its in its lowest energy state so can't do that and because of that does very very strange things.

So for actual atoms, molecules etc, they have a minimum energy that is a bit above strict absolute zero, and when you reach that they do very very strange things eg Bose Einstein Condensates:
http://en.wikipedia.org/wiki/Bose–Einstein_condensate

Thanks
Bill

 

[amrhima]

Bill,

This is exactly what i was looking for, thank you. It seems that things do act strange when they get so cold to keep the uncertainty principle. Do you have any idea why the viscosity becomes so low though? I mean how is it related to the whole topic? The only thing I can think of is that it makes it only slightly more difficult to contain the liquid and know where each atom is.

 

[Drakkith]

So for actual atoms, molecules etc, they have a minimum energy that is a bit above strict absolute zero[/url]

Thanks
Bill

Not true. The minimum energy level is absolute zero. The ground state simply still has energy in it that it cannot give up, even at absolute zero.

 

[bhobba]

Not true. The minimum energy level is absolute zero. The ground state simply still has energy in it that it cannot give up, even at absolute zero.

I can't quite follow the difference.

Since it can't go lower than the ground state that is its minimum energy.

Absolute zero is a physical impossibility for a confined object, as has been discussed.

Thanks
Bill

 

[bhobba]

This is exactly what i was looking for, thank you. It seems that things do act strange when they get so cold to keep the uncertainty principle. Do you have any idea why the viscosity becomes so low though? I mean how is it related to the whole topic? The only thing I can think of is that it makes it only slightly more difficult to contain the liquid and know where each atom is.

Liquid helium is a complex phenomena that's well and truly under the area of condensed matter physics, which I am not into.

If you want to delve into it:
http://hep.ph.liv.ac.uk/~hock/Teaching/2011-2012/6-liquid-helium-4.pdf

Good luck.

Thanks
Bill

 

[Drakkith]

I can't quite follow the difference.

The difference is that at temperatures above absolute zero there are states other than the ground states filled. The minimum energy level isn't slightly above absolute zero, it is absolute zero.

 

[amrhima]

Drakkith,
Does that mean that if a body has absolute zero the particles composing it will be in the ground state which means they will still have some kinetic energy? If I understand correctly, this kinetic energy would be temperature and would raise the body temperature above absolute zero, right?

 

[Enigman]

http://en.wikipedia.org/wiki/Zero-point_energy#Utilization_controversy
n 1900, Max Planck derived the formula for the energy of a single energy radiator, e.g., a vibrating atomic unit:[5]
$\epsilon = \frac{h\nu}{ e^{\frac{h\nu}{kT}}-1}$
where h is Planck's constant, \nu is the frequency, k is Boltzmann's constant, and T is the absolute temperature.
Then in 1913, using this formula as a basis, Albert Einstein and Otto Stern published a paper of great significance in which they suggested for the first time the existence of a residual energy that all oscillators have at absolute zero. They called this residual energy Nullpunktsenergie (German), later translated as zero-point energy. They carried out an analysis of the specific heat of hydrogen gas at low temperature, and concluded that the data are best represented if the vibrational energy is
$\epsilon = \frac{h\nu}{ e^{\frac{h\nu}{kT}}-1} + \frac{h\nu}{2}$
According to this expression, an atomic system at absolute zero retains an energy of ½hν.

...

http://www.britannica.com/EBchecked/topic/656682/zero-point-energy
Zero-point energy- vibrational energy that molecules retain even at the absolute zero of temperature. Temperature in physics has been found to be a measure of the intensity of random molecular motion, and it might be expected that, as temperature is reduced to absolute zero, all motion ceases and molecules come to rest. In fact, however, the motion corresponding to zero-point energy never vanishes.
Zero-point energy results from principles of quantum mechanics, the physics of subatomic phenomena. Should the molecules ever come completely to rest, their component atoms would be precisely located and would simultaneously have precisely specified velocities, namely, of value zero. But it is an axiom of quantum mechanics that no object can ever have precise values of position and velocity simultaneously (see uncertainty principle); thus molecules can never come completely to rest.

 

[amrhima]

Thank you, Enigman,

but if the absolute zero temperature cannot be reached (first law of thermodynamics) then the uncertainty principle would not be at risk even if the temperature was a measure of the energy of molecules, so based on the reason of "saving" the uncertainty principle alone I don't think it would be sufficient to say that absolute zero temperature can still have moving particles. Am I correct?

 

[Enigman]

You are thinking of it classically. Classically, yes at absolute zero the molecules do not move and have no energy. But we are talking about quantum scale and classical physics no longer holds. Absolute zero then implies a state where the system has the lowest possible energy which is the zero-point energy.
Yes, absolute zero can not be reached practically by thermodynamic means. But that doesn't mean that in a theoretical system the uncertainty principle wouldn't still hold.
Note: I am by no means an expert and my statements are not really borne through confidence, if someone could verify (or refute) them it would be great.

 

[amrhima]

Enigman,

I am also an undergraduate and I haven't acquired much prejudice yet! I hope to keep it that way, I never want to be too sure!

Thank you for your answer, but I think the impossibility of the absolute zero is not a practical difficulty, but an actual law of nature. If that is true, then it could be because the lowest energy state gives rise to temperature, so I would like someone to confirm or deny that as well.

 

[atyy]

Thank you for your answer, but I think the impossibility of the absolute zero is not a practical difficulty, but an actual law of nature. If that is true, then it could be because the lowest energy state gives rise to temperature, so I would like someone to confirm or deny that as well.

The lowest energy state does not give rise to temperature. A system that is exactly in its quantum ground state has zero temperature.

The impossibility of absolute zero is different from the uncertainty principle. The uncertainty principle holds even if absolute zero is reached - it holds even for quantum ground states.

 

[bhobba]

Does that mean that if a body has absolute zero the particles composing it will be in the ground state which means they will still have some kinetic energy? If I understand correctly, this kinetic energy would be temperature and would raise the body temperature above absolute zero, right?

Absolute zero is simply not possible as has been pointed out.

Drakkith however is correct - and I appreciate him clarifying his comment. As you lower temperature more and more atoms, molecules etc are in their ground state - but due to the statistical nature of QM some are not regardless of how low the temperature is. Most will be, but a few will not, and that few will get smaller the lower the temperature. But since absolute zero is impossible there is no way for all of them to be in the ground state.

The post of the article about zero-point energy didn't tell the full truth - which is why links to encyclopedia articles, and even Wikipedia needs a bit of care - I give such links myself, but often they do not tell the whole 'truth'. First, this is QFT which as I have said before is a whole new ball game - much much more difficult. Secondly zero point energy is predicted to be, wait for it, infinite. Obviously it can't be, and this was one of the first examples of infinities that plagues QFT and one reason why its a lot more difficult conceptually and mathematically. Renormalisation is required to tame these infinities and if you want to investigate that minefield good luck:
http://arxiv.org/pdf/hep-th/0212049.pdf

I have been through that paper many times, as well as other discussions on it. The bottom line is these infinities strongly suggest these theories are not fundamental, but an approximation to other, as yet, unknown theories that hopefully do not have these issues. This is the modern Effective Field Theory approach developed by Ken Wilson he got a Nobel prize for:
http://en.wikipedia.org/wiki/Effective_field_theory

Because of that what the ZPE tells us about the question of absolute zero is, I suspect, not fully certain. We must await future developments on understanding the physics at and below the Plank scale, because that is what is thought is necessary to fully understand such things, and what's going on there is what our current theories are approximations to, and the ultimate cause of these 'problems'.

Thanks
Bill