Quantum Temperature Relations?

[evanlee]

I was talking with a friend earlier today about the idea that at absolute zero, particles essentially stop moving. I know that this makes sense since temperature is defined as average kinetic energy, which, if this equals 0, implies no movement. That made me think, however, about the uncertainty inherent in quantum mechanics and wondered what a particle or set of particles at absolutely zero would mean? If momentum is known to be zero, do these particles solely act as waves? How do changes in temperature affect systems of quantum particles? I have taken quantum mechanics at the undergraduate level, but if there are principles out there that might answer this question, I'd really appreciate some directing!

Thanks!

 

[Bill_K]

evanlee, For a classical system, your statement that "temperature is defined as average kinetic energy" is close but not exactly correct. For a classical ideal gas the average energy per molecule turns out to be <E> = (3/2) kT. But this is a result, not a definition of T! It is true that according to this formula, thermal motion appears to stop at absolute zero.

However as a real system approaches absolute zero it will differ from this formula. Quantum effects become important, and what happens depends on whether the system is composed of bosons or fermions.

But in the first place, near absolute zero most things freeze! In a solid atoms do not move freely about, and thermal motion consists of collective vibrations of the lattice (phonons). This makes an important difference. To talk about the thermal properties of such a system near absolute zero, we may consider the phonons to be a gas, but it's a different kind of gas in which the number of particles is not fixed. For this reason you can no longer talk about the energy per particle, rather the energy per volume. It turns out that as T goes to zero, <E> ~ T⁴. The same is true of a gas of photons (black-body radiation) which are also bosons.

Another type of system is He⁴. As you know He⁴ becomes a superfluid below a critical temperature T_c, about 4 K. At this point the system becomes inhomogeneous, splitting into two phases with a fraction of the particles forming a condensate. This comes closest to your idea that absolute zero means p = 0. Consistent with the uncertainty principle, particles in the condensate are nonlocalized, meaning their wavefunction extends over a macroscopic distance.

The third type of system is a gas of fermions. Two fermions cannot occupy the same state, so here at absolute zero the particles fill up the lowest available states up to an energy E_F, the Fermi energy. For a Fermi gas, at absolute zero the energy per particle does *not* go to zero.

 

[maverick_starstrider]

I was talking with a friend earlier today about the idea that at absolute zero, particles essentially stop moving. I know that this makes sense since temperature is defined as average kinetic energy, which, if this equals 0, implies no movement. That made me think, however, about the uncertainty inherent in quantum mechanics and wondered what a particle or set of particles at absolutely zero would mean? If momentum is known to be zero, do these particles solely act as waves? How do changes in temperature affect systems of quantum particles? I have taken quantum mechanics at the undergraduate level, but if there are principles out there that might answer this question, I'd really appreciate some directing!

Thanks!

As has been pointed out, temperature is only approximately related to the average kinetic energy of a system (a fact that high school textbooks/teachers seem to be unaware of). And yes, you are correct, there IS a lowest non-zero kinetic energy at absolute zero. It is often called the zero-point energy.

 

[RedCliff]

Hello, Bill,

You pointed out that <KE> = 3/2 kT is not the definition of the temperature, but the derived result from classical mechanics. However, what is the strict definition of temperature . I seemly read a post from a web saying that "absolute temperature (T) is the indicator of particles distribution at different energy level.". This seems coming from statistical picture. At absolute 0, all particles have on zero-point energy.

How do you think?

Thanks!

John

evanlee, For a classical system, your statement that "temperature is defined as average kinetic energy" is close but not exactly correct. For a classical ideal gas the average energy per molecule turns out to be <E> = (3/2) kT. But this is a result, not a definition of T! It is true that according to this formula, thermal motion appears to stop at absolute zero.

However as a real system approaches absolute zero it will differ from this formula. Quantum effects become important, and what happens depends on whether the system is composed of bosons or fermions.

But in the first place, near absolute zero most things freeze! In a solid atoms do not move freely about, and thermal motion consists of collective vibrations of the lattice (phonons). This makes an important difference. To talk about the thermal properties of such a system near absolute zero, we may consider the phonons to be a gas, but it's a different kind of gas in which the number of particles is not fixed. For this reason you can no longer talk about the energy per particle, rather the energy per volume. It turns out that as T goes to zero, <E> ~ T⁴. The same is true of a gas of photons (black-body radiation) which are also bosons.

Another type of system is He⁴. As you know He⁴ becomes a superfluid below a critical temperature T_c, about 4 K. At this point the system becomes inhomogeneous, splitting into two phases with a fraction of the particles forming a condensate. This comes closest to your idea that absolute zero means p = 0. Consistent with the uncertainty principle, particles in the condensate are nonlocalized, meaning their wavefunction extends over a macroscopic distance.

The third type of system is a gas of fermions. Two fermions cannot occupy the same state, so here at absolute zero the particles fill up the lowest available states up to an energy E_F, the Fermi energy. For a Fermi gas, at absolute zero the energy per particle does *not* go to zero.

 

[Bill_K]

RedCliff, It's not the energy that goes to zero at absolute zero, it's the entropy. And the definition of absolute temperature is in terms of the entropy.

The First Law of Thermodynamics is dU = T dS - P dV. We may therefore define T = (∂U/∂S)_V. Or sometimes this is written 1/T = (∂S/∂U)_V.

 

[DrDu]

Quantum-mechanically you define T=0 to be the point where only the lowest energy eigenstate of the system is populated while at higher temperatures also higher energy states have a probability to be populated. These probabilities go under the name Boltzmann, Fermi-Dirac or Bose-Einstein statistics.