What is the Temperature of an Electron?

[eminent_youtom]

Hi there,
I was wondering from few weeks that what is a temperature? According to the classical idea, temperature is caused by the vibration (oscillation) of molecules and atoms. So, is it appropriate to ask the temperature of a single electron or few electrons? till what level temperature exists? what actually is temperature? what happens to the atoms, especially electrons, when we decrease the temperature of the system to the absolute zero, like if we cool it to near absolute zero temeprature?

 

[kanato]

Temperature is a statistical quantity. The formal definition is dS/dE, the change in entropy with respect to energy holding volume and particle number constant. Typically, a practical definition comes from the fact that the atoms, molecules, or whatever particles in your system have average kinetic energy <KE> = 3/2 kT, where k is Boltzmann's constant. (This does not apply in the case of a highly degenerate fermi gas, such as electrons in a solid.) Note too that the average specifically means to average over the KE of all the particles in your system, so it is not correct to take the KE of a random particle in the sample, multiply by 2/(3k) and call that its temperature.

In summary, it does not make sense to ask what the temperature of a single electron, atom, molecule, or even a handful of such particles. So you ask, at what scale does temperature 'kick in' where it makes sense to talk about temperature? Probably around a few million particles, although I don't think there is any set rule. I would say that if you had some large system, and you cut in in half, and whatever means you had for measuring the temperature gives you the same result for both halves without large fluctuations, then you're ok talking about temperature.

Classically speaking, if you lower the temperature to absolute zero, the atoms stop moving. This is not actually the case in quantum mechanics, because zero point motion would be left (you can think of it as an expression of the uncertainty principle; if you knew that p=0 from zero temperature, and the position was fixed, that would violate the UP, so things have to keep moving).

Electrons don't change their behavior much at low temperatures, thanks to the exclusion principle. Basically, they are required to fill available states from the lowest up, with only one to each state. The highest filled state defines the energy scale for electrons, in temperature units it's several(/ten-) thousand K. At any temperature that's a fair amount below this temperature, they act basically the same. So cooling to absolute zero from room temperature (300K) does virtually nothing to them.

 

[Mapes]

Kanato, I believe you mean T = dE/dS (great answer otherwise).

 

[kanato]

Kanato, I believe you mean T = dE/dS (great answer otherwise).

Heh whoops, yeah, that's what I meant :)

 

[eminent_youtom]

In summary, it does not make sense to ask what the temperature of a single electron, atom, molecule, or even a handful of such particles.

Thank you kanato for your insightful answer.

Does energy has a temperature? i am talking about the pure energy without any mass. otherwise if there are masses, certainly energy is going to to increase the temperature of the masses.
The main reason behind my question is the temperature of black holes. Stephen Hawking argued that pair of virtual particles are created near the event horizon [quantum fluctuation due to uncertainty principle], due to which black holes are radiating energy. Which means black holes have some temperature. It is widely accepted fact.

HOWEVER, black holes are such a system where spacetime fabric is wrapped in such a way that, tremendous masses are squeezed in a small volume. i.e. all the atoms and particle are squeezed together, where even atom doesn't exists in a pure atomic form. According to NASA, http://www.gsfc.nasa.gov/scienceques2002/20030516.htm"
now my question is, how is it possible to have vibration of atoms in black holes? what is allowing them to vibrate (if they exists)? Although the mathematics behind the temperature of black holes are well estlablished, couldn't it be just a theoretical idea? because, particles are almost freezed inside the black hole.
so far, have scientists detected any electromagnetic wave radiated by black holes?

 

[kanato]

I don't know what pure energy is, without mass, if it's not light. A system of photons does have temperature. Temperature is related to the energy and entropy of a system, so if the system has internal degrees of freedom (different ways to distribute the energy to the particles inside the system) then it has a temperature.

You question on black holes is better directed at another board, probably the
High Energy, Nuclear, Particle Physics board, or the Cosmology board. I think the interior structure of a black hole is probably something that remains highly speculative at this point, but I don't know.

 

[Gokul43201]

so far, have scientists detected any electromagnetic wave radiated by black holes?

Yes. To my knowledge, the x-ray signature (caused by ionization of collapsing atoms) of black holes is one of the easiest ways to detect them.