Is temperature quantized like energy?

From what I remember about quantum mechanics, energy can only take on certain values--integer multiples of a certain minimally-measurable quantity. Or, perhaps more specifically, that it can only be *known or measured* (at the present time) that energy takes on integer multiples of this quantum value.

So what does this have to do with temperature? Well, from what I remember about thermodynamics, temperature is an average of the kinetic energy of molecules vibrating. If this energy can only take on integer multiples of some quantum value *c*, then the arithmetic average can only take on rational values of *c*, or if cleverer methods are used, roots of roots of roots (etc.) of rational values of *c*.

What does this have to do with anything?

Well, when students learn about the Intermediate Value Theorem in real analysis, they are often introduced to the following problem:

Consider the continuous function over the globe mapping each point on Earth to the temperature at that point. Use the Intermediate Value Theorem to prove that there must exist two antipodal points that have the same temperature.

However, if temperature is quantized, then you could never for instance get the value of pi**c* or e**c*, since these numbers are transcendental and therefore are not the root of any polynomial with rational coefficients. Therefore no temperature function could be continuous over an interval [a,b] if f(a) is less than (and f(b) greater than) a transcendental multiple of c.

Re: Is temperature quantized like energy?

I'm not entirely certain, but I can only suppose so.

Temperature is a function of energy, and energy is quantized, therefore it follows that temperature is not a continuum but quantized spectrum.

As for applying calculus to the real world, no, it's not even clear if spacetime really is a continuum. It's been suggested that spacetime is actually quantized at the Planck length. But nobody really knows at the moment.

Re: Is temperature quantized like energy?

I would have to second GP here, given that temperature is related to infra-red radiation which is of course quantized, I would have to imagine that it is as well.

Re: Is temperature quantized like energy?

Yes, for any given molecule or atom, thermal energy is determined by vibrational states that are related to the mass of the atom(s) and, if applicable, the distance between them. I forget the exact formula, but do remember that there are many different vibrational modes based on what you would expect from a three-dimensional interaction, e.g. towards and away from each other, up and down relative to one another, etc.

Re: Is temperature quantized like energy?

Quote:

Originally Posted by

**Zorak**
Yes, for any given molecule or atom, thermal energy is determined by vibrational states that are related to the mass of the atom(s) and, if applicable, the distance between them. I forget the exact formula, but do remember that there are many different vibrational modes based on what you would expect from a three-dimensional interaction, e.g. towards and away from each other, up and down relative to one another, etc.

In the case of a gas, and let's make this unrealistically simple, helium gas (I.e. No chemistry) the definition of temperature is interesting. It has to do with the average energies of atoms. For helium, unlike molecules, there are no rotational or vibrational energy states. There are only electronic energy states and translational energy. Certainly the electronic states are quantized, but can translational energy be quantized? Can the velocity of an atom be quantized? Certainly it is not at convention length scales, but is translation ("simple" motion) quantized at Plank length scales? Or is this question physically meaningless because of the uncertainty principle?

Re: Is temperature quantized like energy?

Quote:

Originally Posted by

**CliveStaples**
From what I remember about quantum mechanics, energy can only take on certain values--integer multiples of a certain minimally-measurable quantity. Or, perhaps more specifically, that it can only be *known or measured* (at the present time) that energy takes on integer multiples of this quantum value.

So what does this have to do with temperature? Well, from what I remember about thermodynamics, temperature is an average of the kinetic energy of molecules vibrating. If this energy can only take on integer multiples of some quantum value *c*, then the arithmetic average can only take on rational values of *c*, or if cleverer methods are used, roots of roots of roots (etc.) of rational values of *c*.

What does this have to do with anything?

Well, when students learn about the Intermediate Value Theorem in real analysis, they are often introduced to the following problem:

Consider the continuous function over the globe mapping each point on Earth to the temperature at that point. Use the Intermediate Value Theorem to prove that there must exist two antipodal points that have the same temperature.

However, if temperature is quantized, then you could never for instance get the value of pi**c* or e**c*, since these numbers are transcendental and therefore are not the root of any polynomial with rational coefficients. Therefore no temperature function could be continuous over an interval [a,b] if f(a) is less than (and f(b) greater than) a transcendental multiple of c.

Five years later:

I would say it's pretty clear that temperature is not quantized. Note that the quantization of energy in quantum mechanics is for specific systems (e.g. atomic levels, or highly idealized situations like harmonic oscillators). But, for instance, I can write down *a coherent state with any energy*. So, no, temperature is, in general, a continuous parameter.

Still, the temperature of a system with a specific potential, like a large N system of harmonic oscillators probably, would seem to be quantized. But QFT allows for continuous ranges of, for instances, energies emitted from photons when electrons/positrons collide and emit photons. Since atoms formed after the beginning of the universe, (and thus not all photons come from being emitted from atomic orbitals), it seems like there's no reason to think that the temperature of a given material is quantized.

Additionally, take a photon of energy E. It hits an atom, and the atom ejects a pair of electrons. First, if there's two electrons, then they can have any energies of any kind, E_1 and E_2, so long as E_1 + E_2 is satisfied. So if you measure only the first electron, it can have ANY range of energies. You might then ask what happens with the universe's total temperature, but of course, it's quite possibly infinitely large so that's complicated, but also it's important to note as I said above that atoms were formed during the periods between baryogenesis (formation of protons, neutrons form) and BBN (protons/neutrons form into nuclei). Before that, there weren't atomic orbital potentials (there were very different potentials coming from QCD), thus there was no quantization condition at the early universe. Thus, it seems highly unlikely that the energy in the universe is quantized either locally (where particles can have continuous energy, even assuming quantized orbitals) or globally (from the cosmological argument regarding baryogenesis and BBN).

Re: Is temperature quantized like energy?

yes,temperature can be quanticized like energy

Re: Is temperature quantized like energy?

Temperature is a pretty broad term, first what units K, C or F if talking proper SI unit than K. No it cannot be quantized as per quantum mechanics but can be measured along with dissipation if we are talking thermal reaction (release/transformation of energy as heat). Temperature has many sources not 1 constant source, source of thermal reaction can be quantified and we can predict temperature of energy release but temperature itself is not really a thing (substance) but a reactionary effect on or from substances (stuff,things,energies,ect...). As far as "*antipodal points that have the same temperature*" well of course there will be same at a certain frame but not throughout all frames. Especially on earth since our weather system is so highly dynamic, (many interactive values) that cannot be predicted with certainty. The odds given reference to enough time yes antipodal points on Earth can have the same temperature in a given moment but not in all frames.

Also as for "*However, if temperature is quantized, then you could never for instance get the value of pi*c or e*c, since these numbers are transcendental and therefore are not the root of any polynomial with rational coefficients. Therefore no temperature function could be continuous over an interval [a,b] if f(a) is less than (and f(b) greater than) a transcendental multiple of c.*" you are entirely correct if we are talking about thermal reaction above 0 kelvins, any substance below starts following QM. That is why I said temperature is broad, are we talking temperature external to or internal to, both have a mind (so to say) of their own. If an internal temperature drops to 0 kelvin a superfluid becomes of gasses such as helium 3 and takes on many characteristics of the quantum realm. 1) Atoms become harder to distinguish. 2) The matter acts as if one complete substance not made up of individual atoms. 3) Observation of becomes a generalized quantum interaction based on probabilities over certainties of behavior in space and time and more.

Now you can quantize as I said amount of energy turned to heat but not necessarily temperature because temperature is a measurement not a actual thing. Change of thermal reactions yes and we use temperature to indicate such but temperature itself is just that a measured change in thermal dynamics (energetic properties) of a medium. It is like asking if a ruler can be quantized it is a abstract integer placement to gain measurements of effects, nothing more. Hope this helps.