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Temperature

Zhigang Suo's picture

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Li Han's picture

Hi Zhigang, according to the note, temperature can be defined as 1/T=dln(#of state)/dU for a system that can only interact with environment by heat. What is the temperature of H atom in ground state since U is discrete? Is it a legitimate question at all? Thanks.

Li Han

Zhigang Suo's picture

When a small system like a hydrogen atom is in contact with a reservoir (i.e., a large system), the temperature of the reservoir is well defined.  The small system flips from one state to another, and exchanges energy with the reservoir.  We say that the small system is held at a temperature (i.e., the temperature of the reservoir).  We can work out the probability for the small system to be in each of its state (i.e., the Boltzmann distribution).

The temperature of the small system itself, however, is undefined.

You might be interested in this thread of discussion on temperature

Li Han's picture

What about one H atom in thermal contact with a H2 molecule? Can we still talk about the temperature here without referring to another HUGE reservior?

Li Han

Zhigang Suo's picture

I can think of two issues.

  1. When two small systems are brought together, they may react.  If they do, a quantum state of the composite is no longer a combination of a quantum state of one system and a quantum state of the other system.  Consequently, the number of quantum states of the composite is no longer the product of the numbers of quantum states of the two systems.  The fundamental postulate is still correct:  all quantum states of the composite are equally probable.  But now this postulate will not lead to a definition of temperature for each system, simply because the two systems have lost identity in the reaction.
  2. Even when the two systems do not react, the number of quantum states for a small system like a hydrogen atom is a function of energy with large gaps.  The energy transfer between the two systems is no longer a continuous variable.  You cannot define temperature by derivative.  Using the fundamental postulate, you can work out the probability of each partition of energy between the two systems.  In this case, you do not need the concept of temperature to answer any question you can post.
Henry Tan's picture

This may be a naïve question.

Is the number of quantum states a quantity that is experimentally measurable, or just a theoretical concept? This is fundamentally important if we want to set the foundation of thermodynamics on this.

Zhigang Suo's picture

Yes, the number of quantum states is routinely measured experimentally, as described on p.9 of the notes on temperature.

Zhigang,

  This seems circular?  To make the measurement you have to invoke lots of other theory about energy and temperature etc. which requires knowledge of the states.  Perhaps there is a clear logical sequence but I do not see it easily.  As another point, isn't the number of quantum state dependent upon the resolution at which you decide to describe your system?   If there are internal dofs, then there are hidden states and one can never know how many hidden variables there are in a system.  Am I missing something?

 -sanjay

 

Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Zhigang Suo's picture

Sanjay:  Not sure which part you feel to be circular.  I list the main points in the notes here:

  1. I cited two ways to determine the amount of heat added to a system.  I also cited a way to determine absolute temperature.
  2. By the definition of the absolute temperature, dS=dU/T.  Once you can determine heat dU and temperature T, you can determine S incrementally, up to an additive constant.
  3. The symbol S means log (number of quantum states).

The additive constant in S may go back to your cncern of resolution.  In practice S is set to be zero for perfect crystal at T = 0.  Tables of experimental values of S are available.

Zhigang: 

I guess my not very well articulated point was: Can the quantum states of a system by enumerated (experimentally) without relying upon thermodynamic concepts like S, U, and T (which in a way rely upon the enumeration of the quantum states)?  I fully understand and believe that the theory is self-consistent and agree with what you wrote.  My question really is can you make the measurement of the number of quantum states without using any results that rely upon their existance. 

-sanjay

 

Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Henry Tan's picture

As so far to my understanding, all those numbers of quantum states directly got (WITHOUT REFERRING TO THERMODYNAMICS) were from quantum mechanics calculation itself.

No experiment can directly COUNT the number of quantum states. Hopefully I am wrong.

Zhigang Suo's picture

Dear Sanjay and Henry:  Glad we agree on the practice of thermodynamics.  We might differ on the interpretion or wording of the practice.  Such differences are fun, and can enhance our understanding of the practice.

I myself am perfectly happy to let experimental measurements "count" the number of quantum states.  Thus, an experiment functions as an analog computer, and does better and faster than a digital computer in telling us the number of quantum states of an everyday system.

Whenever we measure temperature and heat, we are doing experimental quantum mechanics.  Of course, such measurements do not tell us the shape of electron cloud, or the state of spins, or even what the system consists of, but such measurements do tell us the number of quantum states of the system.

It is wonderful to have a method to determine the number of quantum states of a system without knowing what the system consists of.  We all have intimate knowelege of such a statement if the phrase "the number of quantum states" is replaced with "stiffness".

Weijie Liu's picture

Hi, Zhigang, in the note, it says: The three properties are connected through a beautiful relation: dS = dU/T .

However, in physics we define entropy by dS = dQ/T, which Q is heat. (The Feynman Lectures on Physics, Volume I)

 

Then, how do get the formula in function of internal energy instead of Heat energy?

Best regards,

Zhigang Suo's picture

I have just updated the notes on temperature to get ready for the undergraduate course in the Fall.  

Using dS = dQ/T to define entropy is difficult.  It requires a specific scale of temperature.  In the notes I followed two different routes to the concept of temperature.  We regard entropy as a concept more premitive than temperature.  See notes on entropy

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