# I Quantum thermodynamics of single particle

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1. Oct 20, 2016

### Konte

Hello everybody,

I have two questions:

1) Is it possible to define a temperature for single particle (or atom or molecule)? If so, how?

2) How to model with quantum Hamiltonian an exchange of energy between a single atom (or molecule) and a reservoir at given temperature $T$ ?

Thank you everybody.

Konte

2. Oct 20, 2016

### Bystander

3. Oct 20, 2016

### Konte

4. Oct 21, 2016

### Demystifier

Yes, by taking
$$\rho=e^{-H/kT}$$
where $H$ is the single-particle Hamiltonian.

5. Oct 22, 2016

### Konte

I still have question:
- in this post https://www.physicsforums.com/threa...echanics-and-temperature.426455/#post-2869963, the forumer xerma mentioned a $\rho = \frac{e^{-H/kT}}{Z}$. What is the difference between those two definitions of $\rho$ (that I suppose both density matrix operator) ?

Thank you very much.

6. Oct 23, 2016

### vanhees71

The latter formula is correct. The statistical operator must be of trace $1$. Thus the statistical operator of the canonical ensemble is
$$\hat{\rho}=\frac{1}{Z} \exp(-\beta \hat{H}), \quad Z=\mathrm{Tr} \exp(-\beta \hat{H}), \quad \beta=\frac{1}{k_{\text{B}} T}.$$

7. Oct 23, 2016

### Konte

How to demonstrate this expression of $\hat{\rho}$?
Because, after searching, I always have another alternative form $\hat{\rho}= \sum_i p_i | \psi_i \rangle \langle \psi_i|$

Thanks.

Konte

8. Oct 23, 2016

### vanhees71

This expression you get from the maximum entropy principle. If you look for all statistical operators that lead to a given expectation value $U=\mathrm{Tr} (\hat \rho \hat{H})$ of the energy and minimize the entropy,
$$S=-k_{\text{B}} \mathrm{Tr} \hat{\rho} \ln \hat{\rho},$$
you get to the canonical statistical operator (of course you need the normalization $\mathrm{Tr} \hat{\rho}=1$ as another constraint).

The "alternative form" is just the expansion of the statistical operator in terms of its eigenvectors. Note that there can be also generalized eigenvectors if the operator has a continuous spectrum.

9. Oct 23, 2016

### Konte

@vanhees71
Ok, thank you for this interesting answer. Could you indicate me some lectures that can help to understand and make it clearer for a novice as me, please?
I suppose, all of those concepts are valid and applicable for the case of a single system (like single atom or single molecule) ?

Thanks.
Konte

10. Oct 23, 2016

### vanhees71

Well, these are very general concepts. My favorite book, using the information-theoretical approach, is

A. Katz, Principles of Statistical Mechanics, W. H. Freeman and Company, San Francisco and London, 1967.

Very good is also Landau&Lifshitz, vol. 5 or Reif.

11. Oct 23, 2016

### Konte

Thanks a lot.

Konte

12. Oct 24, 2016

### Konte

Hello,

I'm back, just because a little doubt persist on my understanding:
Is the operator $\hat{\rho} = \frac{e^{-\beta \hat{H}}}{Z}$ still meaningful even for describing a pure state?
Thanks

Konte

13. Oct 24, 2016

### vanhees71

Of course not. An equilibrium state is only a pure state for $T \rightarrow 0$. Note that using the equilibrium (canonical) distribution means that you look at a single particle within a heat bath at temperature $T$!

14. Oct 24, 2016

### Konte

Thank you,

So even for a single particle within a heat bath at $T$°, the concept of mixed states is meaningful?

In other words, single particle is describable as a canonical ensemble once it is "surrounded" by a heat bath at fixed $T$° (equilibrium)?

Konte.