Quantum theory allows us to assign a finite value to entropy and calculate it as a function of Planck's constant. Constant entropy is included in the calculation and this allows us to quantify the predictions of quantum theory. The second law of thermodynamics establishes the existence of entropy as a function of the state of the thermodynamic system, that is, "the second law is the law of entropy." In an isolated system, the entropy either remains unchanged or increases (in nonequilibrium processes), reaching a maximum when thermodynamic equilibrium is established (the law of increasing entropy). Different formulations of the second law of thermodynamics found in the literature are specific consequences of the law of increasing entropy

DR Natalia: please have a look at the link:

http://www.math.uit.no/seminar/Preprints/06-06-PJVL.pdf

"Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in such a way as to illustrate many of the subjects that have appeared."

Chapter Entropy in Quantum Mechanics and Applications to Nonequilibr...

Kind regards--

You can see Von-Neuman Entropy

S = - (l1 *loge l1 +... + lm*loge lm), here S denotes the

Von Neumann entropy, loge stands for logarithm base e , l1,l2,...lm are the eigenvalues of the density matrix description of the quantum system, it is zero for every pure state density matrix description and attains maximum value when the density matrix is (1/m)*Identity matrix( m by m ).

Thank you so much.

Dear Natalia,

Excellent books on this subject are

Robert McEliece

The theory of Information and Coding,

Canbridge University Press;

and

Michael A. Nielsen, Isaac L. Chuang,

Quantum computation and quantum information,

Cambridge : Cambridge University Press, 2000

The wave function describes a pure state, with which one would not think much entropy could be associated (if you look at the post by Debopam Gosh above). Nevertheless, there is quite a large literature on this, which indicates that there is a bit more to the story. One example is this article, Article Entanglement used to identify critical systems

I don't know if this is what you had in mind (?).

Thank you.

The H theorem, or rise of entropy as you approach an isolated system in equilibrium, uses the entropy def.

S= -kB (sum over i) P(i) ln P(i)

which can be associated with QM. This is proved in

Appendix A12 of Reif,

Fundamentals of statistical and thermal physics. Mac Graw Hill

The wavefunction is just a means to compute the probability density of the quantum system; it's from the probability density that the entropy is computed in the usual way. And the probability density is the ``interesting'' quantity to study, not the wavefunction, since many wavefunctions define the same density. The only difference between quantum fluctuations and any other fluctuations is in how the probabilities are computed; once the probability density has been computed, the mathematical properties of the probability density are the same, whatever the origin of the fluctuations. Among such properties is the entropy.

So one way to define what a ``sensible'' system is, is by the property that the probability density, ρ(x,t) is a non-negative function on its space of states such that S(t)=-Trx (ρ(x,t)ln ρ(x,t)), is finite; i.e. where the trace exists. Therefore, for any ``sensible'' system, in fact, the density can be written as ρ(x,t)=|ψ(x,t)|2. So it's, always, possible to introduce a wavefunction; it's not, however, necessarily, useful.

And for all systems a probability density ρ(x,t) satisfies a local conservation law: dρ/dt=-div J(x,t). So part of the description of the system involves defining the probability current density J(x,t).

[For classical systems it turns out that J is a local function of ρ(x,t) (e.g. J=-D grad ρ(x,t)-U(ρ), where U is a vector-valued function of the density ); the interesting property of quantum systems is that J(x,t) can't be written as a local expression in terms of ρ(x,t). (Here ``local'' means an expression that contains a finite number of derivatives.) It can be written, however, as a local expression in terms of ψ(x,t), which is the solution of the Schrödinger equation, that's a partial differential equation. That's why the wavefunction becomes useful for quantum systems, isn't, however, that useful for classical systems. However, in the path integral formulation, that maps quantum fluctuations of one system to classical fluctuations of another system, there's no need to talk about the wavefunction at all; but it is useful to talk about the entropy of the corresponding probability distribution, which turns out to be the entropy of the quantum system.]

Quantum theory allows us to assign a finite value to entropy and calculate it as a function of Planck's constant. Constant entropy is included in the calculation and this allows us to quantify the predictions of quantum theory. The second law of thermodynamics establishes the existence of entropy as a function of the state of the thermodynamic system, that is, "the second law is the law of entropy." In an isolated system, the entropy either remains unchanged or increases (in nonequilibrium processes), reaching a maximum when thermodynamic equilibrium is established (the law of increasing entropy). Different formulations of the second law of thermodynamics found in the literature are specific consequences of the law of increasing entropy

DR Natalia : Please have a look at the following useful links:

http://www.math.uit.no/seminar/Preprints/06-06-PJVL.pdf

Article Entropy Production in Quantum is Different

Regards--

Thank you very much.