The short answer to the main question is yes. As regards the other question, thermodynamics without quantum theory was incomplete, as state sums encountered in the thermodynamic properties were essentially undefined. It is only through quantum mechanics that we know what the state sums actually amount to and thus resolve a fundamental problem in the classical thermodynamics. For details, you may wish to consult the book Statistical Mechanics, 2nd edition (John Wiley & Sons, New York, 2004), by Kerson Huang. A more mathematically-oriented book is Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon Press, Oxford, 1970), by Oliver Penrose. Consult also Statistical Physics, Part 1, 3rd edition (Pergamon Press, Oxford, 1980) by Landau and Lifshitz.

The short answer to the main question is yes. As regards the other question, thermodynamics without quantum theory was incomplete, as state sums encountered in the thermodynamic properties were essentially undefined. It is only through quantum mechanics that we know what the state sums actually amount to and thus resolve a fundamental problem in the classical thermodynamics. For details, you may wish to consult the book Statistical Mechanics, 2nd edition (John Wiley & Sons, New York, 2004), by Kerson Huang. A more mathematically-oriented book is Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon Press, Oxford, 1970), by Oliver Penrose. Consult also Statistical Physics, Part 1, 3rd edition (Pergamon Press, Oxford, 1980) by Landau and Lifshitz.

There are many aspects when the quantum theory has practical effects on thermodynamics. 

Since from its fundamentals, such as the Gibbs paradox, which solution is automatic in quantum theory. Moreover, the translational partition function of free particles is proportional to the cube of the thermal De Broglie length. Then there is the internal partition function, which is the sum over levels of the Boltzmann distribution. The discrete nature of atomic and molecular level is a quantum effects. 

There are some important effects coming out from the exclusion principles, such as the  separation of ortho and para H2, which is a pure quantum effects.

(see doi:10.1016/j.ijhydene.2012.03.103)

Other aspects can be investigated by the use of quantum statistics like Fermi-Dirac or Bose-Einstein.

In some cases only quantum approaches are possible like in ultra-dense plasmas (see the book Kremp et al, Quantum Statistic of Nonideal plasma) using the formalism of second quantization, or the properties of neutron stars.

Without quantum theory, statistical mechanics cannot exists, because the classical mechanics cannot be quantitative in its prediction.

Classical Thermodynamics in its real sense whether it reversible or not doesn't have any thing to do with the quantum mechanics. Thermodynamics like algebra or Newtonian Mechanics,  it is a mathematical science, which  strictly relies on few  axioms, and some of them like the second law, which was   rigorously proven by Caratheodory that is closely related to the integrability of Pfaffian differential form of  the first law of thermodynamics as generalized by  him for  the multivariable systems.

Caratheodory' axiom that  is  generalization of  the Second Law clearly proves the following statement: Arbitrary near to any prescribed initial state there exist states which can not be reached from the initial state as a result of adiabatic processes.

 Combining Caratheodory's generalization of the first and the second laws, one can prove that  there exist the solution of first law denoted by    Del Q=SUMk   Xk  dxkhavea solution with property of     mu Del Q =d Phi,  where Phi is called entropy, which is a function of thermodynamics variables.  The function mu is apart from a multiplicative constant,  a function of the empirical temperature of the system. It is written as 1/T where T  is the absolute temperature.  As EINSTEIN mentioned few times in his life, the thermodynamics next to the algebra is the mathematically most sound scientific discipline, which is self-consistent and it is universal in every respect. 

The people especially  the physicists  they are use to go back and fort between thermodynamics and statistical mechanics because of that there is a great confusion in the terminologies. The first of  the classical quantum mechanics is conservative system in its real sense. How can you talk about in the internal entropy production for a quantum mechanical system? if your Hamiltonian is time invariant without  squeezing one's brain  or eye barrow?

Best Regards!  Please by  the way;   l love statistical mechanics since I learned how to manipulate permutations and combinations algebra  since I was in Naval High School  in early fifties. It is a beautiful scientific discipline having its own merit.

Yes, in combination with statistical thermodynamics. Quantum mechanics can give you the energy levels of a system, statistical thermodynamics can use them to compute thermodynamic properties as ensemble averages.

It is thus possible to generate intermolecular interaction potentials ab initio, and then to use them in computer simulations in order to compute the equation of state.

Dear Ulrich,  you are talking about the Statistical Thermodynamics as elaborated  by Fowler and Guggenheim, and its applications. You are  perfectly  alright, what you are commenting about its invaluable usage in practice. and it was a compulsory graduate course in the Metallurgy and Chemistry Departments in late fifties and early sixties ( Hill's Book) at Stanford. Best Regards.  

@ Tarik Ömer Oğurtani - No one, and certainly not I, object to thermodynamics, thermodynamic functions and thermodynamic variables. Problem arises when one sets out to calculate the thermodynamic functions in terms of thermodynamic variables, considering that the system under investigation consists of interacting microscopic building blocks called atoms. It is here that quantum mechanics proves indispensable. Of course, once one is given a thermodynamic function, such as the Helmholtz free energy, one can perfectly forget about quantum mechanics and its existence.

Lastly, since you refer to Einstein, it was notably Einstein who by making use of quantum mechanics solved the age-old problem of the specific heat "anomaly" at low temperatures (recall the erroneous classical Dulong and Petit law).

I haven't  seen any body even attempted to formulate the statistical thermodynamics to employ  the  Gibbs characteristic function as a Lagrangian, which should be minimized under the isothermal and isobaric conditions for the stability of thermodynamics systems.  All statistical Thermodynamics as clearly stated by Fowler and Guggenheim in their highly respected book on the subject that their treatment relies on the Helmholtz characteristic function for a good reason. Since the arguments of this function is;  Temperature, volume and mole numbers for multi-component thermodynamic systems. In the case of solid continua,   the volume should be replaced by strain tensor as state variable. But still one can easily used Helmholtz free energy as Lagrangian in the elastic theory of fracture (Eshellby, 1957) or in the formulation of irreversible thermodynamics of surfaces and interfaces by Ogurtani (2006, 2008),  compared to the Gibbs free energy , which can't be used as a Lagrangian since one can not find any any vectorial  function like the  displacement,  partial derivatives of which  may be defined the elements of the stress tensor similar to the strain tensor. These are bottle necks of  not only thermodynamics but also the statistical mechanics. 

Dear Behnam;  I would like to thank you giving me the lecture on the quantum mechanics what it can do and can't do,  very briefly and concisely.  Admittedly my knowledge on the topic has got little bid rusty since I left Stanford 1964. Then  I haven't touched NMR and the related topics any more. But I though quantum mechanics last 50 years as two semesters  graduate course, and still teaching it occasionally if I could find interested Ph.D. students.  By the way Albert never believed in quantum mechanics. He was completely against the probabilistic interpretation of Neil Born, what yes! he used it when he saw it was convenient for him. After all he was a pragmatic person!

Dear Tarik, I did not give anyone, including you, "lecture". This being a public forum, and not a private space for conversation between two individuals, one necessarily has to place one's arguments in some general background. My above comment was a direct response to your opening statement, that "Classical Thermodynamics in its real sense whether it reversible or not doesn't have any thing to do with the quantum mechanics." While I concurred with the fact, contained in your statement, that one can deal with the classical thermodynamics without reference to quantum mechanics, I emphasized that in actual fact the main objects of the classical thermodynamics vitally depend on quantum mechanics for being calculated.

As regards Einstein, it is true that Einstein did not consider quantum mechanics a complete theory, however it is an undeniable fact that he made some of the earliest and most fundamental contributions to this theory (the photoelectric effect being another one). Remarkable, when one considers that he took the idea of bosons from Bose and directly applied it to the atomic vibrations of a solid and thereby solved one of the major puzzles of the classical thermodynamics (regarding the specific heat "anomaly" at low temperatures to which I referred in my earlier comment on this page).

By the way I recommend every body the book entitled ' Statistical Thermodynamics' by Erwin Schrödinger' Cambridge University Press, 1962. A course of Seminar Lectures, delivered in January-March 1944 at the School of Theoretical  Physics , Dublin,  IAS. This book is only 95 pages, and I bough it from Stanford Book Store  in 1962.  That was an excellent books I have ever had in my life so concise and clear text!  At  the same I had Tolman's   book 'The Principles of Statistical Thermodynamics'  1959 6th edition, which was appeared first 1938.  Probably the best book ever appeared since then. It has 661 pages. I had no trouble reading them.  I can assure you if you have permutation and combination algebra  in your sack plus if you are gifted in mathematics by bird,  then you could menage  any special statistics problems you face  in material science and solid state physics  using your pencil and eraser! and a sheet of white paper!  Schrödinger's notes are quite sufficient to get you confidence back to normal on this topic. 

Quantum statistical mechanics is a fundamental science that is designed to answer such questions (and they were answered many years ago already). Statistical thermodynamics of quantum systems is derived rigorously in the framework of quantum stat. mechanics. If you would like to really understand quantum statistical mechanics, you have to become strong in a number of math disciplines. If you are not that deeply interested, then any reasonably written handbook for engineers will help. Notably, you have to read modern handbooks, because there have been many new developments and new understanding of various quantum phenomena, and also change in math notations since Schrodinger. If you are to become a strong scientist  or engineer, I would advise you to begin with a strong graduate course in quantum mechanics, such as Cohen-Tannoudji C., Diu B., Laloe F. Quantum mechanics, vols. 1 and 2, and then move to an equally strong graduate course in quantum statistical mechanics, such as A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems, McGraw Hill, NY. Good luck!

@Behnam Farid, thanks for your clear answer.  I agree that quantum mechanics, in many(!) cases naturally gives a way to count microstates in the partition function. In classical statistical mechanics, that "counting" must be done by defining a natural measure on statespace. The best known example is probably the (q,p)-configuration space of a particle which carries a natural measure, the Liouville measure invariant under the Hamiltonian flow (being  the co-tangent space of a manifold or more generally a symplectic space). It seems to be an unfortunate but widespread misunderstanding that quantum mechanics is ultimately discrete. Surely many spectral problems lead to discrete spectrum, with finite dimensional eigen spaces, but the free particle is an easy counterexample, where the spectrum of the Hamiltonian is not discrete and it can hardly be called a mathematical curiosity. For more interesting example, many solid state physics problems give rise to continuous band structures.

Does quantum mechanics give a density of states measure on such a band spectrum from first principles (e.g. by properly accounting finite temperature effects on a finely spaced but discrete spectrum that exist at T = 0) or is this something that has to be put in by hand in the particular (quantum mechanical) model of the solid ? 

The global Gibbs free energy  involve mechanical interaction of the system with its surroundings through the surface tractions and  long range body forces  Even by employing the  methods of thermodynamics whether it is  irreversible or not,   one has great difficulties to formulate a simple problem such  as the stability of surfaces and interfaces of  deformable bodies even under the static loading. One shouldn' t forget natural processes involve without acception the  dissipation of energy. i.e., the positive definite entropy production!  How can you inset this nonconservative elements into the quantum mechanics calculations even though we don't how to handle it properly  in the classical Lagrangian mechanics. I AM SORRY GUYS.

@ Rogier Brussee - You are most welcome. You raise several very good points. Regarding the Liouville measure, it was in fact what Gibbs introduced in describing what is known as the Gibssian ensemble. But we know that classically we cannot escape the Gibbs paradox. Interestingly, the constant h that we introduce classically for normalization in the phase-space description of Hamilton systems, proves to be the constant of Planck when we go over to quantum mechanics (this is apparent from the Sackur-Tetrode equation -- as an aside, the interested may wish to consult the biography of the forgotten genius Hugo Tetrode in the English Wikipedia). Regarding the discrete spectrum, you are absolutely right. It is truly regrettable that the relationship ε = h ν is often incorrectly interpreted as signifying discrete spectra in quantum-mechanical systems.

Regarding your question, when one describes the quantities of interest in terms of appropriate Green functions, one does not need to worry about measure, as measure is automatically accounted for. The density-of-states at arbitrary temperatures, to which you refer, is deduced from the imaginary part of the thermal single-particle Green function along the real axis. If however one describes quantities of interest in terms of many-body states, one encounters the practical problem of massively degenerate excited energies; even for uniform systems, enumerating and indexing these states amounts to a fundamental practical problem. In the Green-function formulation, all the indices and coordinates that do not directly couple to the observable of interest, are automatically traced out. In dealing with many-particle systems, a common approach is use of the micro-canonical ensemble. What I mean by this, I have described in Appendix C of the attached paper.

Article On the Luttinger theorem concerning number of particles in t...

@Behnam Farid, thank you !  I have to digest this. 

You are welcome Rogier,

Dear Samuel,  Please don't worry!  no one   even  attempted yet to calculate the enthalpy or the Gibbs free energy of deformable solids under the realistic boundary conditions using  the quantum mechanics of any kind. 

For the isobaric system one has the following connection between the internal entropy production and global Gibbs free energy variations (bulk and surface) for  the isothermal natural changes: T d DELS  /dt   = -  d DELG /dt   > 0  using this expression, which can be obtained from Planck Criterion (1887) one may formulate well posed boundary value problem, and then   may track down the Gibbs free energy evolution by  the help of the computer simulations, namely:  the grain boundary crack initiation  and propagation. See attached paper.

This studies show first time that the transient regime is controlled by the maximum entropy production but the consecutive  stationary non equilibrium regime is under the action  of the minimum entropy production.   This shows that not only minimum entropy production hypothesis as    proposed by Prigogine but also the maximum entropy production advocated  by Shannon, and others may be  operational in dynamical isobaric systems  in practice.