All major branches of physical science have certain features in common. Each has explicit assumptions: basic laws of nature, such as Newton's Laws of Motion. Each also has implicit assumptions: rules which are as fundamental as the explicit assumptions, but which are usually omitted from the lists of postulates. Finally, each field has a set of exemplary problems, models, and results which demonstrate the range of questions which the theory is competent to answer. We have briefly compared the histories of thermodynamics, quantum mechanics, and statistical mechanics. A comparison of their theoretical structures may prove equally revealing.
The explicit assumptions of textbook thermodynamics are usually reduced to the ``three laws of thermodynamics'', though there are equivalent postulations using different numbers of axioms. The laws of thermodynamics are not complete in themselves. To use them, one also needs to have enough knowledge of material systems to recognize the importance of pressure, temperature, and other thermodynamic variables. The existence of an absolute scale for temperature is sometimes elevated into the status of a law, the so-called Zeroth Law of Thermodynamics. Implicit assumptions of thermodynamics are made visible by the usual proof of Gibbs' phase rule, whose inductive cycle fails when one tries to compare the one-component, one-phase system with something simpler. The derivation of Gibbs' Phase Rule, as presented below, invokes an implicit assumption of thermodynamics, namely that a normal single-component, single-phase system has two degrees of freedom.
Thermodynamic calculations are largely confined to giving algebraic relations between different thermodynamic parameters, or, in a few cases, inequalities which limit the possible numerical values of thermodynamic quantities. Some thermodynamic relations, such as the connection between the enthalpy of reaction and the temperature dependence of the corresponding equilibrium constant, might not be intuitively obvious. However, theoretical thermodynamics does not provide numerical values for thermodynamic quantities. If one wants to know the specific heat of argon at one atmosphere and , one must perform a thermodynamic experiment.
The explicit postulates of quantum mechanics are slightly more elaborate than are those of thermodynamics. A system is presumed to be described by a wave-function (probability density) , whose behavior is described by the time-dependent Schrodinger equation
so long as the system is not being observed. Here , the Hamiltonian operator, is constructed by prescription from the system's Hamiltonian, so quantum mechanics presupposes a knowledge of classical Hamiltonian mechanics.
An implicit assumption of quantum mechanics is provided by the ``Copenhagen Interpretation'' and measurement theory, which tell how is related to experiment. These implicit rules provide that when the system is observed, is to be mathematically expanded in terms of eigenfunctions of the operator which corresponds to the variable of observation. The act of observation physically transforms into a single eigenfunction in a probabilistic manner, so that the initial value of predicts the likelihood that an experiment will have each of its possible outcomes.
In addition to these assumptions, quantum mechanics also has exemplary models, such as the isolated harmonic oscillator. To obtain correct spectral wavelengths, a correct subatomic model must be used. Indeed, any calculation of the atomic spectrum of hydrogen is necessarily both a test of the correctness of quantum theory and a test of the correctness of one's model for the hydrogen molecule. For example, it seems unlikely that a quantum mechanical calculation of atomic spectra would get the right wavelengths if an electron were assumed to be a spherical shell of the usual charge, but having the diameter of a grapefruit.
Unlike thermodynamics, quantum mechanics does provide quantitative information about material systems. Atomic and molecular spectra are obtained with high accuracy, though the accuracy decreases as the number of interacting electrons increases. From quantum mechanics, one can also obtain accurate information about the electronic properties of crystals, and in some cases information about the conformation and stability of large molecules. Quantum mechanics and thermodynamics, at least as generally used, do not appear to overlap. Thermodynamics calculations do not entail a belief in the atomic structure of matter, let alone suggest a particular model of atomic structure. On the other hand even an extensive development of quantum mechanics along conventional lines does not appear to imply the existence of temperature as a significant phenomenon.
The link between microscopic and macroscopic treatments of matter is made by statistical mechanics. The major explicit assumptions of statistical mechanics are:
1) for any material system one can set down a complete microscopically explicit description of the system, either in terms of quantum numbers and complex phases, or in terms of Hamiltonian coordinates and their canonical conjugates.
2) the logarithm of the likelihood that a system, in thermal equilibrium, will be found in a particular one of its allowed states is linearly proportional to the total energy of that state.
3) by taking a correctly-weighted average over the accessible states of the system, one obtains correct values for macroscopic (thermodynamic) parameters.
A major implicit assumption of the theory, for which a strong plausibility argument can be made, is that the average value of is , H being the system Hamiltonian, being the inverse temperature in energy units, and A being the system's Helmholtz free energy.
With the techniques discussed below, one may calculate numerical values of equilibrium thermodynamic properties from quantum-mechanical expressions for intermolecular forces. An extension of the theory in principle allows one to deal with systems which are not in thermal equilibrium. The purpose of this text is to provide exemplary problems and results which illustrate the theory's explicit assumptions, reveal the theory's implicit assumptions, and illuminate the range of questions to which statistical mechanics provides answers.