Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.
Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.
Abstract Planck’s original introduction of the quantum idea was based on a statistical mechanical treatment of black body radiation. S. Bose, a physicist working in India, found a new way of deriving Planck’s law. Einstein applied Bose’s method to a gas of massive particles. This led to the prediction of a new physical effect (Bose-Einstein condensation). With the introduction of the exclusion principle by Pauli, Fermi, and Dirac independently introduced a new kind of statistics. Considerations by Einstein and Schrödinger introduced a preliminary form of second quantization.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
The author presents his experience in teaching at a graduate level the quantal exposition of a new non-statistically based paradigm of physics and thermodynamics. This paradigm, called the Unified Quantum Theory of Mechanics and Thermodynamics, applies to all systems large or small (including one particle systems) either in a state of thermodynamic (i.e. stable) equilibrium or not in a state of thermodynamic equilibrium. It uses as its primitives inertial mass, force, and time and introduces the laws of thermodynamics in the most unambiguous and general formulations found in the literature. Starting with a precise definition of system and of state followed by statements and corollaries of the laws of thermodynamics, the thermodynamic formalism is developed without circularity and ambiguity. In this quantal exposition of the new paradigm, a brief review of the formalism of thermodynamics as a general science not limited to stable equilibrium and large (macroscopic) systems as well as a very brief summary of the three prevalent formalisms in classical physics are presented followed by a presentation and development of solutions for a number of elementary problems in quantum physics (e.g., a particle in a box, a harmonic oscillator, a rigid rotor, etc.). These solutions and the maximum entropy principle are then used in a constrained optimization to develop the canonical and grand canonical distributions for Fermi-Dirac and Bose-Einstein types of particles, i.e. for fermions and bosons. This is done without the use of analogies between statistical and thermodynamic results and without additional hypotheses such as the ergodic hypothesis of statistical mechanics. These distributions are then employed under various assumptions (i.e. the Boltzmann, constant-potential, point-particle, and continuous eigenvalue-spectrum approximations) to derive the corresponding thermodynamic property expressions for perfect, semi-perfect (ideal), and Sommerfeld gases as well as for mixtures of ionized and dissociated gases. In a similar fashion but with a change from a single- to a multi-particle partition function and with the addition of various inter-particle potentials for two-particle interactions (e.g., the Lennard-Jones potential, the square-well potential, etc.), expressions for the thermodynamic properties of dense gases are developed and presented.An initial version of this paper was published inJuly of 2006 in the proceedings of ECOS’06, AghiaPelagia, Crete, Greece.
The premise that physical variables must be viewed as random variables, with observed values, assigned as averages over distributions, is shown to yield the formalisms of thermodynamics, Hamiltonian mechanics, and quantum theory. A general mathematical argument first establishes the validity of an embedding of th space of physical variables in a larger manifold of "hidden" variables and a general structure for the distribution of these variables. Then, by consideration of various ways of specifying the preparation of a system for measurement, the formalisms of physics cited above are deduced. It is also shown that the generalized Langevin equation follows from these premises and is applicable to a wide class of phenomena. This formulation yields a new definition for compatible observables as used in quantum theory and demonstrates, in a natural way, that physical laws should be invariant under orthogonal transformations of certain variables, which in one circumstance include the complex orthogonal transformations of the Lorentz form, as in special relativity. In summary, it appears that most of the formal structure of physical theory is a consequence of the statistical nature of physical variables.
Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)
Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the fundamental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that represents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.