Quantum and Classical Statistical Systems in Terms of Constraints

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.)