Relative Probabilities in Quantum and Classical Statistical Mechanics

 Classical mechanics in the ideal gas scenario and quantum mechanics make use of a set of energy levels which a single particle may occupy. Thus the problem becomes to determine the probability associated with ei i.e. P(ei). In this note we take the approach that relative probabilities depend on the energy gap in question. For example, consider a quantum two level system where ei values are provided by the time-independent Schrodinger equation. From the point of view of statistics, these are simply a set of ei values. Their associated physics is contained in the Schrodinger equation. For the quantum two level system one must find the relative probabilities PR(e1) and PR(e2). Next, imagine a quantum system with many levels and assume for the sake of argument there are levels  e1,e2 and  ei,e(i+1) such that e2-e1=e(i+1)-ei. If the mechanism is the same for jumping from e1 to e2 and from ei to e(i+1) (i.e. absorbing a photon of the same energy),  one might argue that the relative probabilities PR(e2)/PR(e1) =  PR(e(i+1))/PR(ei). From this one may obtain the Maxwell-Boltzman result exp(-ei/T)).  Note that levels do not need to be adjacent. For example in a large level system one might consider e1 and e3 and any ei and ej such that e3-e1 = ej - ei.

   In some situations this may need to be modified (e.g. fermions and bosons have enhancements and there may be level related issues), but as a first step we argue that insisting on relative probabilities being dependent on ei-ej has important consequences. Furthermore, in such a case the normalization constant for these relative probabilities i.e. C(T)=  Sum over i  exp(-ei/T) is important. Taking ln(C(T) and then taking the derivative d/dT yields an expression proportional to the average energy. An expression for dE/dT dT, however, would lead to a second order differential equation in ln(C(T)). If one introduces a new function S such that E-TS = -Tln(C), then one may show that dE/dT = T dS/dT i.e. entropy S emerges.