Maximization Of Arrangements and Quantum Bound States

 The Maxwell-Boltzmann MB distribution is often derived by maximizing the number of possible arrangements of particles with different energies subject to the constraint of a fixed number of particles and fixed energy. Using the canonical approximation that only average energy is conserved and using Stirling’s approximation for large number factorials leads to the Maxwell-Boltzmann distribution. The ln of total number of arrangements using Stirling’s approximation is linked to a math form called Shannon’s entropy i.e. - Sum over i f(ei) ln(f(ei)).

  Interestingly this same MB result may be obtained through reaction balance which makes no use of the idea of arrangements or entropy or maximization. It is concerned with including all possible physical reactions (for example two body reactions between all energy particles) subject to conservation of energy. In this sense it deals directly with the dynamics of the problem while the arrangement approach does not. The dynamical approach considers all possible reactions with notions of independence or lack of information and ensures the system remains unchanged. Why are they the same? We argue that it is because the “recipe” for the probability of two body reactions namely n(e1)n(e2) is of the same form as the leading factor of  n(e1)! n(e2)!.

   In the case of quantum mechanics one may apply the dynamical approach to obtain the time-independent Schrodinger equation. In such a case one does not need to think in terms of “maximizing” any quantity although one considers all possible reactions with a potential V(x). a(p) factors in W(x) = wavefunction = Sum over p a(p)exp(ipx) are interrelated, but retain their identity after all possible interactions occur. This we argue is the idea for the eigenvalue equation. 

   At the same time, Shannon’s entropy, which is directly linked to the notion of arrangements, is applied to a bound state and possibly linked to thermodynamics i.e. dE = TdS -PdV which may be applied to an adiabatic expansion. Shannon’s entropy is created from existing probabilities in this case which do not arise from explicitly maximizing entropy subject to constraints as the probabilities follow directly from the Schrodinger equation. One might ask why Shannon’s entropy has any relevance if it is the dynamics which determines the value of W(x) and a(p).

It seems, however, that one may knock out a particle in a state p with probability P(p)= a*(p)a(p) or find a particle at x with probability P(x)=W*(x)W(x). These are classical type probabilities linked to the quantum dynamical probabilities a(p) and W(x). It seems Shannon’s entropy is only applied to P(x) and P(p) or a product P(x)P(p) which may give it physical meaning in a quantum bound state.

   We examine these ideas in this note.