Does Quantum Thermodynamics Dictate the Form of the Wavefunction P(x)=W(x)W(x)?

 The Schrodinger equation, a partial differential equation in W(x,t) the wavefunction, was developed by Schrodinger from ideas of L. de Broglie and then  given the interpretation of probability Probability(x,t)= W*(x,t)W(x,t)  by M. Born.

    In this note we ask whether the form P(x)=W*(x)W(x) together with P(p)=a*(p)a(p) such that W(x)=Sum over p a(p)exp(ipx)  follows from quantum thermodynamics. In particular we concentrate on a quantum adiabatic transformation and insist it be one of no entropy change (as in (1)). The quantum spatial density of a particle in a box with infinite potential depends on L as P(x)=   1/L density(x/L) (bound state). Thus Shannon’s entropy simply using P(x) is L dependent. One needs P(x)P(p) in addition to the first term changing by ln(1/L) and the second by ln(L) in order to remove L dependence during an adiabatic expansion. Similar considerations hold for the quantum simple harmonic oscillator (2). Thus the idea of a P(x) and P(p) which are linked (not independent) immediately follows form quantum thermodynamics as does the form of entropy S=Sx+Sp.

   As a result it seems that if one deals with probability, one must not only consider conditional probability, but also a transformation function f(px) which allows for inverse dependencies on p and x on a parameter. These ideas together with P(p/x)=P(x and p) / P(x) ((1a)) and P*(x/p) = P(x and p) / P(p) lead, we argue, to the form P(x)=W*(x)W(x) and P(p)=a*(p)a(p) where W(x)=Sum over p a(p) exp(ipx).

    We considered the approach of conditional probabilities in an earlier note (3), but did not give motivation for using them or the form exp(ipx). Rather we brought them in as assumptions, assuming that there should be both a P(p) and P(x). In this note, we argue that they seem to follow from the idea of quantum thermodynamics and a zero entropy change along an adiabatic path i.e. one which does not change the quantum bound state n.