Logarithm of Quantum Wavefunction as a Partition Function

In quantum mechanics, one focuses on densities such as the probability density W*(x)W(x), where W(x) is the wavefunction, and expectation values W*(x) Operator W(x) (integrated over space). The question then is what is the significance of the wavefunction W(x)?

In this note, we argue that ln(W(x)) can act as a partition function with the added probability factor sin(px) multiplied by f(p) in a Fourier expansion of W(x). The factor sin(px) contains both exp(ipx) and exp(-ipx) and so represents both forward and backward motion which are usually separated in time and  which accountsfor a wavelength. This approach allows one to calculate an average kinetic energy which satisfies: KE + V(x) = E where V(x) is the potential at any point x and E the energy. It is possible that this approach has appeared before in the literature, but we are unaware of any specific paper. We consider the case in which f(p)=sin(px) exp(-p2/a) where a is a constant and show that it is associated with a harmonic oscillator.