# Position Dependent Planck's Constant in the Schrodinger Equation. (arXiv:1812.02325v1 [quant-ph])

There is evidence that Planck's constant shows statistical variations with

altitude above the earth due to Kentosh and Mohageg, and yearly systematic

changes with the orbit of the earth about the sun due to Hutchins. Many others

have postulated that the fundamental constants of nature are not constant

either locally or universally. This work is a mathematical study examining the

impact of a position dependent Planck's constant in the Schrodinger equation.

With no modifications to the equation, the Hamiltonian becomes a non-Hermitian

radial frequency operator. The frequency operator does not conserve

normalization, time evolution is no longer unitary, and frequency eigenvalues

can be complex. The wavefunction must be continually renormalized at each time

in order that operators commuting with the frequency operator produce constants

of the motion. To eliminate these problems, the frequency operator is replaced

by a symmetrizing anti-commutator so that it is once again Hermitian. It is

found that particles statistically avoid regions of higher Planck's constant in

the absence of an external potential. Frequency is conserved, and the total

frequency equals "kinetic frequency" plus "potential frequency". No

straightforward connection to classical mechanics is found, that is, the

Ehrenfest's theorems are more complicated, and the usual quantities related by

them can be complex or imaginary. Energy is conserved only locally with small

gradients in Planck's constant.