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.

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