  The use of fractional momentum operators and fractionary kinetic energy used
to model linear damping in dissipative systems such as resistive circuits and a
spring-mass ensambles was extended to a quantum mechanical formalism. Three
important associated 1 dimensional problems were solved: the free particle
case, the infinite potential well, and the harmonic potential. The wave
equations generated reproduced the same type of 2-order ODE observed in
classical dissipative systems, and produced quantized energy levels. In the
infinite potential well, a zero-point energy emerges, which can be fitted to
the rest energy of the particle described by special relativity, given by
relationship $E_r=mc^2$. In the harmonic potential, new fractional creation and
destruction operators were introduced to solve the problem in the energy basis.
The energy eigenvalues found are different to the ones reported by earlier
approaches to the quantum damped oscillator problem reported by other authors.
In this case, a direct relationship between the relativistic rest energy of the
particle and the expected value of the fractionary kinetic energy in the base
state was obtained. We conclude that there exists a relationship between
fractional kinetic energy and special relativity energies, that remains unclear
and needs further exploration, but also conclude that the current form of
transforming fractionary momentum operators to the position basis will yield
non-observable imaginary momentum quantities, and thus a correction to the way
of transforming them needs to be explored further.
