On the Extension of Linear Damping to Quantum Mechanics through Fractionary Momentum Operators Pt. I

Abstract

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.