Informação de Fisher e entropia de Shannon de osciladores com massa dependente da posição

Abstract

In this work we study from both classical and quantum point of view the position dependent mass harmonic oscillator (PDMHO). Classically, we use the Legendre transformation to find the Hamiltonian of the system. Next, we define two functions, and , to simplify the hamiltonian of the PDMHO. By using the Poisson algebra we find the expressions for the position and moment. At last, by using a canonical transformation we relate the equations of the PDMHO to those of the simple harmonic oscillator (SHO). Quantically, we write the Hamiltonian of the PDMHO in terms of the operators and . Next, we consider that these operators satisfy the same algebra that those of the SHO. By assuming that both the classical and quantum PDMHO have the same form, we are able to find a simple form for the PDMHO Hamiltonian. Finally, by transforming the Schrödinger equation (SE) of the PDMHO into that of the SHO, we can write the wave function of the PDMHO in terms of that of the SHO. We will study two time-dependent systems, namely and , we observe that as , they tend to a simple harmonic oscillator. For each system we find the position and momentum (classical study), as well as the wave-function (quantum study). For both systems we analyze the the position e momentum uncertainty, the product uncertainty, the fisher information and Shannon entropy, for the ground state, as a function of the parameter.