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Expectation Value Based Phase Space Trajectories of Autonomous Quantum Systems For Initial Wave Functions Composed of a Finite Number of Eigenstates

Last modified: 2020-01-28

#### Abstract

This work considers the phase space conceptuality and its

quantum counterpart by using fluctuationlessness limit and

fluctuation corrections. We confine ourselves to the cases

where the motion of the system under consideration is started

by an initial wave function which is composed of certain

number of Hamiltonian eigenfunctions as a linear combination

of these eigenfunctions with certain constant linear combination

coefficients. At the first attempt we prefer to use an important

theorem of Mathematical Fluctuation Theory. This theorem, which

may be called "Fluctuationlessness Theorem" and has a proof,

states that the expectation value of a function of certain

operators is equal to the image of the expectation values of

the argument operators under this very same function at the

absence of the all fluctuations. This approximation enables us

to define a planar phase space whose coordinates are the

expectation values of the momentum and position operators. We have constructed sufficiently many expectation value equations and

then we have obtained the trajectories of the system in this planar

phase space, which is analogous to its classical mechanics

counterpart, by solving these equations. Cosinusoidal temporal

behaviors in these equations, cause periodical motions for the

system point in phase space on certain multinomial curves. Planar,

two dimensional, phase space can be extended to higher dimensional spaces by introducing certain fluctuations such that in the case of infinite number of discrete state eigenfuctions this extended space becomes denumerably infinite dimensional. Certain illustrations will be presented during the relevant session of MS1 for rather simple systems like Hydrogen Atom and Harmonic Oscillator.

quantum counterpart by using fluctuationlessness limit and

fluctuation corrections. We confine ourselves to the cases

where the motion of the system under consideration is started

by an initial wave function which is composed of certain

number of Hamiltonian eigenfunctions as a linear combination

of these eigenfunctions with certain constant linear combination

coefficients. At the first attempt we prefer to use an important

theorem of Mathematical Fluctuation Theory. This theorem, which

may be called "Fluctuationlessness Theorem" and has a proof,

states that the expectation value of a function of certain

operators is equal to the image of the expectation values of

the argument operators under this very same function at the

absence of the all fluctuations. This approximation enables us

to define a planar phase space whose coordinates are the

expectation values of the momentum and position operators. We have constructed sufficiently many expectation value equations and

then we have obtained the trajectories of the system in this planar

phase space, which is analogous to its classical mechanics

counterpart, by solving these equations. Cosinusoidal temporal

behaviors in these equations, cause periodical motions for the

system point in phase space on certain multinomial curves. Planar,

two dimensional, phase space can be extended to higher dimensional spaces by introducing certain fluctuations such that in the case of infinite number of discrete state eigenfuctions this extended space becomes denumerably infinite dimensional. Certain illustrations will be presented during the relevant session of MS1 for rather simple systems like Hydrogen Atom and Harmonic Oscillator.