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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
Metin Demiralp

Last modified: 2020-01-28


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.