Abstract
The probability representation of quantum mechanics where the system states are identified with fair probability distributions is reviewed for systems with continuous variables (the example of the oscillator) and discrete variables (the example of the qubit). The relation for the evolution of the probability distributions which determine quantum states with the Feynman path integral is found. The time-dependent phase of the wave function is related to the time-dependent probability distribution which determines the density matrix. The formal classical-like random variables associated with quantum observables for qubit systems are considered, and the connection of the statistics of the quantum observables with the classical statistics of the random variables is discussed.
Highlights
The goal of this work is to discuss some aspects of the new formulation of quantum mechanics where system states are described by the probability distributions
We reviewed the probability representation of quantum states and considered the application of this approach on the examples of harmonic oscillator states and qubit states
In the case of systems with continuous variables, like the harmonic oscillator, we obtained the relation of the probability representation of quantum states with the path integral method
Summary
The goal of this work is to discuss some aspects of the new formulation of quantum mechanics where system states are described by the probability distributions. The evolution of the quantum states described in the conventional formulation of quantum mechanics by the Schrödinger equation for the wave function or by the von Neumann equation for the density operator, as well as by the Gorini–Kossakowskii–Sudarshan–Lindblad (GKSL) equation [9,10], is described by the kinetic equations for the probability distributions identified with the quantum states. For the harmonic oscillator state, the symplectic tomographic probability distribution can be introduced using the fractional Fourier transform of the wave function y|ψ = ψ(y). For the harmonic oscillator, we derive the integral expression for the Green function (propagator), which describes the evolution of symplectic tomographic probability distribution as follows:.
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