Conventional Quantum Statistics with a Probability Distribution Describing Quantum System States
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nventional Quantum Statistics with a Probability Distribution Describing Quantum System States V. I. Man’koa, b, *, O. V. Man’koa, c, **, and V. N. Chernegaa, *** a
bMoscow
Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia Institute of Physics and Technology (State University), Dolgoprudny, Moscow oblast, 141701 Russia c Bauman Moscow State Technical University, Moscow, 105005 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received December 20, 2019; revised January 16, 2020; accepted January 29, 2020
Abstract—The review of a new probability representation of quantum states is presented, where the states are described by conventional probability distribution functions. The invertible map of the probability distribution onto density operators in the Hilbert space is found using the introduced operators called a quantizer– dequantizer, which specify the invertible map of operators of quantum observables onto functions and a product of the operators onto an associative product (star product) of the functions. Examples of a quantum oscillator and a spin-1/2 particle are considered. The kinetic equations for probabilities, specifying the evolution of the states of a quantum system, which are equivalent to Schrödinger and von Neumann equations, are derived explicitly. DOI: 10.1134/S1063779620040486
INTRODUCTION In quantum mechanics, quantum field theory [1], and quantum statistics, the physical system states are described by wavefunctions [2], density matrices [3, 4], as well as by vectors in a Hilbert space [5]. In classical statistical mechanics, the system states are described by probability distributions [6] in the phase space. The evolution of classical states is described by the Liouville equation for the probability density f (q, p, t ) in the phase space of the system or by the Boltzmann equation [7, 8]. The evolution of pure quantum states, identified with wavefunctions of systems with a Hamiltonian, is described by the Schrödinger equation, while the von Neumann equation describes the evolution of a density matrix of mixed states. The description of classical states by probability distribution functions and the description of quantum states by wavefunctions or by density matrices are very different. The description of quantum states intuitively requires the additional interpretation from the standpoint of equivalence with the classical picture of physical phenomena. In this connection, the other descriptions of quantum states were suggested, e.g., with the employment of the quasiprobability distributions in the phase space of systems: Wigner functions [9], Husimi–Kano functions [10, 11], and Glauber–Sudarshan functions [12, 13]. The description of states by a function in the phase
state was proposed by Blokhintsev [14]. All these quantum state descriptions deal with functions in the formal phase space of the system, but these functions are not the probability distributions. It is not possible to describe a particle state by the join
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