Abstract
The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states.
Highlights
The superposition principle of quantum states, identified with the wave functions introduced by Schrödinger [1], is the fundamental property of quantum systems
The foundation of quantum mechanics and the role of superposition principle were discussed for quantum states [3,4], which are associated with the density matrices of density operators acting in the Hilbert space, introduced in [5,6]
The aim of this paper is to demonstrate how such a phenomenon as interference of quantum states, described by the superposition principle of complex wave functions, is considered in the probability representation of quantum mechanics [10,11,12,13,14,15,16,17,18,19,20,21]
Summary
The superposition principle of quantum states, identified with the wave functions introduced by Schrödinger [1], is the fundamental property of quantum systems. The aim of this paper is to demonstrate how such a phenomenon as interference of quantum states, described by the superposition principle of complex wave functions, is considered in the probability representation of quantum mechanics [10,11,12,13,14,15,16,17,18,19,20,21]. Since the density operators ρψ1 =| ψ1 ψ1 | and ρψ2 =| ψ2 ψ2 | provide the expression for the density operator of superposition state | ψ ψ |, where vector | ψ is a linear combination of vectors | ψ1 and | ψ2 , the superposition principle can be formulated as an addition rule of the probabilities (quantum-state tomograms).
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