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Journal of Wuhan Institute of Technology

Quantum mechanics analysis of eigenstate in original EPR paradox

Tan Yuan-he

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The eigenstate existing in the problem of EPR paradox is introduced.The commutation of related operators is analyzed and the wave-function in coordinate representation of the eigenstate is verified and its physical signification is discussed.The eigenstate is expended in the direct product space of the moment eigenstates of a two-particle system,subsequently the wave-function in momentum representation is obtained.The relation of the form of the wave-function in momentum representation and the selection of the origin of coordinates,including its physical meaning was analyzed.Results showed that the eigenstate was a entangled state of the coordinate and momentum.

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•Feb 1, 2003•0 citations
The eigenstate existing in the problem of EPR paradox is introduced.The commutation of related operators is analyzed and the wave-function in coordinate representation of the eigenstate is verified and its physical signification is discussed.The eigenstate is expended in the direct product space of the moment eigenstates of a two-particle system,subsequently the wave-function in momentum representation is obtained.The relation of the form of the wave-function in momentum representation and the selection of the origin of coordinates,including its physical meaning was analyzed.Results showed that the eigenstate was a entangled state of the coordinate and momentum.
The Einstein–Podolsky–Rosen (EPR) wavefunction is a two-particle wavefunction that is a simultaneous eigenfunction of the relative coordinate and the centre-of-mass momentum. The Fock space representation of this function is easily obtained as a direct product of an eigenfunction of the relative coordinate and an eigenfunction of the centre-of-mass momentum. Straightforward generalization to an arbitrary number of particles is shown to be feasible, and a classification of multiparticle entangled states is outlined. The relation between entanglement and inseparability is pointed out, and the status of the EPR states within the much broader family of entangled states is emphasized.
The EPR paradox is shown to be a result of the wrong definitions in quantum mechanics of such physical quantities as momentum and position. The corrected definitions show that momentum and position of particles can be precisely defined simultaneously and uncertainty relations do not prevent it. Therefore, with correct definition of physical quantities in quantum mechanics, the paradox disappears. The Bohm Aharonov version of the EPR paradox is illustrated by the example of measuring the polarization of photon pairs. The common view of the two photon state radiated by a source is presented. The common Bell’s inequality is discussed and the simplest version of it for a specific hidden variable is derived. A possibility of an experimental study of whether a photon and its polarization are preexistent before measurements is considered.
It is shown that there is one purely deterministic outcome when measurement is made on the state function chosen by EPR to describe the combined two-particle system – the distance between the two particles is preserved the same. Further, it is shown that, surprisingly, the Ψ-function designed according to QM leads to the following paradox – despite the fact that the two particles move in opposite directions, in time the distance between them becomes shorter and shorter. As is known, Einstein, Podolsky and Rosen (EPR) [1] observe two particles which have interacted in the past (for 0 Tt< <) but the interaction between them has ceased for times Tt>. Nevertheless, both particles, even at times Tt> EPR consider to be described by one common state-function, namely 1 2 ( ,)x xΨ. The authors have chosen to explore one particular form of that state-function, namely ( ) 1 2 01 2 (), i x x x pdpex x − −Ψ = ∫ =. (1) This function may be rewritten as ( ) 2 0 11 2 (), i ix x p x pdpe ex x − +Ψ = ∫ = = (2) and then 1 i x p e = can be considered as the orthonormal basis of an observable A pertaining to particle 1 (position of particle 1) while the exponents