Three-dimensional representation of the many-body quantum state

Journal of Molecular Modeling, Sep 2018

Using the trajectory conception of state, we give a simple demonstration that the quantum state of a many-body system may be expressed as a set of states in three-dimensional space, each associated with a different particle. It follows that the many-body wavefunction may be derived from a set of waves in 3-space. Entanglement is represented in the trajectory picture by the mutual dependence of the 3-states on the trajectory labels.

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Three-dimensional representation of the many-body quantum state

Journal of Molecular Modeling September 2018, 24:269 | Cite as Three-dimensional representation of the many-body quantum state AuthorsAuthors and affiliations Peter Holland Open Access Original Paper First Online: 03 September 2018 Received: 18 July 2018 Accepted: 16 August 2018 120 Downloads Part of the following topical collections:International Conference on Systems and Processes in Physics, Chemistry and Biology (ICSPPCB-2018) in honor of Professor Pratim K. Chattaraj on his sixtieth birthday Abstract Using the trajectory conception of state, we give a simple demonstration that the quantum state of a many-body system may be expressed as a set of states in three-dimensional space, each associated with a different particle. It follows that the many-body wavefunction may be derived from a set of waves in 3-space. Entanglement is represented in the trajectory picture by the mutual dependence of the 3-states on the trajectory labels. KeywordsQuantum state Many-body system Spacetime trajectory Entanglement Identical particles  This paper belongs to Topical Collection International Conference on Systems and Processes in Physics, Chemistry and Biology (ICSPPCB-2018) in honor of Professor Pratim K. Chattaraj on his sixtieth birthday Motivation for the spatial trajectory conception of the quantum state A curious dichotomy between theory and practice pervades the history of quantum mechanics. On the one hand, the theory is supposed to be about ‘measurements’, procedures whose outcomes are the eigenvalues of self-adjoint Hilbert space operators that represent the observables ‘measured’. The role of the ‘state’ of a physical system, a vector ψ(x) (in the position representation) in the Hilbert space, is to encode the probabilities of the empirical outcomes. The ψ conception of state has been adopted almost universally since the advent of quantum theory. Both the formalism and the debates over the theory’s meaning are routinely couched in its terms, including by those who seek to discern causal mechanisms underlying the statistical ψ calculus. On the other hand, in real laboratories rather than in theoreticians’ heads, measurements are about the determination of position—of a meter pointer, of a symbol in a printout, of an oscilloscope track, ... The statistical regularities predicted by the theory are tested, in the end, by sequences of individual position experiments (amplified to the macroscopic level). Empirical physical assertions about a quantum system are either about or are inferences from the measurement of position. When we ‘measure spin’ we infer that quantity from the discrete spatial domains impacted by a beam of identically prepared systems on a detecting screen, with the cumulative density of the successive impacts indicating the probability distribution. For all the talk of operators, Hilbert space and entanglement, we have to map our abstract multidimensional theoretical analysis into assertions about the (likely) locations of moving objects in three-dimensional physical space; that is, in the first instance, into statements about three-dimensional trajectories. The following question therefore presents itself: if our direct connection with the ‘quantum world’ is through the time-varying positions of objects in physical 3-space, in which the objects may legitimately be regarded as part of an ecumenical quantum description even if they comprise macroscopic components, why is the theory not formulated directly in these terms, that is, why is the quantum state not a time-dependent position variable rather than merely a time-dependent encoder of the statistics of position? To couch the theory directly in terms of experimental outcomes would, after all, chime with the instrumentalist views that have dominated most quantal discourse. Of course, these are contentious issues. But it turns out that the basic problem that emerges from these considerations—that of representing the quantum state using position as the state variable—has a simple and apparently uncontentious solution [1] (see [2] for a recent account and further references, and see [3] for a discussion setting the theory in a wider conceptual and historical context). In fact, the model we propose accounts for more than just empirical variables; it provides an alternative conception of the quantum state in general processes, measurements or otherwise. Moreover, the two state pictures, the wavefunction and the trajectory, are not in conflict; they stand in a harmonious complementary relation of codetermination. The wavefunction formulation describes temporal changes in the system’s state at each space point (analogous to the Eulerian picture in fluid mechanics), and the trajectory formulation describes the transport of the system’s state across space (analogous to the fluidical Lagrangian picture). In particular, the paths are conveyors of constant parcels of probability. This extension of the notion of state (...truncated)


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Peter Holland. Three-dimensional representation of the many-body quantum state, Journal of Molecular Modeling, 2018, pp. 269, Volume 24, Issue 9, DOI: 10.1007/s00894-018-3804-7