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Testing de Broglie’s Double Solution in the Mesoscopic Regime

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Abstract

We present here solutions of a non-linear Schrödinger equation in presence of an arbitrary linear external potential. The non-linearity expresses a self-focusing interaction. These solutions are the product of the pilot wave with peaked solitons the velocity of which obeys the guidance equation derived by Louis de Broglie in 1926. The degree of validity of our approximations increases when the size of the soliton decreases and becomes negligible compared to the typical size over which the pilot wave varies. We discuss the possibility to reveal their existence by implementing a humpty-dumpty Stern-Gerlach interferometer in the mesoscopic regime.

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Notes

  1. It is worth noting that, although the potential chosen here presents some analogy with previous proposals aimed at fulfilling de Broglie’s double solution program [10, 11], it differs from them by the fact that it is undissociable from the factorisation ansatz (2). This ansatz has been studied in the past, in the framework of the linear Schrödinger equation actually, by Barut [12] and Bindel [13] for instance, but, to the knowledge of the author, the system of coupled Eqs. (1, 2, 11) has never been considered in the past.

  2. Note that, to simplify the mathematical treatment we assumed here to begin with that the three cartesian components of the barycentre of the pilot wave oscillate in phase. Now, as the Eq. (19) is separable in cartesian coordinates, our results are still valid if we relax this assumption.

  3. The main difference between the present approach and “conventional” semi-classical gravity à la Schrödinger-Newton is that here the source term is supposed to be proportional to \(\mid \Psi \mid ^2\) and not to \(\mid \Psi _L\mid ^2\) as is the case in semi-classical gravity. Our approach however is reminiscent of Penrose’s ideas [16, 35, 36] according to which a fundamental non-linearity explains the collapse of the wave function, excepted that here the non-linearity is not of gravitational origin, and that the collapse of each elementary particle is permanent and occurs over regions of size of the order of the Compton wavelength.

  4. In first approximation, the self-gravitational potential can be estimated making use of \(V^\mathrm {eff}(d)=-Gm^2({3R^2-d^2\over 2R^3})\), which expresses the gravitational potential inside a homogeneous nanosphere of radius R.

  5. Actually, a single humpty dumpty device suffices to falsify our model, because the standard theory predicts that no dephasing appears between the two branches of the humpty dumpty device when only one nanosphere is present. This is not the case with our approach, which predicts a dephasing equal to \(\pm \tau {Gm^2\over \hbar } ({3\over 2R}-{1\over d})\) with d the distance between the spin up and spin down paths and R the radius of the nanosphere. The value of the sign of the dephasing depends on where the (mass/energy of the) nanosphere is located. The localisation of the mass/energy of the nanosphere obeys in turn the Born rule, in virtue of the quantum H-theorem.

  6. For what concerns the criticism according to which a deterministic equation cannot mimic the indeterministic nature of the measurement process, it is nullified by Valentini’s H-theorem. Deterministic chaos leads to indeterminism as is well-known [27,28,29].

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  1. CLARTE, Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, Campus de St-Jérôme, 13013, Marseille, France

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Durt, T. Testing de Broglie’s Double Solution in the Mesoscopic Regime. Found Phys 53, 2 (2023). https://doi.org/10.1007/s10701-022-00626-1

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