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arXiv:1105.3800

Crtierion of effective centre-of-mass method in Quantum Mechanics

Bo‐Yuan Ning, Jun Zhuang, Xi‐Jing Ning

Published May 19, 2011• 0 citations
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Abstract

In describing the motion of atoms and clusters, we face with choosing quantum mechanics or classical mechanics under different conditions. In principle, there exist two criteria for this choice, but they do contradict in some cases though they are in agreement for other cases. Actually, this problem is closely related with the effective centre-of-mass method, the underlying application of quantum mechanics. It is shown that quantum mechanics must be selected for particle's motion when the de Broglie wave length of the mass centre is larger than the particle size, and in such case the effective centre-of-mass can be used in Quantum Mechanics. In order to test this conclusion, an easy-manufactured experiment is suggested.

DOI
10.48550/arxiv.1105.3800

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In describing the motion of atoms and clusters, we face with choosing quantum mechanics or classical mechanics under different conditions. In principle, there exist two criteria for this choice, but they do contradict in some cases though they are in agreement for other cases. Actually, this problem is closely related with the effective centre-of-mass method, the underlying application of quantum mechanics. It is shown that quantum mechanics must be selected for particle's motion when the de Broglie wave length of the mass centre is larger than the particle size, and in such case the effective centre-of-mass can be used in Quantum Mechanics. In order to test this conclusion, an easy-manufactured experiment is suggested.
We investigate the implication of the nonlinear and non-local multi-particle Schrödinger–Newton equation for the motion of the mass centre of an extended multi-particle object, giving self-contained and comprehensible derivations. In particular, we discuss two opposite limiting cases. In the first case, the width of the centre-of-mass wave packet is assumed much larger than the actual extent of the object, in the second case it is assumed much smaller. Both cases result in nonlinear deviations from ordinary free Schrödinger evolution for the centre of mass. On a general conceptual level we include some discussion in order to clarify the physical basis and intention for studying the Schrödinger–Newton equation.
We investigate the implication of the nonlinear and non-local multi-particle Schrodinger-Newton equation for the motion of the mass centre of an extended multi-particle object, giving self-contained and comprehensible derivations. In particular, we discuss two opposite limiting cases. In the first case, the width of the centre-of-mass wave packet is assumed much larger than the actual extent of the object, in the second case it is assumed much smaller. Both cases result in nonlinear deviations from ordinary free Schrodinger evolution for the centre of mass. On a general conceptual level we include some discussion in order to clarify the physical basis and intention for studying the Schrodinger-Newton equation.
From the invariance properties of the Schrodinger equation and the isotropy of space we show that a generic (non-relativistic) quantum system is endowed with an ``external'' motion, which can be interpreted as the motion of the centre of mass, and an ``internal'' one, whose presence disappears in the classical limit. The latter is caused by the spin of the particle, whatever is its actual value (different from zero). The quantum potential in the Schrodinger equation, which is responsible of the quantum effects of the system, is then completely determined from the properties of the internal motion, and its ``unusual'' properties have a simple and physical explanation in the present context. From the impossibility to fix the initial conditions relevant for the internal motion follows, finally, the need of a probabilistic interpretation of quantum mechanics.
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To clarify the discussion as to whether or not a quantum mechanical treatment of the motion of electrons in magnetic fields will lead to a formula for em\frac{e}{m}me​, differing from that obtained by classical electro-dynamics the following problem was solved. A uniform magnetic field in the zzz-direction exists in the half-space x>0x>0x>0. A plane monochromatic de Broglie wave, travelling in the positive xxx-direction representing electrons of arbitrary energy, impinges normally on the plane x=0x=0x=0. Solutions of the wave equation were found fulfilling appropriate boundary conditions at the plane x=0x=0x=0. Currents are calculated quantum mechanically and compared with the corresponding classical expressions. It was found that for electrons possessing energies of the order of magnitude used in deflection experiments, no observable deviations from classical results are predicted. Another quantum mechanical effect is diffraction at slits. Simple approximate calculations show that this effect can produce a fractional error in em\frac{e}{m}me​ of the order of the de Broglie wave-length divided by the slit width. These results are opposite to the conclusions reached by Page. We may remark that he solved a problem of "stationary states" which does not represent the actual experiments.