5 1 2 D O C . 5 1 6 S C H R Ö D I N G E R S W A V E M E C H A N I C S 516. “Does Schrödinger’s Wave Mechanics Completely Determine the Motion of a System, or Only Statistically?”[1] [Berlin, 5 May 1927][2] It is known that currently the opinion prevails that a complete temporal-spatial description of the motion of a mechanical system according to quantum mechanics does not exist. It should supposedly make no sense, for instance, to specify the in- stantaneous configuration and the instantaneous velocities of an atom’s electrons.[3] In contrast, it will be shown in the following that Schrödinger’s wave mechanics does suggest unambiguous assignment of the system’s motions to any solution to the wave equation. Whether these assignments do justice to the facts can then be investigated by the computation of special cases. Let ψ be a solution to a Schrödinger equation belonging to a given potential en- ergy function Φ (1) ¢If a system with just one degree of freedom is involved, then ψ defines at every point of the trajectory the kineti² If ψ is given, then by (1), is defined at each configuration point, i.e., the ki- netic energy L. If a system of just one degree of freedom is involved, then L defines the velocity only ambiguously. The motion is fully defined if the condition is added that the velocity should change only continuously. For systems with several de- grees of freedom, this method fails, because the direction of the motion is not known. However, the following consideration leads to the goal. We assume that it is possible to assign unambiguously different directions to function ψ at each point of the n-dimensional configuration space n, and to decom- pose the kinetic energy into n summands, each of which is unambiguously assigned to one of those directions. One could then also assign to each of these directions a velocity in that direction corresponding to that summand. The resultant of all these velocities would then be the system’s velocity vector in configuration space. Now I would like to follow this thought through. The symbol in (1), according to Schrödinger, refers to a metric of the con- figuration space, which is characterized[4] by . (2) Then Δψ 8π2- h2 --------( E Φ)ψ + 0. = E Φ Δψ 2L gμνqμqν · · ds2 dt2 ------- - = =
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