On a Parallelism between Classical Mechanics and Quantum Mechanics

概要

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Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound systems in a rather general form by introducing reasonably "quantum-mechanical momentum" from the Schrodinger equation. The following are proved: The "quantum-mechanical momentum" which is nearly equal to the corresponding classical momentum in "classical region" can be defined in the whole range of validity of the coordinate for the bound systems. In the same way as in classical mechanics, in quantum mechanics one can construct an associated wave accompanying a motion of a quantum particle and obeying the Schrodinger equation. The uniqueness condition on the associated wave throughout an orbit leads to the boundary condition on the wave function which is composed of the associated wave and to "quantum-mechanical quantization condition", equivalently. In addition, all these quantum-mechanical quantities are approximately equal to the corresponding classical ones in the "classical region" of the coordinate. In particular it is demonstrated that even for a bound system of a singular (central) potential exists the ground state with the lowest energy eigenvalue. It is proved in one dimension that to any physical system obeying the Schrodinger equation always exists the corresponding classical system.
理論物理学刊行会の論文
1965-08-25