On a Dynamical Treatment of a Quantum

Yoshio MURATA

Published Mar 31, 1969• 0 citations

Abstract

A dynamical treatment of a quantum in a potential is proposed by using the formulation of quantum mechanics with classical pictures and defining the quantum mechanical (abbreviated to q. m.) principal function. It is shown in the case of Coulomb potential that the q. m. momentum with some parameters can be uniquely determined and the q. m. principal function can be obtained in principle. On this occasion a new dynamical postulate is introduced. In addition to the above, it is qualitatively demonstrated that the q. m. quantities in the classical region are approximately equal to the classical ones and are larger than those in the non-classical region. Accordingly we may consider the effect of the so-called quantum potential to be small in comparison with that of the usual potential.

Related papers

量子の動力学的取扱いについて

良夫村田, Yoshio MURATA

Mar 31, 19690 citations

The Relation between Classical Mechanics and Quantum Mechanics

Yoshio MURATA

By using the formulation of quantum mechanics with classical pictures, the motion of a quantum is analysed for linear harmonic oscillator. On this occasion it is postulated that the quantum mechanical momentum of the quantum is approximately equal to the corresponding classical mechanical momentum in the region of validity of the coordinates. According to this analysis it is shown that the quantum mechanical motion well corresponds to the classical motion, and that the wave equation determines the associated wave along the whole orbit of the quantum in the configuration space and the uniqueness of the wave throughout the orbit plays an important role in quantum mechanics. In addition to the above, an attempt is made to make clear the well-known fact that classical mechanics with the quantization gives approximately the same results as quantum mechanics.

Mar 31, 19670 citations

On a Parallelism between Classical Mechanics and Quantum Mechanics

Yoshio Nishiyama

Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound system in a rather general form by introducing reasonably “quantum-mechanical momentum” from the Schrödinger 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 Schrödinger 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 Schrödinger equation always exists the corresponding classical system.

Progress of Theoretical PhysicsAug 1, 19650 citations

Proposal for the Quantum Mechanical Orbit Equation

Yoshio Nishiyama

The orbit equation for the motion of a quantum in a potential is proposed by defining uniquely the quantum mechanical momentum from the Schrodinger wave equation of a one-particle system and using the Hamilton-Jacobi formalism. On this occasion it is necessary to introduce a new dynamical postulate, “initial condition”, which is independent of the quantization condition as well as of the Schrodinger wave equation. The momentum is determined in such a way that it is approximately equal to the corresponding classical mechanical momentum in the classical region of validity of the coordinates. The momentum in the non-classical region is in general very small in comparison with that in the classical region. The quantum-mechanical motion well corresponds to the classical motion. If a system is macroscopic, the motion of the particle is described by Newtonian mechanics and the quantization condition is of no validity. The “quantization of magnetic flux” derives from the quantization condition. It is closely related to the energy level structure of the atomic system. The above argument is applied to the scalar electromagnetic and sound wave equations to give the dynamical motion of a photon and a phonon without recourse to the so-called second quantization method.

Progress of Theoretical PhysicsSep 1, 19650 citations

Schroedinger and Hamilton-Jacobi equations

M. V. Lokajíček

Time-dependent Schroedinger equation represents the basis of any quantum-theoretical approach. The question concerning its proper content in comparison to the classical physics has not been, however, fully answered until now. It will be shown that there is one-to-one physical correspondence between basic solutions (represented always by one Hamiltonian eigenfunction only) and classical ones, as the non-zero quantum potential has not any physical sense, representing only the "numerical" difference between Hamilton principal function and the phase of corresponding wave function in the case of non-inertial motion. Possible interpretation of superposition solutions will be then discussed in the light of this fact. And also different interpretation alternatives of the quantum-mechanical model will be newly analyzed and new attitude to them will be reasoned.

arXiv:quant-ph/0611176Nov 16, 20060 citations

On the Formulation of Quantum Mechanics associated with Classical Pictures

Takehiko Takabayasi

By re-expressing the generalized Schrodinger equation in its coordinate representation in terms of amplitude and phase of the state vector, a quantum-mechanical change of a system can be made to correspond to an ensemble of classical motions of the system to which is added some internal potential. This ensemble may be taken as the statistical. This enables an alternative formulation of quantum mechanics with which is associated that classical picture. Such a formulation leads, however, if developed straightforwardly according to the picture, to a theoretical scheme quite different from that of ordinary quantum mechanics concerning the problems other than the equation of motion. Furthermore, this formulation is not applicable to Fermions when spin, or exclusion principle is taken into account. Bohm's recent renewed form of the statistical interpretation of quantum mechanics referring to this formulation is criticized in several points. On this formulation we can construct a singular ensemble which is shown to correspond to the transformation kernel of the wave function to infinitesimally later instant, and thus a connection between this formulation and that of Feynman's is taken out. Hydrodynamical picture formally equivalent to the statistical picture in one-body problems is also considered, and from such analogy certain formal generalization of Schrodinger equation is suggested.

Progress of Theoretical PhysicsAug 1, 1952340 citations