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The Relation between Classical Mechanics and Quantum Mechanics

Yoshio MURATA

Published Mar 31, 1967• 0 citations
OpenAlex

Abstract

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.

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量子の動力学的取扱いについて

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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.

•Mar 31, 1969•0 citations
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.
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.
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.
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.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
In this paper the relationship of classical physics and quantum physics is studied by introducing a partial differential equation, which describes classical mechanics, but looks very similar to the Schrodinger wave equation of quantum mechanics. This work is largely based on David Bohm's causal interpretation of quantum mechanics, but is in some sense complementary to it. In Bohm's theory the Schrodinger wave equation is used to derive classical looking equations of motion for quantum physics. Here exactly the opposite is done. The equations of classical physics are put into a form resembling the Schrodinger equation of quantum physics.
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.