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A Statistical Derivation of Non-Relativistic Quantum Theory

U. Klein

Published Feb 22, 2012• 2 citations
Publisher

Abstract

A previous derivation of the single-particle Schr\"odinger equation from statistical assumptions is generalized to an arbitrary number $N$ of particles moving in three-dimensional space. Spin and gauge fields are also taken into account. It is found that the same statistical assumptions that imply Schr\"odinger's equation determine also the form of the gauge coupling terms, and the form of the corresponding local (Lorentz) forces. An explanation for the role of the electrodynamic potentials, as statistical representatives of the Lorentz force, is given. For a single particle, spin one-half is introduced as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, using the twofold number of variables, leads to Pauli's equation. The new spin term is again the statistical representative of the corresponding local force. For a $N$-particle system, spin is introduced as a consequence of gauge coupling.The classical limit $\hbar \to 0$ of Schr\"odinger's equation and closely related questions of interpretation of the quantum mechanical formalism are discussed.

DOI
10.5772/33075

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Progress of Theoretical Physics•Aug 1, 1952•340 citations
A previous derivation of the single-particle Schr"odinger equation from statistical assumptions is generalized to an arbitrary number NNN of particles moving in three-dimensional space. Spin and gauge fields are also taken into account. It is found that the same statistical assumptions that imply Schr"odinger's equation determine also the form of the gauge coupling terms, and the form of the corresponding local (Lorentz) forces. An explanation for the role of the electrodynamic potentials, as statistical representatives of the Lorentz force, is given. For a single particle, spin one-half is introduced as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, using the twofold number of variables, leads to Pauli's equation. The new spin term is again the statistical representative of the corresponding local force. For a NNN-particle system, spin is introduced as a consequence of gauge coupling.The classical limit ℏ→0\hbar \to 0ℏ→0 of Schr"odinger's equation and closely related questions of interpretation of the quantum mechanical formalism are discussed.
A previous derivation of the single-particle Schrödinger equation from statistical assumptions is generalized to an arbitrary number NNN of particles moving in three-dimensional space. Spin and gauge fields are also taken into account. It is found that the same statistical assumptions that imply Schrödinger's equation determine also the form of the gauge coupling terms, and the form of the corresponding local (Lorentz) forces. An explanation for the role of the electrodynamic potentials, as statistical representatives of the Lorentz force, is given. For a single particle, spin one-half is introduced as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, using the twofold number of variables, leads to Pauli's equation. The new spin term is again the statistical representative of the corresponding local force. For a NNN-particle system, spin is introduced as a consequence of gauge coupling.The classical limit ℏ→0\hbar \to 0ℏ→0 of Schrödinger's equation and closely related questions of interpretation of the quantum mechanical formalism are discussed.
A previous one-dimensional derivation of Schrödinger's equation from statistical assumptions is generalized to three spatial dimensions, gauge fields, and spin. It is found that the same statistical assumptions that imply Schrödinger's equation determine also the form of the gauge coupling terms, and the form of the corresponding local (Lorentz) forces. An explanation for the role of the electrodynamic potentials, as statistical representatives of the Lorentz force, is given. Spin one-half is introduced as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, using the twofold number of variables, leads to Pauli's equation. The new spin term is again the statistical representative of the corresponding local force. The classical limit ℏ→0\hbar \to 0ℏ→0 of Schrödinger's equation and closely related questions of interpretation of the quantum mechanical formalism are discussed.
In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrödinger field theory, obtaining the Schrödinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrödinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.
In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrodinger field theory, obtaining the Schrodinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrodinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.
An Ising-type classical statistical model is shown to describe quantum fermions. For a suitable time-evolution law for the probability distribution of the Ising-spins our model describes a quantum field theory for Dirac spinors in external electromagnetic fields, corresponding to a mean field approximation to quantum electrodynamics. All quantum features for the motion of an arbitrary number of electrons and positrons, including the characteristic interference effects for two-fermion states, are described by the classical statistical model. For one-particle states in the non-relativistic approximation we derive the Schrödinger equation for a particle in a potential from the time evolution law for the probability distribution of the Ising-spins. Thus all characteristic quantum features, as interference in a double slit experiment, tunneling or discrete energy levels for stationary states, are derived from a classical statistical ensemble. Concerning the particle-wave-duality of quantum mechanics, the discreteness of particles is traced back to the discreteness of occupation numbers or Ising-spins, while the continuity of the wave function reflects the continuity of the probability distribution for the Ising-spins.
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.