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量子論におけるレトロディクションII

K.-I. Izawa, T. Taniguchi

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

We continue to investigate retrodictive aspects of quantum theory based on the previous paper numbered I. We now proceed to consider quantum mechanics of a single interacting particle in one-dimensional space. Under successive mearsurements of particle position and momentum as a wave packet, we argue that a piecewise approximation to the particle motion, when applicable, results in the classical retrodiction just as it results in the classical prediction.

Related papers

量子論におけるレトロディクション I

K.-I. Izawa, T. Taniguchi

We consider retrodictive aspects of quantum theory. As a first step, we adopt quantum mechanics of a single free particle in one-dimensional space. Under successive mearsurements of particle position and momentum as a wave packet, we derive estimation on retrodictive probability, which happens to coincide with the expression for predictive probability into the past.

•Jan 1, 2017•0 citations

A simple means for deriving quantum mechanics

Eric Tesse

A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.

quant-pharXiv:2602.04524•Feb 5, 2026•0 citations

量子レトロディクションと履歴

K.-I. Izawa, T. Taniguchi

We further see the content and consistency of our claims on retrodictive aspects of quantum theory based on the previous papers, which deal with quantum mechanics of a single particle in one-dimensional space under successive mearsurements of particle position and momentum as a wave packet. The content of retrodiction can be best understood as statements on the past recorded in the present state. Conditional probabilites with the past and future states specified are consistently obtained through the ordinary predictive and/or our retrodictive probability formula in any combined way.

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Quantum mechanics

Stephen W. McKnight, Christos Zahopoulos

The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.

Cambridge University Press eBooks•Aug 10, 2015•0 citations

Quantum Mechanics of a Single Particle in One-Dimensional Space I

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CRC Press eBooks•Dec 7, 2018•0 citations

On the theory of wave packets

Dmitry V. Naumov

In this paper we discuss some aspects of the theory of wave packets. We consider a popular noncovariant Gaussian model used in various applications and show that it predicts too slow a longitudinal dispersion rate for relativistic particles. We revise this approach by considering a covariant model of Gaussian wave packets, and examine our results by inspecting a wave packet of arbitrary form. A general formula for the time dependence of the dispersion of a wave packet of arbitrary form is found. Finally, we give a transparent interpretation of the disappearance of the wave function over time due to the dispersion—a feature often considered undesirable, but which is unavoidable for wave packets. We find, starting from simple examples, proceeding with their generalizations and finally by considering the continuity equation, that the integral over time of both the flux and probability densities are asymptotically proportional to the factor 1/| x | 2 in the rest frame of the wave packet, just as in the case of an ensemble of classical particles.

arXiv:1309.1717•Physics of Particles and Nuclei Letters•Dec 1, 2013•46 citations
We continue to investigate retrodictive aspects of quantum theory based on the previous paper numbered I. We now proceed to consider quantum mechanics of a single interacting particle in one-dimensional space. Under successive mearsurements of particle position and momentum as a wave packet, we argue that a piecewise approximation to the particle motion, when applicable, results in the classical retrodiction just as it results in the classical prediction.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
We further see the content and consistency of our claims on retrodictive aspects of quantum theory based on the previous papers, which deal with quantum mechanics of a single particle in one-dimensional space under successive mearsurements of particle position and momentum as a wave packet. The content of retrodiction can be best understood as statements on the past recorded in the present state. Conditional probabilites with the past and future states specified are consistently obtained through the ordinary predictive and/or our retrodictive probability formula in any combined way.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
In this paper we discuss some aspects of the theory of wave packets. We consider a popular noncovariant Gaussian model used in various applications and show that it predicts too slow a longitudinal dispersion rate for relativistic particles. We revise this approach by considering a covariant model of Gaussian wave packets, and examine our results by inspecting a wave packet of arbitrary form. A general formula for the time dependence of the dispersion of a wave packet of arbitrary form is found. Finally, we give a transparent interpretation of the disappearance of the wave function over time due to the dispersion—a feature often considered undesirable, but which is unavoidable for wave packets. We find, starting from simple examples, proceeding with their generalizations and finally by considering the continuity equation, that the integral over time of both the flux and probability densities are asymptotically proportional to the factor 1/| x | 2 in the rest frame of the wave packet, just as in the case of an ensemble of classical particles.