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Modern Physics Letters A

BEHAVIOR OF CHARGED PARTICLE MOTION ALONG A MAGNETIC FIELD AND A RETARDING POTENTIAL-EVIDENCE FOR THE EXISTENCE OF DISCRETE ENERGY STATES IN CLASSICAL MECHANICAL DOMAIN

Ram K. Varma, A. M. Punithavelu

Published Jan 20, 1993• 17 citations
Publisher

Abstract

We report here experimental results which point to a rather unexpected, astonishing behavior of charged particle motion in a magnetic field in the presence of a retarding electric potential. An electron beam, when energy scanned with respect to its ‘parallel’ energy using a retarding potential, the response of the transmitted current as a function of the retarding potential is found to be oscillatory as against the monotonic response, expected according to the standard initial value paradigm of classical mechanics. The unexpected ‘peaks’ and ‘dips’ in the transmitted current plots are interpreted as “allowed” and “forbidden” energy states of the system. The peaks (or dips) are found to fit very well with a theoretical relationship which motivated the search for such a behavior; also “quantum numbers” for the peaks have been identified. A very astonishing aspect of the relationship is the dependence of the energies of the allowed states on the length of the box.

DOI
10.1142/s0217732393000179

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BEHAVIOR OF CHARGED PARTICLE MOTION IN A MAGNETIC FIELD AND A RETARDING POTENTIAL II – OBSERVATIONS OF THE EXISTENCE OF "FORBIDDEN STATES" AND AHARANOV-BOHM ANALOG IN CLASSICAL MECHANICAL DOMAIN

Ram K. Varma, A. M. Punithavelu

The observations made earlier1 on the existence of discrete 'forbidden' (and allowed) states in a classical mechanical system of charged particle motion in a magnetic field are extended to include the region where the applied retarding potential exceeds the potential equivalent of the electron energy. The electron current flowing to the ground from the anode of the electron gun is found to exhibit strong dips as the potential on the grid of the detector system kept at a distance along the magnetic field, is swept from a large negative value to zero. The observed anode current 'dips' are quite enigmatic since the electrons under the conditions (grid potential far exceeding the cathode potential) are unable to reach the grid, yet they respond to the grid potential changes in quite an unexpected manner. This observation is thus reminiscent of the Aharanov-Bohm effect in quantum mechanics. The dips are moreover found to fit a relation, a modified form of that given in the earlier reported results.

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We describe an experiment to test a quantum-like theory which predicts quantum-like behavior for an ensemble of electrons in a classical configuration with static magnetic and electric fields. Some of the earlier experiments had supporting evidence for anomalous, quantum-like effects in such a situation showing systematic modulations of electron current when a retarding potential is varied, even though the quantum wavelength of the electrons in such a configuration was less than a billionth of the spatial width of the potential well. Our experiment conclusively rules out any nonclassical, quantum-like behavior in electron transmission through simple electric barriers, when magnetic fields are present. We identify secondary electrons generated at various electrodes as the main source of apparent anomalous behavior. We also present a classical derivation of the quantum-like equation describing the modulations.

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TOPOLOGICAL MANIFESTATIONS IN CLASSICAL MECHANICS: DISCRETE ALLOWED AND FORBIDDEN STATES OF MOTION

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Consequences of the topology of the configuration space of a Hamiltonian dynamical system are considered for a coherent system of trajectories. It is shown that when the space is multiply-connected and therefore the action integral is multivalued, the allowed states of motion (labeled by the initial data) are constrained to a discrete set by the requirement that the action be single-valued. One thus obtains a quantum-like discretization of allowed states of motion even in classical mechanics. Such discrete “allowed” and “forbidden” states have indeed been observed in the classical mechanical system of charged particles in a magnetic field. The relationship of this formalism with a Schrödinger-like formalism for the latter problem given earlier is discussed.

Modern Physics Letters A•Dec 21, 1994•5 citations

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Double-wave function quantum theory is applied to describe the motion of charged particle in both uniform magnetic field and three-dimensional harmonic oscillat or potential field.Quantum results and classical limit results are derived respe ctively.A comparison between the classical limit results and those of classical mechanics is made.

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On the behaviour of the solutions of certain Schredinger equations for vanishing potentials

Tovie Leon Rome

In studying the diamagnetism of free electrons in a uniform magnetic field it was found that reducing the field to zero in the wavefunction did not yield the experimentally indicated free particle plane wave wavefunction. However, solving the Schroedinger Equation resulting from setting the field equal to zero in the original equation did yield a plane wave wavefunction. This paradox was not found to be peculiar to the case of a charged particle in a uniform magnetic field but was found to occur in a number of other systems. In order to gain an understanding of this unexpected behavior, the following systems were analyzed: the one-dimensional square well potential; a charged, spinless particle in a Coulomb field and in a uniform electric field; a one-dimensional harmonic oscillator; and a charged, spinless particle in a uniform magnetic field. From these studies the following were obtained: conditions for determining the result of reducing the potential in a wavefunction; the condition under which the potential of a system may be switched off while maintaining the energy of the system constant; the relationship between the result of physically switching off a potential, the result of reducing it in the wavefunction, and the solution of the Schroedinger Equation obtained by decreasing the potential to zero in the original wave equation; and a general property of any wavefunction with respect to reducing any parameter within this wavefunction.

•Jan 1, 2012•0 citations
We report here experimental results which point to a rather unexpected, astonishing behavior of charged particle motion in a magnetic field in the presence of a retarding electric potential. An electron beam, when energy scanned with respect to its ‘parallel’ energy using a retarding potential, the response of the transmitted current as a function of the retarding potential is found to be oscillatory as against the monotonic response, expected according to the standard initial value paradigm of classical mechanics. The unexpected ‘peaks’ and ‘dips’ in the transmitted current plots are interpreted as “allowed” and “forbidden” energy states of the system. The peaks (or dips) are found to fit very well with a theoretical relationship which motivated the search for such a behavior; also “quantum numbers” for the peaks have been identified. A very astonishing aspect of the relationship is the dependence of the energies of the allowed states on the length of the box.
The observations made earlier1 on the existence of discrete 'forbidden' (and allowed) states in a classical mechanical system of charged particle motion in a magnetic field are extended to include the region where the applied retarding potential exceeds the potential equivalent of the electron energy. The electron current flowing to the ground from the anode of the electron gun is found to exhibit strong dips as the potential on the grid of the detector system kept at a distance along the magnetic field, is swept from a large negative value to zero. The observed anode current 'dips' are quite enigmatic since the electrons under the conditions (grid potential far exceeding the cathode potential) are unable to reach the grid, yet they respond to the grid potential changes in quite an unexpected manner. This observation is thus reminiscent of the Aharanov-Bohm effect in quantum mechanics. The dips are moreover found to fit a relation, a modified form of that given in the earlier reported results.
We describe an experiment to test a quantum-like theory which predicts quantum-like behavior for an ensemble of electrons in a classical configuration with static magnetic and electric fields. Some of the earlier experiments had supporting evidence for anomalous, quantum-like effects in such a situation showing systematic modulations of electron current when a retarding potential is varied, even though the quantum wavelength of the electrons in such a configuration was less than a billionth of the spatial width of the potential well. Our experiment conclusively rules out any nonclassical, quantum-like behavior in electron transmission through simple electric barriers, when magnetic fields are present. We identify secondary electrons generated at various electrodes as the main source of apparent anomalous behavior. We also present a classical derivation of the quantum-like equation describing the modulations.
Consequences of the topology of the configuration space of a Hamiltonian dynamical system are considered for a coherent system of trajectories. It is shown that when the space is multiply-connected and therefore the action integral is multivalued, the allowed states of motion (labeled by the initial data) are constrained to a discrete set by the requirement that the action be single-valued. One thus obtains a quantum-like discretization of allowed states of motion even in classical mechanics. Such discrete “allowed” and “forbidden” states have indeed been observed in the classical mechanical system of charged particles in a magnetic field. The relationship of this formalism with a Schrödinger-like formalism for the latter problem given earlier is discussed.
In studying the diamagnetism of free electrons in a uniform magnetic field it was found that reducing the field to zero in the wavefunction did not yield the experimentally indicated free particle plane wave wavefunction. However, solving the Schroedinger Equation resulting from setting the field equal to zero in the original equation did yield a plane wave wavefunction. This paradox was not found to be peculiar to the case of a charged particle in a uniform magnetic field but was found to occur in a number of other systems. In order to gain an understanding of this unexpected behavior, the following systems were analyzed: the one-dimensional square well potential; a charged, spinless particle in a Coulomb field and in a uniform electric field; a one-dimensional harmonic oscillator; and a charged, spinless particle in a uniform magnetic field. From these studies the following were obtained: conditions for determining the result of reducing the potential in a wavefunction; the condition under which the potential of a system may be switched off while maintaining the energy of the system constant; the relationship between the result of physically switching off a potential, the result of reducing it in the wavefunction, and the solution of the Schroedinger Equation obtained by decreasing the potential to zero in the original wave equation; and a general property of any wavefunction with respect to reducing any parameter within this wavefunction.