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Journal of Modern Physics

Time Quanta Present in Simple Quantum Systems

Stanisław Olszewski

Published Jan 1, 2021• 0 citations
Publisher

Abstract

Differences of the time periods in two independent quantum systems are examined on a semiclassical level. The systems are the electron in the hydrogen atom and a free-electron particle moving in a one-dimensional potential box, respectively. It is demonstrated that in both systems the relativistic correction to the time interval can be expressed as a multiple of the same quantum of time. The size of the quantum is proportional to the ratio of the Planck’s constant and the rest energy of the electron particle.

DOI
10.4236/jmp.2021.127062

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Journal of Modern Physics•Jan 1, 2021•0 citations
Differences of the time periods in two independent quantum systems are examined on a semiclassical level. The systems are the electron in the hydrogen atom and a free-electron particle moving in a one-dimensional potential box, respectively. It is demonstrated that in both systems the relativistic correction to the time interval can be expressed as a multiple of the same quantum of time. The size of the quantum is proportional to the ratio of the Planck’s constant and the rest energy of the electron particle.
The energy emitted by an electron in course of its transition between two quantum levels can be considered as a dissipated energy. This energy is obtained within a definite interval of time. The problem of the size of the time interval necessary for transitions is examined both on the ground of the quantum approach as well as classical electrodynamics. It is found that in fact the emission time approaches the time interval connected with acceleration of a classical velocity of the electron particle from one of its quantum levels to a neighbouring one.
Recently it was shown that classical "relativistic" particle dynamics was implicit in physics going back to Maxwell. The demonstration utilized a simple modification of a 1906 thought experiment by which Einstein established the mass equivalence of energy, independently of the relativity postulates. The modified thought experiment leads directly to correct expressions for the momentum and kinetic energy of a particle at high speed. In the present paper, one additional ingredient is added, namely, the Bohr quantization rule of the old quantum theory. It is found that a semi-classical model of a hydrogen atom and a hydrogen molecule ion exhibit, respectively, time dilation and length contraction.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.
Definitions of the mechanical parameters entering the Bohr model of the hydrogen atom allowed us to calculate the time intervals connected with the electron transitions between the nearest-neighbouring energy levels in the atom. This is done in a strictly non-probabilistic way. The time results are compared with those derived earlier on the basis of the classical Joule-Lenz law for the energy emission adapted to the case of the electron transfer in the quantum systems. A similar formalism has been next applied to the harmonic oscillator and a particle moving in the one-dimensional potential box.
The paper presents a non-probabilistic approach to the time interval associated with the energy emission produced by the electron transition in a quantum system. The calculations were performed for the hydrogen atom and the electron particle in a one-dimensional potential box. In both cases, the rule of conservation of the electron momentum has been applied. The results, limited to the time intervals of transitions between two neighbouring quantum energy levels, occur to be much similar to those obtained earlier with the aid of the Joule-Lenz energy emission theory.