The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="...The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em></span>), with the commutation relation <img src="Edit_28f5b839-7de4-41e5-9ed8-69dc1bf72c2c.bmp" alt="" />, and using a Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s like equation on these variable, and associating a linear operator to a constant of motion <em>K</em> (<em>x, v, t</em>) of the classical system, The comparison with the quantization in the space (<em>x, p</em>) is done with the usual Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s equation for the Hamiltonian <em>H</em><span style="white-space:normal;">(</span><em style="white-space:normal;">x, p, t</em><span style="white-space:normal;">)</span>, and with the commutation relation <img src="Edit_cca7e318-5b35-4c55-8f09-6089970ce9a2.bmp" alt="" />. It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization has fewer oscillations than the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, the average energy of the system is higher in (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">,展开更多
A new method to construct coherent states of a time-dependent forced harmonic oscillator was given. The close relation to the classical forced oscillator and the minimum uncertainty relation were investigated. The app...A new method to construct coherent states of a time-dependent forced harmonic oscillator was given. The close relation to the classical forced oscillator and the minimum uncertainty relation were investigated. The applied periodic force (off-resonance case), in general, will attenuate the AA phase.展开更多
The newly developed YY model contains a set of constitutive rules to describe the structures of atomic nuclei and subatomic particles, by using two elementary sub-quark particles, the Yin and Yang fermions of charge 1...The newly developed YY model contains a set of constitutive rules to describe the structures of atomic nuclei and subatomic particles, by using two elementary sub-quark particles, the Yin and Yang fermions of charge 1/3 forming all the particles of the Standard Model. This model suggests a modular structure of the universe, in which two elementary constituents recursively form all the matter. The advantage of this hypothesis is that it provides a total symmetry and a noticeably clear conceptual understanding. Moreover, it justifies the cosmological formation of a limited number of atoms, e.g., H and Li with their isotopes, considering that matter can be produced as a free agglomerate of semi-stable neutrons, which would lead to the feeding of baryonic matter in the universe. In this current article, some further theoretical aspects are proposed as an evolution of the YY model. They cover correlation paths between interacting quarks, the considerations of color forces between yin-yang elementary elements. Moreover, an agreement of the YY model with the Teplov approach based on harmonic quarks and oscillators is established, and the mass of Yin and Yang is considered. Two example nuclei are used for the analysis: a radioactively stable deuteron (containing a neutron and a proton) and a possible semi-stable dineutron (roughly “consisting of two neutrons”), which is purely theoretical, represent a very natural and legal nuclear state within YY model. Based on the results obtained here, some indications are given for a possible simple experimental verification providing proof for the stability or instability of the dineutron.展开更多
<strong>Motivation:</strong> We study the asymptotic-type dynamics of various real pointlike objects that one models by a variety of differential equations. Their response to an external force one defines ...<strong>Motivation:</strong> We study the asymptotic-type dynamics of various real pointlike objects that one models by a variety of differential equations. Their response to an external force one defines solely by the trajectory of a single point. Its velocity eventually stops changing after cessation of the external force. The response of their acceleration to the long-term external force is slow and possibly nonlinear. <strong>Objective:</strong> Our objective is to present technique for making simplified models for the long-term dynamics of pointlike objects whose motion interacts with the surroundings. In the asymptotic-type long-term dynamics, the time variable <em>t</em> ∈ (<em>t<sub>m</sub></em>, +∞) and<em> t<sub>m</sub></em> > 0 is large, say <img src="Edit_6f0f7522-7319-4b30-a451-0453ff0f75d3.bmp" alt="" />! <strong>Method:</strong> We apply Taylor series expansion to differential equations to model the acceleration of pointlike object whose response to the long-term external force is not instantaneous and possibly nonlinear. <strong>Results:</strong> We make simplified models for the long-term dynamics of pointlike objects by Taylor polynomials in time derivatives of the external force. <strong>Application:</strong> We interpret the relativistic Lorentz-Abraham-Dirac equation as an equation for modeling the long-term dynamics, where <em>t</em> ≥ <em>t<sub>m</sub></em> <span style="white-space:nowrap;">≫</span> 0. This interpretation resolves the conceptual and usage controversy surrounding its troublesome application to determine the trajectory of a radiating charged particle, thus contributing to the development of more adequate modeling of physical phenomena.展开更多
文摘The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em></span>), with the commutation relation <img src="Edit_28f5b839-7de4-41e5-9ed8-69dc1bf72c2c.bmp" alt="" />, and using a Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s like equation on these variable, and associating a linear operator to a constant of motion <em>K</em> (<em>x, v, t</em>) of the classical system, The comparison with the quantization in the space (<em>x, p</em>) is done with the usual Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s equation for the Hamiltonian <em>H</em><span style="white-space:normal;">(</span><em style="white-space:normal;">x, p, t</em><span style="white-space:normal;">)</span>, and with the commutation relation <img src="Edit_cca7e318-5b35-4c55-8f09-6089970ce9a2.bmp" alt="" />. It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization has fewer oscillations than the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, the average energy of the system is higher in (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">,
文摘A new method to construct coherent states of a time-dependent forced harmonic oscillator was given. The close relation to the classical forced oscillator and the minimum uncertainty relation were investigated. The applied periodic force (off-resonance case), in general, will attenuate the AA phase.
文摘The newly developed YY model contains a set of constitutive rules to describe the structures of atomic nuclei and subatomic particles, by using two elementary sub-quark particles, the Yin and Yang fermions of charge 1/3 forming all the particles of the Standard Model. This model suggests a modular structure of the universe, in which two elementary constituents recursively form all the matter. The advantage of this hypothesis is that it provides a total symmetry and a noticeably clear conceptual understanding. Moreover, it justifies the cosmological formation of a limited number of atoms, e.g., H and Li with their isotopes, considering that matter can be produced as a free agglomerate of semi-stable neutrons, which would lead to the feeding of baryonic matter in the universe. In this current article, some further theoretical aspects are proposed as an evolution of the YY model. They cover correlation paths between interacting quarks, the considerations of color forces between yin-yang elementary elements. Moreover, an agreement of the YY model with the Teplov approach based on harmonic quarks and oscillators is established, and the mass of Yin and Yang is considered. Two example nuclei are used for the analysis: a radioactively stable deuteron (containing a neutron and a proton) and a possible semi-stable dineutron (roughly “consisting of two neutrons”), which is purely theoretical, represent a very natural and legal nuclear state within YY model. Based on the results obtained here, some indications are given for a possible simple experimental verification providing proof for the stability or instability of the dineutron.
文摘<strong>Motivation:</strong> We study the asymptotic-type dynamics of various real pointlike objects that one models by a variety of differential equations. Their response to an external force one defines solely by the trajectory of a single point. Its velocity eventually stops changing after cessation of the external force. The response of their acceleration to the long-term external force is slow and possibly nonlinear. <strong>Objective:</strong> Our objective is to present technique for making simplified models for the long-term dynamics of pointlike objects whose motion interacts with the surroundings. In the asymptotic-type long-term dynamics, the time variable <em>t</em> ∈ (<em>t<sub>m</sub></em>, +∞) and<em> t<sub>m</sub></em> > 0 is large, say <img src="Edit_6f0f7522-7319-4b30-a451-0453ff0f75d3.bmp" alt="" />! <strong>Method:</strong> We apply Taylor series expansion to differential equations to model the acceleration of pointlike object whose response to the long-term external force is not instantaneous and possibly nonlinear. <strong>Results:</strong> We make simplified models for the long-term dynamics of pointlike objects by Taylor polynomials in time derivatives of the external force. <strong>Application:</strong> We interpret the relativistic Lorentz-Abraham-Dirac equation as an equation for modeling the long-term dynamics, where <em>t</em> ≥ <em>t<sub>m</sub></em> <span style="white-space:nowrap;">≫</span> 0. This interpretation resolves the conceptual and usage controversy surrounding its troublesome application to determine the trajectory of a radiating charged particle, thus contributing to the development of more adequate modeling of physical phenomena.
