{"title":"位置作为自变量与量子力学中 1/2 时间分数导数的出现","authors":"Marcus W. Beims, Arlans J. S. de Lara","doi":"10.1007/s10701-024-00787-1","DOIUrl":null,"url":null,"abstract":"<div><p>Using the position as an independent variable, and time as the dependent variable, we derive the function <span>\\({\\mathcal{P}}^{(\\pm )}=\\pm \\sqrt{2m({\\mathcal{H}}-{\\mathcal{V}}(q))}\\)</span>, which generates the space evolution under the potential <span>\\({\\mathcal{V}}(q)\\)</span> and Hamiltonian <span>\\({\\mathcal{H}}\\)</span>. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (<span>\\(-{\\mathcal{H}}\\)</span>). While the classical dynamics do not change, the corresponding Quantum operator <span>\\({{{\\hat{\\mathcal P}}}}^{(\\pm )}\\)</span> naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of <span>\\({{{\\hat{\\mathcal P}}}}^{(\\pm )}\\)</span> and the kinetic energy <span>\\({\\mathcal{K}}_{0}\\)</span> (separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are <span>\\(\\pm {\\hbar {\\omega}} /2\\)</span>, and the corresponding pair of states are coupled for <span>\\({\\mathcal{K}}_{0}\\ne 0\\)</span>. No time evolution is present for <span>\\({\\mathcal{K}}_{0}=0\\)</span>, and the ground state with energy <span>\\({\\hbar {\\omega}} /2\\)</span> is stable. For <span>\\({\\mathcal{K}}_{0}>0\\)</span>, the ground state becomes coupled to the state with energy <span>\\(-{\\hbar {\\omega}} /2\\)</span>, and <i>this coupling</i> allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of <span>\\({\\mathcal{K}}_{0}=k{\\hbar {\\omega}}\\)</span> (<span>\\(k=1,2,\\ldots\\)</span>). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case <span>\\({\\mathcal{K}}_{0}=0\\)</span> leads to plane-waves-like solutions at the threshold. Above the threshold (<span>\\({\\mathcal{K}}_{0}>0\\)</span>), we obtain a plane-wave-like solution, and for the bounded states (<span>\\({\\mathcal{K}}_{0}<0\\)</span>), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.</p></div>","PeriodicalId":569,"journal":{"name":"Foundations of Physics","volume":"54 4","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics\",\"authors\":\"Marcus W. Beims, Arlans J. S. de Lara\",\"doi\":\"10.1007/s10701-024-00787-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Using the position as an independent variable, and time as the dependent variable, we derive the function <span>\\\\({\\\\mathcal{P}}^{(\\\\pm )}=\\\\pm \\\\sqrt{2m({\\\\mathcal{H}}-{\\\\mathcal{V}}(q))}\\\\)</span>, which generates the space evolution under the potential <span>\\\\({\\\\mathcal{V}}(q)\\\\)</span> and Hamiltonian <span>\\\\({\\\\mathcal{H}}\\\\)</span>. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (<span>\\\\(-{\\\\mathcal{H}}\\\\)</span>). While the classical dynamics do not change, the corresponding Quantum operator <span>\\\\({{{\\\\hat{\\\\mathcal P}}}}^{(\\\\pm )}\\\\)</span> naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of <span>\\\\({{{\\\\hat{\\\\mathcal P}}}}^{(\\\\pm )}\\\\)</span> and the kinetic energy <span>\\\\({\\\\mathcal{K}}_{0}\\\\)</span> (separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are <span>\\\\(\\\\pm {\\\\hbar {\\\\omega}} /2\\\\)</span>, and the corresponding pair of states are coupled for <span>\\\\({\\\\mathcal{K}}_{0}\\\\ne 0\\\\)</span>. No time evolution is present for <span>\\\\({\\\\mathcal{K}}_{0}=0\\\\)</span>, and the ground state with energy <span>\\\\({\\\\hbar {\\\\omega}} /2\\\\)</span> is stable. For <span>\\\\({\\\\mathcal{K}}_{0}>0\\\\)</span>, the ground state becomes coupled to the state with energy <span>\\\\(-{\\\\hbar {\\\\omega}} /2\\\\)</span>, and <i>this coupling</i> allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of <span>\\\\({\\\\mathcal{K}}_{0}=k{\\\\hbar {\\\\omega}}\\\\)</span> (<span>\\\\(k=1,2,\\\\ldots\\\\)</span>). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case <span>\\\\({\\\\mathcal{K}}_{0}=0\\\\)</span> leads to plane-waves-like solutions at the threshold. Above the threshold (<span>\\\\({\\\\mathcal{K}}_{0}>0\\\\)</span>), we obtain a plane-wave-like solution, and for the bounded states (<span>\\\\({\\\\mathcal{K}}_{0}<0\\\\)</span>), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.</p></div>\",\"PeriodicalId\":569,\"journal\":{\"name\":\"Foundations of Physics\",\"volume\":\"54 4\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Foundations of Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10701-024-00787-1\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10701-024-00787-1","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics
Using the position as an independent variable, and time as the dependent variable, we derive the function \({\mathcal{P}}^{(\pm )}=\pm \sqrt{2m({\mathcal{H}}-{\mathcal{V}}(q))}\), which generates the space evolution under the potential \({\mathcal{V}}(q)\) and Hamiltonian \({\mathcal{H}}\). No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (\(-{\mathcal{H}}\)). While the classical dynamics do not change, the corresponding Quantum operator \({{{\hat{\mathcal P}}}}^{(\pm )}\) naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of \({{{\hat{\mathcal P}}}}^{(\pm )}\) and the kinetic energy \({\mathcal{K}}_{0}\) (separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are \(\pm {\hbar {\omega}} /2\), and the corresponding pair of states are coupled for \({\mathcal{K}}_{0}\ne 0\). No time evolution is present for \({\mathcal{K}}_{0}=0\), and the ground state with energy \({\hbar {\omega}} /2\) is stable. For \({\mathcal{K}}_{0}>0\), the ground state becomes coupled to the state with energy \(-{\hbar {\omega}} /2\), and this coupling allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of \({\mathcal{K}}_{0}=k{\hbar {\omega}}\) (\(k=1,2,\ldots\)). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case \({\mathcal{K}}_{0}=0\) leads to plane-waves-like solutions at the threshold. Above the threshold (\({\mathcal{K}}_{0}>0\)), we obtain a plane-wave-like solution, and for the bounded states (\({\mathcal{K}}_{0}<0\)), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.
期刊介绍:
The conceptual foundations of physics have been under constant revision from the outset, and remain so today. Discussion of foundational issues has always been a major source of progress in science, on a par with empirical knowledge and mathematics. Examples include the debates on the nature of space and time involving Newton and later Einstein; on the nature of heat and of energy; on irreversibility and probability due to Boltzmann; on the nature of matter and observation measurement during the early days of quantum theory; on the meaning of renormalisation, and many others.
Today, insightful reflection on the conceptual structure utilised in our efforts to understand the physical world is of particular value, given the serious unsolved problems that are likely to demand, once again, modifications of the grammar of our scientific description of the physical world. The quantum properties of gravity, the nature of measurement in quantum mechanics, the primary source of irreversibility, the role of information in physics – all these are examples of questions about which science is still confused and whose solution may well demand more than skilled mathematics and new experiments.
Foundations of Physics is a privileged forum for discussing such foundational issues, open to physicists, cosmologists, philosophers and mathematicians. It is devoted to the conceptual bases of the fundamental theories of physics and cosmology, to their logical, methodological, and philosophical premises.
The journal welcomes papers on issues such as the foundations of special and general relativity, quantum theory, classical and quantum field theory, quantum gravity, unified theories, thermodynamics, statistical mechanics, cosmology, and similar.