位置作为自变量与量子力学中 1/2 时间分数导数的出现

IF 1.2 3区 物理与天体物理 Q3 PHYSICS, MULTIDISCIPLINARY Foundations of Physics Pub Date : 2024-07-26 DOI:10.1007/s10701-024-00787-1
Marcus W. Beims, Arlans J. S. de Lara
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引用次数: 0

摘要

将位置作为自变量,时间作为因变量,我们得出了函数 ({\mathcal{P}}^{(\pm )}=pm \sqrt{2m({\mathcal{H}}-{\mathcal{V}}(q))}\ ),它产生了势能 ({\mathcal{V}}(q)\)和哈密顿({\mathcal{H}}\)下的空间演化。没有使用参数化。典型共轭变量是时间和减去哈密顿(\(-{\mathcal{H}}\))。虽然经典动力学没有改变,但相应的量子算子 \({{\hat{mathcal P}}}}^{(\pm )}\) 自然会导致 1/2 分数时间演化,这与最近提出的量子力学时空对称形式主义是一致的。利用狄拉克程序,变量分离是可能的,而与位置无关的狄拉克两耦合方程取决于 1/2 分数导数,与时间无关的狄拉克两耦合方程则导致与力成正比的势的正负移动。这两个方程耦合(±)解({{\hat{\mathcal P}}}}^{(\pm )}\),动能({\mathcal{K}}_{0}\)(分离常数)是耦合强度。因此,我们得到了具有有限力的系统的一对耦合态,但不一定是静止态。谐振子(HO)的势移为 \(\pm {\hbar {\omega}} /2/\),相应的一对状态耦合为 \({\mathcal{K}}_{0}ne 0/\)。({/mathcal{K}}_{0}=0/)不存在时间演化,能量为\({/hbar {\omega}} /2/)的基态是稳定的。当 \({\mathcal{K}}_{0}>0\) 时,基态与能量为 \(-{\hbar {\omega}} /2\) 的态耦合,这种耦合允许描述 HO 中更高的激发态。HO的能量量化导致了\({\mathcal{K}}_{0}=k{\hbar {\omega}}\) ((k=1,2,\ldots\))的量化。对于一维氢原子,势移变成了虚移,并且与位置有关。解耦情况(\({\mathcal{K}}_{0}=0\)导致在阈值处出现类似平面波的解。在阈值以上(\({/mathcal{K}}_{0}>0\),我们得到一个平面波样的解,而对于有界态(\({/mathcal{K}}_{0}<0\),波函数变得与精确解相似,但被挤压得更靠近原子核。
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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.

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来源期刊
Foundations of Physics
Foundations of Physics 物理-物理:综合
CiteScore
2.70
自引率
6.70%
发文量
104
审稿时长
6-12 weeks
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.
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