论文标题

中央对称形状不变电位的统一方案

A Unified Scheme of Central Symmetric Shape-Invariant Potentials

论文作者

Koohrokhi, Taha, Izadpanah, Abdolmajid, Gerayloo, Mitra

论文摘要

大多数物理系统,无论是经典的还是量子机械的,都表现出球形对称性。角动量(表示为$ \ ell $)是一种保守数量,当粒子在中心力的影响下移动时,它出现在离心电位中。这项研究介绍了一种形式主义,其中$ \ ell $扮演着统一的角色,将可解决的中心电势整合到超电势中。该框架表明,库仑电势是作为同质($ r $独立于)各向同性超电势的直接结果。相反,$ \ ell $独立的中央超电势在3维谐波振荡器(3-DHO)电位中产生。此外,局部$ \ ell $依赖性的中央超电势产生适用于有限范围相互作用(例如分子或核次系统)的潜力。此外,我们讨论了对任意$ d $尺寸的概括,并研究了超电势的属性,以确定何时断裂或不间断的超对称性。该方案还解释说,三维中的自由粒子波函数是从超对称的自发分解中获得的,并阐明了正面3-DHO电位(作为颠倒的电势)如何具有负能量谱。我们还提出了中央超电球和超级零部队的复杂的等光变形,它们可以在动态平衡中具有有趣的开放系统应用程序。最后,作为一种实际应用,我们应用这种形式主义来为迪特伦指定新的有效潜力。

Most physical systems, whether classical or quantum mechanical, exhibit spherical symmetry. Angular momentum, denoted as $\ell$, is a conserved quantity that appears in the centrifugal potential when a particle moves under the influence of a central force. This study introduces a formalism in which $\ell$ plays a unifying role, consolidating solvable central potentials into a superpotential. This framework illustrates that the Coulomb potential emerges as a direct consequence of a homogenous ($r$-independent) isotropic superpotential. Conversely, a $\ell$-independent central superpotential results in the 3-Dimensional Harmonic Oscillator (3-DHO) potential. Moreover, a local $\ell$-dependent central superpotential generates potentials applicable to finite-range interactions such as molecular or nucleonic systems. Additionally, we discuss generalizations to arbitrary $D$ dimensions and investigate the properties of the superpotential to determine when supersymmetry is broken or unbroken. This scheme also explains that the free particle wave function in three dimensions is obtained from spontaneous breakdown of supersymmetry and clarifies how a positive 3-DHO potential, as an upside-down potential, can have a negative energy spectrum. We also present complex isospectral deformations of the central superpotential and superpartners, which can have interesting applications for open systems in dynamic equilibrium. Finally, as a practical application, we apply this formalism to specify a new effective potential for the deuteron.

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