论文标题
具有对称三重孔结构的确切可解决的六势
Exactly Solvable Sextic Potential Having Symmetric Triple-Well Structure
论文作者
论文摘要
在本文中，我们介绍了一个完全可以解决的六势家族，并且首次使用超对称方法的整个能量谱和波源。自三十年前以来，有人建议所有“添加剂”或“翻译”形状不变的超电势，由两种功能组合形成，并且它们的列表已经被大多数教科书中可用的众所周知的确切可解决的潜力所耗尽，而且还有其他。我们已经设计了一个新的超级电球家族，该家族由三个函数（两个单一和一个有理由）的线性组合形成，其中参数函数的变化在四个参数中是线性的。这个带有超电势$ W（x，a，b，d，g）的新电位家族= ax^3 + bx - \ frac {dx} {1 + gx^2} $将扩展到恰好可溶解的schrödinger方程的列表。我们已经表明，结合状态的能量在量子数中是合理的。此外，通过惯常的做法，谐波振荡器近似于中央孔周围的电势，这是无效的。两个外孔明显影响激发态的概率密度分布。我们注意到三孔电位的种群位于两个外孔中。这些结果具有潜在的应用，可以探索更多的物理现象，例如隧道效应和intsantons动力学。
In this paper, we introduce a family of sextic potentials that are exactly solvable, and for the first time, a family of triple-well potentials with their whole energy spectrum and wavefunctions using supersymmetry method. It was suggested since three decades ago that all "additive" or "translational" shape invariant superpotentials formed by two combination of functions have been found and their list was already exhausted by the well-known exactly solvable potentials that are available in most textbooks and furthermore, there are no others. We have devised a new family of superpotentials formed by a linear combination of three functions (two monomials and one rational) and where the change of parameter function is linear in four parameters. This new family of potentials with superpotential $W(x,A,B,D,G) = Ax^3 + Bx -\frac{Dx}{1+Gx^2}$ will extend the list of exactly solvable Schrödinger equations. We have shown that the energy of the bound states is rational in the quantum number. Furthermore, approximating the potential around the central well by a harmonic oscillator, as a usual practice, is not valid. The two outer wells affect noticeably the probability density distribution of the excited states. We have noticed that the populations of the triple-well potentials are localized in the two outer wells. These results have potential applications to explore more physical phenomena such as tunneling effect, and instantons dynamics.
