تقرير
Constructions of the soluble potentials for the non-relativistic quantum system by means of the Heun functions
| العنوان: | Constructions of the soluble potentials for the non-relativistic quantum system by means of the Heun functions |
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| المؤلفون: | Dong, Shishan, Yanez-Navarro, G., Sanchez, M. A. Mercado, Garcia, C. Mejia, Sun, Guo-Hua, Dong, Shi-Hai |
| سنة النشر: | 2017 |
| المجموعة: | Mathematics Mathematical Physics Quantum Physics |
| مصطلحات موضوعية: | Quantum Physics, Mathematical Physics |
| الوصف: | The Schr\"{o}dinger equation $\psi"(x)+\kappa^2 \psi(x)=0$ where $\kappa^2=k^2-V(x)$ is rewritten as a more popular form of a second order differential equation through taking a similarity transformation $\psi(z)=\phi(z)u(z)$ with $z=z(x)$. The Schr\"{o}dinger invariant $I_{S}(x)$ can be calculated directly by the Schwarzian derivative $\{z, x\}$ and the invariant $I(z)$ of the differential equation $u_{zz}+f(z)u_{z}+g(z)u=0$. We find an important relation for moving particle as $\nabla^2=-I_{S}(x)$ and thus explain the reason why the Schr\"{o}dinger invariant $I_{S}(x)$ keeps constant. As an illustration, we take the typical Heun differential equation as an object to construct a class of soluble potentials and generalize the previous results through choosing different $\rho=z'(x)$ as before. We get a more general solution $z(x)$ through integrating $(z')^2=\alpha_{1}z^2+\beta_{1}z+\gamma_{1}$ directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail. Comment: 11 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1710.07199 |
| رقم الأكسشن: | edsarx.1710.07199 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |