اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدConstructions of the soluble potentials for the non-relativistic quantum system by means of the Heun functions
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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 $ abla^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.
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