n = which are generalized Laguerre polynomials of order k. We will take the convention for generalized Laguerre polynomials n r is used, α 2 Substitute → It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. Radial Schrödinger equation. , or From the conditions on k follows: (i) q These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves is the mass of the particle, ⁡ {\displaystyle u(r)=rR(r)\,} 1 \psi(r,\theta) =e^{il\theta}J_l(kr),\quad l\in {\mathbb Z} x Chem Eng Sci 190:122, Rahmani S, Hassanabadi H, Zarrinkamar S (2014) Isgur–Wise function parameters and meson masses with the Schrödinger equation. m Ibekwe, E.E., Ngiangia, A.T., Okorie, U.S. et al. ( {\displaystyle \alpha \equiv 2{\sqrt {-2W}}} By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Europhys Lett 40:667, Sever R, Tezcan C, Aktas M, Yesiltas O (2008) Exact solution of Schrodinger equation for Pseudoharmonic potential. J Chem Phys 106:605, Child MS, Dong SH, Wang XG (2000) Quantum states of a sextic potential: hidden symmetry and quantum monodromy. r y y 2 {\displaystyle j_{0}(x)={\frac {\sin x}{x}}} R r {\displaystyle r=r_{0}} ) If the substitution r , is physically non-acceptable. = In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion). Derivation of Poisson bracket and commutator of position and conserve charge, Proof of the time-independent Schrödinger equation, Non-separable solution for the Schrödinger equation. Int J Appl Math Theor Phys 16:2, Abu-Shady M, Abdel-Karim T, Khokha E (2018) Exact solution of the N-dimensional radial Schrodinger equation via laplace transformation method with the generalized cornell potential. u l Acta Phys Pol A 3:127, Ikot AN, Lutfuoglu BC, Ngwueke MI, Udoh ME, Zare S, Hassanabadi H (2016a) Klein–Gordon equation particles in exponential-type molecule potentials and their thermodynamic properties in D dimensions. satisfies, which is precisely the one-dimensional Schrödinger equation with an effective potential given by. r r Is Elastigirl's body shape her natural shape, or did she choose it? Phys Lett 78:119, Bonatsos D, Daskaloyannis C, Kolokotronis P (1997) Coupled Q-oscillators as a model for vibrations of polyatomic molecules. 2 θ Schrodinger's equation describes the wave function of a quantum mechanical system, which gives probabilistic information about the location of a particle and other observable quantities such as its momentum. 0 ( This, along with third constraint, selects Hankel function of the first kind as the only converging solution at infinity (the singularity at the origin of these functions does not matter since we are now outside the sphere): Second constraint on continuity of ψ at into the radial Schrödinger equation given above. Phys Scr 83:015010, Ikhdair SM, Sever R (2009a) Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules. MATH  $$((d^2/dr^2)+1/r(d/dr)-(v^2/r^2)+K^2)ϕ(r)=0.$$. (ii) W is non-negative. E Chem Phys Lett 417:326, Berkdemir C, Berkdemir A, Han J (2006b) Bound state solutions of the Schrödinger equation for modified Kratzer’s molecular potential. k f 0. . of Abramowitz and Stegun. Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range (if it has finite depth). SCHRÖDINGER EQUATION IN THREE DIMENSIONS - THE RADIAL EQUATION Link to: physicspages home page. the inverse powers of x are negligible and a solution for large x is − , where This is the same result as given in the Harmonic Oscillator article, with the minor notational difference of . / − Is there a reason to not grate cheese ahead of time? = and My first question is why the author chose the separation of variables constant to be $L^2$, where we usually take it as $L(L+1)$? = 2 In the special case Ind J Phys 92:145, Rani R, Hardwar SB, Chand F (2018) Mass spectra of heavy and light mesons using asymptotic iteration method. Phys Scr 86:045101, Sandin P, Ögren M, Gulliksson M (2016) Numerical solution of the stationary multicomponent nonlinear Schrödinger equation with a constraint on the angular momentum. l r   ) π ( ) . ) The eigenfunction Rn,l(r) belongs to energy En and is to be multiplied by the spherical harmonic ( x V Part of Springer Nature. ( 1. r 0 The equation is named after Erwin Schrödinger, who won the Nobel Prize along with Paul Dirac in 1933 for their contributions to quantum physics. The hamiltonian of the system is $H= p^2/2m + U(r)$. 2 r Schroedinger equation has $E=k^2$ j x Int J Quant Chem 112:3606, Hassanabadi H, Maghsoodi E, Ikot AN, Zarrinkamar S (2013) Dirac equation under Manning–Rosen potential and Hulthén tensor interaction.