Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Franz Schwabl Quantum Mechanics 4th Edit · Problem 6.2
Motion in an Electromagnetic Field
Physics — Quantum Mechanics · associated-laguerre-polynomials
Problem Show that for the associated Laguerre polynomials
L r s ( x ) = ( d / d x ) s L r ( x ) , L^s_r(x) = (d/dx)^s L_r(x) \quad, L r s ( x ) = ( d / d x ) s L r ( x ) , L r ( x ) = e x ( d / d x ) r e − x x r , L_r(x) = e^x (d/dx)^r e^{-x} x^r \quad, L r ( x ) = e x ( d / d x ) r e − x x r , L r s ( x ) = ∑ k = 0 r − s ( − 1 ) k + s [ r ! ] 2 x k k ! ( k + s ) ! ( r − k − s ) ! \displaystyle L^s_r(x) = \sum_{k=0}^{r-s} (-1)^{k+s} \frac{[r!]^2 x^k}{k!\,(k+s)!\,(r-k-s)!} L r s ( x ) = k = 0 ∑ r − s ( − 1 ) k + s k ! ( k + s )! ( r − k − s )! [ r ! ] 2 x k ( − 1 ) m e − x t / ( 1 − t ) ( 1 − t ) m + 1 = ∑ n = 0 ∞ t r ( r + m ) ! L r + m m ( x ) \displaystyle \frac{(-1)^m e^{-xt/(1-t)}}{(1-t)^{m+1}} = \sum_{n=0}^{\infty} \frac{t^r}{(r+m)!} L^m_{r+m}(x) ( 1 − t ) m + 1 ( − 1 ) m e − x t / ( 1 − t ) = n = 0 ∑ ∞ ( r + m )! t r L r + m m ( x ) Browse the problem library