SPHERICAL HARMONICS 33 2 - - w22 [-] all,. (I - m)! (21 + 1) (2.173) Associated Legendre Polynomials Po"(.) = 1 P;(Z) = (I -x')~/' = sin8 Pi(z) = 3z(1 z')~/' = 3cosBsinB P,"(z) = 3( 1 - z') = 3 sin' 8 P,'(z) = -(52 - 1)(1- z')'/' = -(5c0s26 - 1)sinB P,"(z) = 152(1 -z') = 15cos8sin'B Pi(.) = 15( 1 - z')~/' = 15sin3 8. 3 3 2 2 2.5 SPHERICAL HARMONICS We have seen that the solution of Equation (2.17) with respect to the inde pendent variable 4 is given as (4) = Aeim4 + Be-im@. (2.174) Imposing the periodic boundary condition am(4 + 2~) = am(+), it is seen that the separation constant m has to take finteger values. However, in Section 2.4 we have also seen that m must be restricted further to the integer values -1, , 0, , 1. We can now define another complete and orthonormal set as 1 {am($) = -eim@} , m = -1, , 0, , 1. (2.175) 6 This set satisfies the orthogonality relation 2T Jd d4amt (4)*~(4) = fimmf. (2.176) We now combine the two sets {am(4)} and {F;r"(8)} to define a new com- plete and orthonormal set called the spherical harmonics as 21 + 1 (I - m)! 47r (1 +m)! elm+ P,"(cosB), m 2 0. (2.177) 34 LEGENDRE EQUATION AND POLYNOMIALS In conformity with applications to quantum mechanics and atomic spec- troscopy, we have introduced the factor (-l)m. It is also called the Condon- Shortley phase. The definition of spherical harmonics can easily be ex- tended to the negative m values as Y-"(6,4) = (-1)mqm*(Q,4), m 2 0. (2.178) The orthogonality relation of Ym(8, 4) is given as Since they also form a complete set, any sufficiently well-behaved and at least piecewise continuous function g(B,$) can be expressed in terms of qm(8, 4) as 1=0 m=-1 where the expansion coefficients Akare given as (2.180) (2.181) Looking back at Equation (2.13), we see that the spherical harmonics sat- isfy the differential equation (2.182) If we rewrite this equation as the left-hand side is nothing but the square of the angular momentum operator (aside from a factor of ti) in quantum mechanics, which is given as (2.184) In quantum mechanics the fact that the separation constant X is restricted to integer values means that the magnitude of the angular momentum is SPHERlCAL HARMONlCS 35 quantized. From Equation (2.183) it is seen that the spherical harmonics are also the eigenfunctions of the 3' operator. Spherical Harmonics q"(8, $) 36 LEGENDRE EQUATION AND POLYNOMIALS 1=3 Problems 2.1 tial equations: i) Laguerre equation: Locate and classify the singular points of each of the following differen- ii) Harmonic oscillator equation: d2Q& (.) + (. - x2) IFE (X) = 0 dx2 iii) Bessel equation: 2JL(X) + ZJL(X) + (2 - rn2)Jm(X) = 0 iv) d2Y dY 2 (x4 - 2X3 + X2)- + (. - 1) - + 2x = 0 dx2 dx PROBLEMS 37 vi) Chebyshev equation: &Y dY - - 2- dx +n2y = 0 vii) Gegenbauer equation: gc; dCX (x) dx2 dx (1-2)- -(2X+l)x-” ++(n+2X)C~(x)=O viii) Hypergeometric equation: d2y(z) + [c - (a + b + 1)x]- dy(x) - uby(x) = 0 z- d2y(z) + [c - 21- - .y(z) = 0 dx 2(1-x)- dx2 ix) Confluent Hypergeometric equation: dYk) dz2 dz 2.2 find solutions about x = 0: For the following differential equations use the Frobenius method to i> 2x 3 -++x2-++3y=O d2Y dY dx2 dx ii) iii) iv) v) vi) 3 d2Y 2 dY 8 x -++x -+(x3+-x)y=O 9 dx2 dx 3 d2Y 2 dY 3 x -+fx -+(x3+-x)y=O dx2 dx 4 2d2Y dY x - + 3x- + (22 + 1)y = 0 dx2 dx &Y ZdY x3- + 2 - + (8z3 - 9x)y = 0 dx2 dx 2 2d2Y -+x-+x2y=o dY dx2 dx 38 LEGENDRE EQUATION AND POLYNOMIALS vii) d2Y dY Z- + (1 -I.) - +4y = 0 dX2 dx viii) 3 d2Y 2 dY 2% - + 52 - + (x3 - 2x)y = 0 dx2 dx 2.3 Find finite solutions of the equation d2Y dY (1 - 2)- - 2- + n2y = 0 dx2 dx in the interval 3: E [-1,1] for n = integer. 2.4 Consider a spherical conductor with radius a, with the upper hemisphere held at potential Vo and the lower hemisphere held at potential -Vo, which are connected by an insulator at the center. Show that the electric potential inside the sphere is given as 2.5 solutions of Using the Frobenius method, show that the two linearly independent R = 0, are given as 2.6 The amplitude of a scattered wave is given as 00 f(0) = yC(21+ l)(ei6[ sinGl)e(cosB), 1 =o where B is the scattering angle, 1 is the angular momentum, and 61 is the phase shift caused by the central potential causing the scattering. If the total scattering cross section is PROBLEMS 39 show that 2.7 Prove the following recursion relations: 4 (z) = P;+l (z) + (x) - 2x4' (.> P(,l (z) - xP( (z) = (1 + l)q (x) 2.8 Use the Rodriguez formula to prove P; (z) = zP;-, (x) + 14-1 (z) where 1 = 1,2, . 2.9 Show that the Legendre polynomials satisfy the following relations: 9 d - dx [(l - z2)P/(iC)] + 1(1+ 1)9(x) = 0 2.10 Derive the normalization constant, Ni, in the orthogonality relation 1 s_, P1) (z) A (x) &E = "b of the Legendre polynomials by using the generating function. 2.11 Show the integral 40 LEGENDRE EQUATION AND POLYNOMIALS where (1 - n) = (even integer] . 2.12 are given as Show that the associated Legendre polynomials with negative m values (I - m)! (1 + m)! PFrn(Z) = (-1)rn- P;"(z), m 2 0. 2.13 in the interval [-1,1]. 2.14 A metal sphere is cut into sections that are separated by a very thin insulating material. One section extending from 6 = 0 to 8 = 8, at potential Vo and the second section extending from 6 = 60 to 8 = 7r is grounded. Find the electrostatic potential outside the sphere. 2.15 Expand the Dirac delta function in a series of Legendre polynomials The equation for the surface of a liquid drop (nucleus) is given by 22 22 24 T =a (I+E~-+E~-), T2 r4 where 2, €2, and €4 are given constants. Express this in terms of the Legendre polynomials as 2.16 can be expressed in terms of the Legendre polynomials as Show that the inverse distance between two points in three dimensions 1 1 where r< and r, denote the lesser and the greater of r and r', respectively. 2.17 Evaluate the sum &+1 03 s = c -fi(Z). l=O 1+1 Hint: Try using the generating function for the Legendre polynomials. 2.18 Wronskian If two solutions yl(z) and y2(z) are linearly dependent, then their W[Yl(Z),Y2(Z)I = Yl(Z)Y2Z) - d(Z)Y2(Z) PROBLEMS 41 vanishes identically. What is the Wronskian of two solutions of the Legendre equation? 2.19 The Jacobi polynomials Pp'b)(~~~8), where n = positive integer and a, b are arbitrary real numbers, are defined by the Rodriguez formula d" 2"n!(l - x)"(l+ x)b dx" P b'(x) = (-1" - [( 1 - ,)"+a( 1 + Z)"+b] , 1x1 < 1. Show that the polynomial can be expanded as n P27b)(~~~6) = A(n, a, b, k) ks0 Determine the coefficients A(n, a, b, k) for the special case, where a and b are both integers. 2.20 Find solutions of the differential equation d2Y 2~(x - 1)- + (10~ - dx2 dx satisfying the condition y(x) = finite in the entire interval z E [0,1]. Write the solution explicitly for the third lowest value of A. This Page Intentionally Left Blank [...]... ) ! 2 (3. 37) = L (x) , 3. 3 ORTHOGONALITY OF LAGUERRE POLYNOMIALS To show that the Laguerre polynomials form an orthogonal set, we evaluate the integral (3. 38) Using the generating function definition of the Laguerre polynomials we write 03 =E 1-t L n(z) tn (3. 39) n=O and 1 1-s xs ELm OL) = (z) sm (3. 40) n=O We first multiply Equations (3. 39) and (3. 40) and then the result with ep" to write (3. 41) ORTHOGONALiTY... ORTHOGONALiTY OF LAGUERRE POLYNOMlALS 49 Interchanging the integral and the summation signs and integrating with respect to x gives us 2 1 [~We~xLn(x)Lm(x)dx tnsm n,m=O It is now seen that the value of the integral in Equation (3. 38) can be obtained by expanding in powers of t and s and then by comparing the equal powers of tnsm with the left-hand side of Equation (3. 42) If we write I as I= (1-9(1-4 JWexp... n!xr ( n- T ) ! ( T ! ) 2 r=O (3. 23) Laguerre polynomials are defined by setting a0 = 1 in Equation (3. 23) as (3. 24) 3. 2 OTHER DEFINITIONS OF LAGUERRE POLYNOMIALS 3. 2.1 Generating Function of Laguerre Polynomials The generating function of the Laguerre polynomials is defined as -xt (3. 25) To see that this gives the same polynomials as Equation (3. 24), we expand the left-hand side as power series: 1 -,(I... LAGUERRE EQUATION AND POLYNOMIALS 45 go to zero Hence for a finite solution in the interval [O,m],we terminate the series [Eq (3. 10)]by restricting po (energy) to the values po = 2 ( N + 1 + l), N = 0, 1,2, (3. 13) Since 1 takes integer values, we introduce a new quantum number, n, and write the energy levels of a singleelectron atom as En = - Z2me4 2fi2n2 n = 1,2, , ' (3. 14) which are nothing but the... 1 - t )r=o zrtr r=O (3. 26) OTHER D€F/N/T/ONS OF LAGUERRE POLYNOMIALS 47 Using the binomial formula 1 (r + 1)(r + 2)t2 + = l+(r+l)t+ 2! (1 - ty+' + =cr OC) s! ) (r tS, (3. 27) s=o Equation (3. 26) becomes Defining a new dummy variable as n=r+s, (3. 29) we now write and compare equal powers of t Since s=n-r>O, r (3. 31) 5 n; thus we obtain the Laguerre polynomials L, ( x )as (3. 32) 3. 2.2 Rodriguez Formula... Frobenius method, find a n infinite series solution about x = 0 in the interval [0,m] Check t h e convergence of your solution Should your solution be finite everywhere, including the end points of your interval? why? PROBLEMS 67 iv) For finite solutions everywhere in the interval [0,co], what restrictions do you have to impose on the physical parameters of the system v) For 1 = 0,1, and 2 find explicitly... following differential equation for the radial part of the wave function: 2 + d R ( x ) + [E-x x dx &R(X) dx2 where x and E are defined in terms of the radial distance r and the energy E as x=- r Iri vmw and E= E hW/2 and 1 takes integer values 1 = 0,1,2 i) Examine the nature of the singular point at z = 00 ii) Show that in t h e limit as x 00, the solution goes as f iii) Using the Frobenius method, ... RELATIONS AND ORTHOGONALITY If we show the expression inside the square brackets on the right-hand side as J,, , integral I,, will be given as e In, = &GKFhGqK (4.62) Jnm Writing the left-hand side of Equation (4.61) explicitly we get 00 = e2st (s &e-t2+2tx e - 2 + 2 s x e- 1 2 x + t ), , h (4. 63) where we have defined u=x-(s+t) (4.64) Expanding this in power series o f t and s gives us (4.65) Finally,... L - 1 (z) (3. 50) Another useful recursion relation is given as 11-1 (3. 51) r=O Laguerre Polynomials Lo ( x ) = 1 -x + 1 L1 (x)= (1/2!) (9 42 + 2) L2 (XI = (1 /3! ) ( - x 3 + 9z2- 18% 6 ) + L3 (x)= (1/4!) (z4- 16x3+ 722’ - 962 24) L4 (x)= L5(x)= (1/5!) (-z5 + 25x4 - 200x3 + 600~’- 600x + 120) + 3. 4.2 (3. 52) Special Values of Laguerre Polynomials Taking x = 0 in the generating function we find 00 1 ELn... k values, (3. 63) 52 LAGUERRE POLYNOMIALS L; c + n+k (2)= (-1y (n k)! r ! xr k (n+ IC - r ) !( r ! ) 2 - k)! (r (-1y r=k (3. 64) Defining a new dummy variable s as (3. 65) s=r-k, we find the final form of the associated Laguerre polynomials as c n L; ( x )= (-1y s=o 3. 6 3. 6.1 (n+ k)!x" ( n- s ! (k + s)!s! ) (3. 66) PROPERTIES OF ASSOCIATED LAGUERRE POLYNOMIALS Generating Function The generating function . Equations (3. 39) and (3. 40) and then the result with ep" to write (3. 41) ORTHOGONALiTY OF LAGUERRE POLYNOMlALS 49 Interchanging the integral and the summation signs and integrating. of the integral in Equation (3. 38) can be obtained by expanding in powers of t and s and then by comparing the equal powers of tnsm with the left-hand side of Equation (3. 42) the integral (3. 38) Using the generating function definition of the Laguerre polynomials we write 03 = ELn (z) tn n=O 1-t and OL) xs 1 = ELm (z) sm. n=O 1-s (3. 39) (3. 40)