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Appendix: Special Mathematical Functions Appendix: Special Mathematical Functions For completeness this appendix briefly lists without further discussion the definiti­ tions and some important properties of the special functions occurring in the book. More detailed treatments can be found in the relevant literature. The "Handbook of Mathematical Functions" [AS70], the "Tables" by Gradshteyn and Rhyzik [GR65] and the compilation by Magnus, Oberhettinger and Soni [M066] are particularly useful. Apart from these comprehensive works it is worth mention­ ing Appendix B in "Quantum Mechanics I" by Messiah [Mes70], which describes a selection of especially frequently used functions. A.I Legendre Polynomials, Spherical Harmonics The Ith Legendre Polynom P,(x) is a polynomial of degree 1 in x, 1 d' 2 , P,(x) = 2'11 dx'(x - 1), 1 = 0, 1, ... (A.1) It has 1 zeros in the interval between -1 and +1; for even (odd) I, P,(x) is an even (odd) function of x. The associated Legendre junctions P"m(x) , Ixl ~ 1, are products of (1 - x2)m/2 with polynomials of degree 1 - m (m = 0, ... , I) , (A.2) The spherical harmonics Yi,m(fJ, c/» are products of exp (imc/» with polyno­ mials of degree m in sin fJ and of degree 1- m in cos fJ, where the fJ-dependence is given by the associated Legendre functions (A.2) as functions of x = cos fJ. For m ;::: 0, 0 ~ fJ ~ 7r we have Yi,m(fJ, c/» = (_1)m [(2~: 1) ~~: :;!! f/2 P',m(cos fJ)eim.p (A.3) = (_I)m [(2/+1) (/_m)!]1/2 . mfJ dm n( fJ) im.p 47r (1 + m)! sm d(cos fJ)m .q cos e The spherical harmonics for negative azimuthal quantum numbers are obtained via (A.4) 302 Appendix A reflection of the displacement vector x = r sin 0 cos </>, y = r sin 0 sin </>, z = r cos 0 at the origin (cf. (1.67» is achieved by replacing the polar angle 0 by 7r-0 and the azimuthal angle </> by 7r+¢. This does not affect sin 0, but cos 0 changes to - cos O. In the expression (A.3) for Yi,m spatial reflection introduces a factor (_I)I-m from the polynomial in cosO and a factor (_I)m from the exponential function in ¢. Altogether we obtain (A.5) The integral over a product of two spherical harmonics is given by the or­ thonormality relation (1.58). The integral over three spherical harmonics is a pro­ totype example for the Wigner-Eckart theorem, which says that the dependence of the matrix elements of (spherical) tensor operators in angular momentum eigenstates on the component index of the operator and the azimuthal quantum numbers of bra and ket is given by appropriate Clebsch-Gordan coefficients (see Sect. 1.6.1). For the spherical harmonics YL,M as an example for a spherical tensor of rank L we have J Yi:m(Q) YL,M(Q) Yi',m,(Q) dil , , [(2I'+I)(2L+I)]1/2 , = (I, miL, M, 1 ,m ) 47r(21 + I) (1, OIL, 0,1 ,0) (A.6) The special Clebsch-Gordan coefficient (I, OIL, 0, 1',0) is given by [Edm60] (I,OIL,O, 1',0) = v'2T+1 (_1)(I-L-I')/2 X [(J -2l)!(J -2L)!(J -2I')!] 1/2 (J + I)! (J/2)! x (J/2 _ l)!(J /2 _ L)!(J/2 _ I')! (A.7) The sum J = 1 + L + I' of the three angular momentum quantum numbers must be even. The Clebsch-Gordan coefficient (A.7) vanishes for odd J. Explicit expressions for the spherical harmonics up to I = 3 are given in Sect. 1.2.1 in Table 1.1. For further details see books on angular momentum in quantum mechanics, e.g. [Edm60,Lin84]. A.2 Laguerre Polynomials The generalized Laguerre polynomials L~(x), v = 0, 1, ... are polynomials of degree v in x. They are given by (A.8) A.3 Bessel Functions 303 and have v zeros in the range 0 < x < 00. The ordinary Laguerre polynomials LII(x) correspond to the special case 0 =O. In general 0 is an arbitrary real number greater than -1. The binomial coefficient in (A.S) is defined as follows for non-integral arguments: ( z) r(z+ 1) (A.9) y = r(y + 1) r(z - y + 1) . Here r is the Gamma function. It is defined by oo r(z + 1) = L e e-t dt (A. 10) and has the property r(z + 1) = zr(z) (A.11) For positive integers z = n we have r(n + 1) = n!. For half-integral z we can derive r(z) recursively from the value r(1/2) = ..fo via (A.ll). The orthogonality relation for the generalized Laguerre polynomials reads oo -% a a a d r(v+o+ 1) c L e X L,.(x)LII(x) X = , V,.,II. (A.12) o ~ The following recursion relation is very useful, because it enables the numerically efficient evaluation of the Laguerre polynomials for a given index 0: (v+1)L~+1(x) - (2v+o+1-x)L~(x) +(v+O)L~_l(X) = 0 , (A. 13) v = 1, 2, ... Note: The Laguerre polynomials defined by (A.S) correspond to the definitions in [AS70, GR65 und M066]. The Laguerre polynomials in [Mes70] contain an additional factor r(v+o+ 1). A.3 Bessel Functions The Bessel functions of order v are solutions of the following second-order differential equation: 2J2w dw 2 2 z dz2 + z dz + (z - v )w = 0 (A. 14) The ordinary Bessel function JII(z) is the solution which fulfills the following boundary condition near the origin z =0: J (z) %;0 (!Z)" (vf= -1, -2, -3, ...) (A. 15) II r(v + 1) , 304 Appendix As a power series in Z we have (Z)V 00 (_lz2)k Jv(z) = i L k!r(~+k+ 1) (A.I6) k=O For Izl --+ 00 the asymptotic form of Jv is Jv(z) Izl~oo ifcos [z - (v+ 4) i] (A.I7) From the Bessel function J v with the asymptotic behaviour (A.17) and a second solution of (A.14), which oscillates like a sine asymptotically, we can construct two linear combinations asymptotically proportional to exp ±i(z - ... ). They are called the first and second Hanke/ functions, H~l) and H~2). Their asymptotic form is Hv(1) (z) Izl-+oo= y;-;{2 exp {.+1 [z- ( v+i1) '27r]} ' (A.18) (2) Izl-+oo 1) Hv (z) = y;-;(2 exp {.-1 [z - ( v + '2 "27r]} Near z = 0 we have (for ~1I > 0) H~l)(z) = _H~2)(z) = -2.. (~(1I)~, z --+ 0, ~1I > 0 . (A.19) 7r i Z The modified Besse/functions Iv(z) of order 1I are connected to the ordinary Bessel functions by the simple relation iVIv(z)=Jv(iz), (-7r<argz~7r/2) . (A.20) They are hence solutions of the differential equation 2 J2w dw 2 2 Z - + z- - (z + 1I )w = 0 (A.21) dz2 dz ' and their behaviour for small Izl is, as for Jv, z-+o (!zV Iv(z) = r(lI+l) , (lIf-l,-2,-3, ...) (A.22) For Izl --+ 00 the asymptotic form of Iv is Izl~oo ~ I v ()z ;;;--- , (I arg(z) 1< 7r/2) (A.23) y27rz For non-integral values of 1I the modified Bessel functions Iv(z) and Lv(z) defined by (A.20), (A.16) are linearly independent, and there is a linear combi­ nation A.3 Bessel Functions 305 K ( ) = ~ Lv(z) - Iv(z) Z (A.24) v 2 S10.() V7r , which vanishes asymptotically, 1%1::'00 [7r -% K ( ) 37r/2) (A.25) v Z - V2z e , (Iargzl < The Bessel functions of half-integral order v = 1+ 1/2, 1 = 0, 1, ... play an important role as solutions of the radial SchrOdinger equation (1.74) with angular momentum quantum number 1in the absence of a potential. The connection to the radial SchrOdinger equation becomes clear when we write the equations (A.I4) and (A.2I) as differential equations for the function ¢(Z) = v'Z w(z) . (A.26) (A.I4) then becomes (with v = I+!) cP¢ _ 1(1 + 1) ¢ + ¢ = 0 (A.27) dz2 z2 ' and (A.2I) becomes cP¢ _ 1(1 + 1) ¢ _ ¢ =0 (A.28) dz2 z2 If we write z as kr (for E=h2k2/(21-') > 0) or as ",r (for E= -h2",2/(21-') <0), then (A.27) or (A.28) respectively is just the radial SchrOdinger equation (1.74) for V == o. For the modified Bessel function K '+1/ 2 of half-integral order 1+ 1/2 there is a series expansion K () [7r -% ~ (I + k)! -k 1+1/2 z = V2z e ~ k!(1 _ k)! (2z) . (A.29) The derivative of K'+l/2 can be expressed in terms of K'+1/2 and K ,-l/2, d 1+ 1 -d K ,+1/2(Z) = ___2 K ,+1/2(Z) - K ,- 1/2(Z) (A.30) z z The spherical Bessel function j,(z) is defined as j/(z) = IfJ ,+l/2(Z) . (A.3I) For small z we have, according to (A.I5), • %--+0 zl J/(Z) = (21 + I)!! (A.32) 306 Appendix and asymptotically according to (A.l7) . Izl-'oo. ( 7r) ZJl(z) = SIll Z -12 (A.33) With (A.26) we see that zj,(z) is a solution of (A.27), the radial SchrOdinger equation for positive energy. The linearly independent solution, which differs asymptotically from (A.33) in that the sine is replaced by a cosine, is Z n,(z), where n, is the spherical Neumann function, (A.34) For the derivatives of the spherical Bessel functions we have the simple formula d.() .
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