"Hermite FunctionsHERMITEFUNCTIONSHERMITE DIFFERENTIAL EQUATIONThe Hermite Differential Equation is 2 d y dy? 2x ? 2p y ? 0 ……….(1)2 dx dx Let us assume that the solution to Eq. (1) is represented by the power series ? k ?r k k ?1 k ?2 k ?3 k ?4 k ?r ….(2)y ? a x ? a x ? a x ? a x ? a x ? a x ? ... ? a x ? ... r 0 1 2 3 4 r ? r ?0 Thusdy k ?1 k k ?1 k ?r ?1 ? a kx ? a ?k ? 1 ?x ? a ?k ? 2 ?x ?......... ? a ?k ? r ?x ?......... o 1 2 r dx dy k k ?1 k ?2 k ?r ? ? ? ? ? ? x ?a kx ? a k ?1 x ? a k ? 2 x ?......... ? a k ? r x ?......... o 1 2 r dx 2 d y k ?2 k ?1 k ?a k ?k ?1 ?x ?a ?k ?1 ?kx ? a ?k ? 2 ? ?k ?1 ?x ? ........ o 1 2 2 dx k ?r ?2 ? a ?k ? r ? ?k ? r ?1 ?x ?........... rPutting in Eq. (1), we have k ?2 k ?1 k k ?r ?2 ? ? ? ? ? ? ? ? ? ? ? ? a k k ?1 x ? a k ?1 kx ? a k ? 2 k ?1 x ? ... ? a k ? r k ? r ?1 x ? .... o 1 2 r k k ?1 k ?2 k ?r ? 2a kx ? 2a ?k ?1 ?x ? 2a ?k ? 2 ?x ?......... ? 2a ?k ? r ?x ?........ o 1 2 r k k ?1 k ?2 k ?3 k ?r ? 2pa x ? 2pa x ? 2pa x ? 2pa x ? .. ? 2pa x ? ....... ?0 o 1 2 3 rork ?2 k ?1 k ?a k ?k ?1 ? ?x ? ?a ?k ?1 ?k ?x ? ?a ?k ? 2 ? ?k ?1 ? ? 2a k ? 2 pa ?x ? .... o 1 2 o o k ?r ? ?a ?k ? r ? 2 ? ?k ? r ?1 ? ? 2a ?k ? r ? ? 2 pa ?.x ? ...... ?0 r ?2 r r Equating the coefficient of each power of x to zero, we getk ??a k ?k ?1 ? = 0(coefficient of x )o Asa ? 0, soo Mathematical Methods of Physics Page 46 Hermite Functions k ?k ?1 ? = 0So thatk ? 0 ?? or k ? 1 ? k ?? a ?k ?1 ?k = 0(coefficient of x )1 For? k ? 1, a ? 0 1? ? and k ? 1 a mayormaynotbezero 1 ? k a ?k ? 2 ? ?k ?1 ? ? 2a k ?2 pa = 0(coefficient of x )2 o o ?k ? p ?a ? 2a2 o ?k ? 2 ? ?k ?1 ? ...k+r a ?k ? r ? 2 ? ?k ? r ?1 ? ? 2a ?k ? r ? ? 2 pa = 0(coefficient of x )r ?2 r r ?k ? r ? p ? a ? 2ar ?2 r ?k ? r ? 2 ? ?k ? r ?1 ? For k = 0?r ? p ? a ? 2a……………..(3)r ?2 r ?r ? 2 ? ?r ?1 ? For k = 1?r ?1 ? p ? a ? 2a……………..(4)r ?2 r ?r ? 2 ? ?r ? 3 ? Case IWhen k = 0, a may or may not be zero.Taking a = 0,we have from 1 1 Eq. (3)?r ? p ?a ? 2a r ?2 r ? ? ? ? r ? 2 r ?1 For r = 0Mathematical Methods of Physics Page 47 Hermite Functions? ? p ?a ? 2a2 o 1.2 For r = 1?1 ? p ?a ? 2a3 1 2.3 Sincea = 0, so a = 0. Likewise a = a =…………=0.1 3 5 7 For r = 2?2 ? p ? ? ? p ? ?2 ? p ? 2a ? 2a ? 2 a4 2 o 3.4 4! For r = 4?4 ? p ? ? ? p ? ?2 ? p ? ?4 ? p ? 3a ? 2a ? 2 a6 4 o 5.6 6 ! And in general ?n ? p ?a ? 2a(n – even)n ?2 n ?n ? 2 ? ?n ?1 ? The series solution to the Hermite equation when k = 0 is therefore 2 4 6 y ? a ? a x ? a x ? a x ...................................o 2 4 62 3 2 ? ? p ? 2 ? ? p ? ?2 ? p ? 2 ? ? p ? ?2 ? p ? ?4 ? p ? 2 4 6 y ? a ? a x ? a x ? a x ? ..............(5) o o o o 1.2 4! 6 ! Since in this solution x has even powers, we call it as y (x).even Case IIWhen k = 1, a = 0.Taking a = 0.FromEq. (4)1 1 ?r ?1 ? p ?a ? 2ar ?2 r ?r ? 2 ? ?r ? 3 ? For r = 0?1 ? p ? ?1 ? p ? 1a ? 2a ?2 a2 o o 2.3 3! For r = 1Mathematical Methods of Physics Page 48 Hermite Functions?2 ? p ?a ? 2a3 1 3.4 Sincea = 0, so a = 0. Likewise a = a =…………=0.1 3 5 7 For r = 2?3 ? p ? ?1 ? p ? ?3 ? p ? 2a ? 2a ? 2 a4 2 o 4.5 5 ! For r = 4?5 ? p ? ?1 ? p ? ?3 ? p ? ?5 ? p ? 3a ? 2a ? 2 a6 4 o 6.7 7 ! And in general ?n ?1 ? p ?a ? 2a(n – even)n ?2 n ?n ? 2 ? ?n ? 3 ? The series solution to the Hermite equation when k = 1 is therefore 3 5 7 y ? a x ? a x ? a x ? a x ...................................o 2 4 6? ? ?1 ? p ? ?1 ? p ? ?3 ? p ? ?1 ? p ? ?3 ? p ? ?5 ? p ? 1 3 2 4 3 7 y ? a x ? 2 x ? 2 x ? 2 x ? ....... .......(6) o ? ? 3! 5! 7 ! ? ? Since in this solution x has odd powers, we call it as y (x).odd For p = 0,y (x) = a ,and y (x) series diverges.even o odd For p = 1,y (x) = a x, and y (x) series diverges.odd o even 4 ? ? 2 For p = 2,y (x) = a ,and y (x) series diverges.even o 1 ? x odd ? ? 2 ? ? 4 ? ? 3 For p = 3,y (x) = a ,and y (x) series diverges.odd o x ? x even ? ? 6 ? ? and so on.Thus all convergent solutions are polynomial in x and are called HermitePolynomials if a is selected so that the coefficient of the highest power of xo Mathematical Methods of Physics Page 49 Hermite Functionsn in each solutionequals 2 . Withthis condition, the solution y (x) is calledn Hermite PolynomialH (x):n o 0 For p = 0,y = a x ; a = 2 = 1 ? H (x) = 1 o o o o 1 For p = 1,y = a x; a = 2 = 2? H (x) = 2x1 o o 1 2 2 2 For p = 2,y = a ; -2a = 2 , ? a = 2 ?H (x)=?1 ? 2x ? ?4x ? 2 ? 2 o o o 2 2 2 3 ? ? 3 For p = 3,y = a ;- a = 2 ,? a = ??123 o x ? x o o ? ? 3 3 ? ? 2 3 ? ? 3? H (x) = 12 H (x) ? ?8x ?12x ?3 x ? x 3? ? 3 ? ? 4 4 ? ? 4 2 4 For p = 4, y = a ;a = 2 ,? a = ?124 o 1- 4 x ? x o o ? ? 3 3 ? ? 4 ? ? 2 4 4 2? H (x) = 12 H (x) =4 1- 4 x ? x 4 ?16 x 48 x ?12 ? ? ? 3 ? ? and so on.HERMITE FUNCTIONSThe Hermite DE is of the form2 d y dy ? 2x ? 2 p y ? 0…….(7)2 dx dxwhere p is a parameter.Consider an equation that is closely related to the Hermite equation2 d ? 2 ? ? ? ? x ? ? ? 0 ……(8)2 dxLet2 ?x / 2 ? ? e y…….(9)Then2 2 d ? dy ?x / 2 ?x / 2 ? e ? xyedx dx Mathematical Methods of Physics Page 50 Hermite Functions2 2 2 2 2 2 d ? d y dy ? ? ?x / 2 ?x / 2 ?x / 2 2 ?x / 2 and ? e ?2xe ? e ? x e y? ? 2 2 dx ? ? dx dx Putting in Eq. (7), 2p =? – 1, we have 2 d y dy ? 2x ? ? ? ?1 ?y ? 0 …….(10) 2 dx dx Moreover, using Eq. (9), Eq. (8) becomes2 2 2 2 2 2 d y dy ? ? ?x / 2 ?x / 2 ?x / 2 2 ?x / 2 2 ?x / 2 e ?2xe ? e ? x e y ? ? ? ? x ?e y ? 0? ? 2 dx ? ? dx 2 ? ? 2 d y dy ?x / 2 2 2 e ?2x ? ?1 ? x ?y ? ? ? ? x ?y ? 0? ? 2 dx dx ? ? ? ? 2 d y dyOr? 2x ? ? ? ?1 ?y ? 0 2 dx dx which is similar to Eq. (10). Hence the general solution to Eq. (8) is 2 ?x / 2 ? ? e yHence if ? is of the form1 + 2n,withn > 0, then the solution of Eq. (8) willbe a constant multiple of the function ? (x) so that n 2 ?x / 2 ? (x) ? e H (x)n n where H (x) is the Hermite polynomial of degree n and? (x) is the Hermiten n function of order n.GENERATING FUNCTION AND RODRIGUE’S FORMULA FORHERMITE POLYNOMIALSThe generating function for the Hermite polynomials is? n 2 t 2xt ? t g(x, t) ? e ? H ?x ? ………..(11)? n n! n ?0 n Hermitepolynomials are the coefficients of t in the expansion of this equation.Mathematical Methods of Physics Page 51 Hermite FunctionsRewriting Eq. (10) as 2 3 n 2 2 t t t t x ?(x ? t) g(x, t) ?e e ? H ?x ? ? H ?x ? ? H ?x ? ? H ?x ? ? ........ ? H ?x ? o 1 2 3 n 1! 2! 3! n! Orn n ? ? ? 2 ? 2 ? g(x, t) ? H ?x ? ?(x ? t) x n ? e e ? n! ? H (x)….(12)? ? ? ? n n n n! ? ?t ? ? ?t ? ? ? ? ? t ?0 t ?0 ? ? Puttingt = x – p so that for t = 0,x = p,and ? ??t ?p Andn n n 2 2 n 2 ? ? ? ? ? ? ? ?(x ? t) ? p ? ( ?1) ? pe ? e e? ? ? ? n n n ? ? ? ? ?t ?t ?p n n n 2 2 2 ? ? d ? ? ? ? ? ? ?(x ? t) n ?x n ?xe ? ( ?1) e ? ( ?1) e….(13)? ? ? ? ? ? n n n ? ? ? ? ? ? ?t ?x dx From (12) and (13), we have n 2 2 d ? ? x n ?x H (x) ?e ( ?1) e….(14)? ? n n ? ? dx Eq. (14) is called Rodrigue’s formula and enables us to calculate Hermitepolynomials of various degrees. H (x) ? 1o 2 2 d ? ? x 1 ?x H (x) ? e ( ?1) e ? 2x? ? 1 dx ? ? 2 2 2 d ? ? x 2 ?x 2 H (x) ? e ( ?1) e ? ?4x ?2 ?? ? 2 2 ? ? dx 3 2 2 d ? ? x 3 ?x 3 ? ?H (x) ? e ( ?1) e ? 8x ?12x ? ? 3 3 ? ? dx Mathematical Methods of Physics Page 52 Hermite Functions4 2 2 d ? ? x 4 ?x 4 2 and so on.H (x) ? e ( ?1) e ? ?16x ?8x ?12 ? ? ? 4 4 ? ? dx RECURRENCE FORMULAE FOR H (x)n ? H ?x ? ? 2n H (x) ….(15)n n ?12xH (x) ? 2nH (x) ? H (x) ….(16)n n ?1 n ?1 ? H ?x ? ? 2xH (x) ? H (x)….(17)n n n ?1 Proof: ? H ?x ? ? 2n H (x)n n ?1 The generating function for Hermite polynomial is ? n 2 t 2xt ? t e ? H ?x ?n ? n! n ?0 Differentiating with respect to x? n 2 t 2xt ? t 2t e ? H ? ?x ?n ? n! n ?0 ? ? n n t t 2t H ?x ? ? H ? ?x ?n n ? ? n! n! n ?0 n ?0 ? ? n ?1 n t t 2 H ?x ? ? H ? ?x ?n n ? ? n! n! n ?0 n ?0 Replacing n by n – 1 on LHS,? ? ? H ?x ? H ?x ? n ?1 n n n2 t ? t? ? ?n ?1 ?! n! n ?0 n ?0 n Comparing the coefficients of t on both sides? ? ? H (x) H x n ?1 n 2 ??n ?1 ?! n! Mathematical Methods of Physics Page 53 Hermite Functionsn! H (x) n ?1 ? 2 ? H ?x ?n ?n ?1 ?! ? H ?x ? ? 2n H (x)n n ?12xH (x) ? 2nH (x) ? H (x)n n ?1 n ?1 The generating function for Hermite polynomial is ? n 2 t 2xt ? t e ? H ?x ?? n n! n ?0 Differentiating with respect to t? 2 nH ?x ? 2xt ? t n n ?1 2 ?x ?t ?e ? t? n! n ?0 ? 2 H ?x ? 2xt ? t n n ?1 2 ?x ?t ?e ? t ?n ?0 is meaningles s ?? ?n ?1 ?! n ?1 ? ? n t H ?x ? n n ?12 ?x ?t ? H ?x ? ? tn ? ? n! ?n ?1 ?! n ?0 n ?1 ? ? ? H ?x ? H ?x ? H ?x ? n n n n ?1 n n ?1 2x t ? 2 t ? t? ? ? n! n! ?n ?1 ?! n ?0 n ?0 n ?1 n Comparing the coefficients of t on both sidesH ?x ? H ?x ? H ?x ? n n ?1 n ?1 2x ? 2 ?n! ?n ?1 ?! n ! 2xH ?x ? ? 2n H ?x ? ? H ?x ?n n ?1 n ?12xH ?x ? ? 2n H ?x ? ? H ?x ? n n ?1 n ?1 H ? ?x ? ? 2xH (x) ? H (x)n n n ?1 Putting (16) in (15), we have Mathematical Methods of Physics Page 54 Hermite Functions?2xH (x) ? H (x) ? H (x)n n ?1 nH ? (x) ? 2xH (x) ? H (x) Hence the proof.n n n ?1 n Comparing the coefficients of t on both sidesH (x) H ? ?x ? n ?1 n 2 ??n ?1 ?! n! n! H (x) n ?1 ? 2 ? H ?x ?n ?n ?1 ?! ? H ?x ? ? 2n H (x)n n ?1 RECURRENCE FORMULAE FOR? (x)n2n ? (x) ? x ? (x) ? ? ? (x)….(18)n ?1 n n 2x ? (x) ? 2n ? (x) ? ? (x) ….(19)n n ?1 n ?1 ? ? (x) ? x ? (x) ? ? (x) ….(20)n n n ?1 Proof: UsingHermite function 2 ?x / 2? (x) ? e H (x)n n Differentiating with respect to x2 2 ? ?x / 2 ?x / 2 ? (x) ? e H ? (x) ? xe H (x)n n nUsing Eq. (15)2 2 ? ?x / 2 ?x / 2 ? (x) ? e 2n H (x) ? xe H (x)n n ?1 n ? ? (x) ? 2n ? (x) ? x ? (x)n n ?1 n or ? 2n ? (x) ? x ? (x) ? ? (x)n ?1 n nMathematical Methods of Physics Page 55 "