Legendre polinomlari

§4.3. Rodrigues formulasi Ta’rif (80)-bo’yicha Hn(x) = d n dtn e −t 2+2xt t=0 . Shu formulani qulai ko’rinishga keltirish uchun e −t 2+2xt = e −(t−x) 2+x 2 deb olamiz, unda Hn(x) = d n dtn e −t 2+2xt t=0 = e x 2 d n dtn e −(t−x) 2 t=0 = = e x 2 (−1)n d n dxn e −(t−x) 2 t=0 = (−1)n e x 2 d n dxn e −x 2 (84) formulaga kelamiz. Bu - Hermite polinomlari uchun Rodrigues formulasi. 1.23-mashq. Rodrigues formulasidan foydalanib Hn(x) ni n = 0, 1, 2 lar uchun toping va ularni (81)-formulalar bilan solishtiring. §4.4. Differensial tenglama (83)-ni (82)-ga olib borib qo’yamiz va hosil bo’lgan munosabatdan x bo’yicha hosila olamiz: Hn+1(x) = 2xHn(x) − H ′ n (x) ⇒ H ′ n+1(x) = 2Hn(x) + 2xH′ n (x) − H ′′ n (x). Bu tenglikning chap tomonida (83)-ni yana bir marta ishlatsak H ′′ n (x) − 2xH′ n (x) + 2nHn(x) = 0 (85) tenglamaga kelamiz. Bu - Hermite tenglamasi.
§4.5. Hermite polinomlarining ortogonalligi va normasi Quyidagi munosabat o’z-o’zidan oydindir: e −x 2 g(x, t)g(x, s) = e −x 2 e −t 2+2xte −s 2+2xs = ∑ ∞ n=0 ∑ ∞ m=0 e −x 2 Hn(x)Hm(x) t n s m n! m! . Qulay ko’rinishga keltiraylik: e −x 2 e −t 2+2xte −s 2+2xs = e −(x−(s+t))2+2st . 36 Chap va o’ng tomondan x bo’yicha integral olamiz: ∫ ∞ −∞ dx e−(x−(s+t))2+2st = ∑ ∞ n=0 ∑ ∞ m=0 t n s m n! m! ∫ ∞ −∞ dx e−x 2 Hn(x)Hm(x). Chap tomondagi integral oson topiladi: √ πe2st = ∑ ∞ n=0 ∑ ∞ m=0 t n s m n! m! ∫ ∞ −∞ dx e−x 2 Hn(x)Hm(x). Eksponentaning ta’rifi bo’yicha e 2st = ∑ ∞ n=0 (2st) n n! . Demak, ∫ ∞ −∞ dx e−x 2 Hn(x)Hm(x) = { 0, n ̸= m; 2 nn! √ π, n = m. (86) Kvant mexanikasida garmonik ossillator masalasini yechganimizda to’lqin funksiya Hermite polinomlari orqali ifodalanadi: ψn(x) = e −x 2/2Hn(x) √ 2 nn! √ π . (87) Yuqoridagi formula bilan solishtirsak, ∫ ∞ −∞ dx ψn(x)ψm(x) = δnm (88) ekanligini ko’ramiz. 1.6-misol. Yuqorida aytganimizdek Hermite polinomlari chiziqli ossillatorning kvant analizida uchraydi. Bir o‘lchamli Schr¨odinger tenglamasi − h¯ 2 2m d 2ψ(x) dx2 + [U(x) − E] ψ(x) = 0 ga U(x) = 1 2 kx2 potensialni kiritamiz. Bunday potensial F = −U ′ (x) = −kx chiziqli qaytaruvchi kuchga olib keladi. Bu tenglamada k = mω2 √ va ξ = mω/hx¯ almashtirishlar bajarilsa, Schr¨odinger tenglamasi ψ ′′(ξ) + ( 2E hω¯ − ξ 2 ) ψ(ξ) = 0 37 ko‘rinishga keladi. Olingan tenglamada ψ(ξ) = e −ξ 2/2H(ξ) almashtirish bajarilsa quyidagini olamiz: H ′′(ξ) − 2ξH′ (ξ) + ( 2E hω¯ − 1 ) H(ξ) = 0. (89) Hosil bo‘lgan tenglamaning yechimini Frobenius metodi bo‘yicha qidiramiz: H(ξ) = ∑ n cnξ n = c0 + c1ξ + c2ξ 2 + · · · Qulaylik uchun a = 2E/(¯hω) − 1 belgilash kiritilsa, cn koeffisientlar uchun quyidagi rekurrent munosabat kelib chiqadi: cn+2 = a − 2n (n + 1)(n + 2)cn. |cn/cn+2| nisbat katta n larda cheklangan emas, demak, bu cheksiz qator yaqinlashuvchi bo‘lmaydi. Shuning uchun qatorni a = 2n tanlash asosida ntartibli polinomga aylantiramiz. Bu esa birinchidan, (89)-tenglamani Hermite tenglamasi (85)-ga aylantiradi, ikkinchidan kvantlangan ossillatorning yaxshi ma’lum bo‘lgan energetik sathlarini beradi: En = ( n + 1 2 ) hω.