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6.2: Хвильові функції жорсткого ротатора називаються сферичними гармоніками
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Розв'язками рівняння атома водню Шредінгера є функціями, які є добутком сферичної гармонічної функції та радіальної функції.
5.11: Многочлени Лежандра
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У цьому розділі ми розглянемо достатньо про многочлени Лежандра, щоб бути корисними в наступному розділі.
1.14: Многочлени Лежандра
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Ви виявите, що коефіцієнт of $r^l$ - це поліноміальний вираз в $x$ градусі $l$ . $(1 - 2rx + r^2 )^{-1/2} = P_0 (x) + P_1 (x) r + P_2 (x) r^2 + P_3 (x) r^3 ... \label{1.14.2} \tag{1.14.2}$ Коефіці...Ви виявите, що коефіцієнт of $r^l$ - це поліноміальний вираз в $x$ градусі $l$ . $(1 - 2rx + r^2 )^{-1/2} = P_0 (x) + P_1 (x) r + P_2 (x) r^2 + P_3 (x) r^3 ... \label{1.14.2} \tag{1.14.2}$ Коефіцієнти послідовної степені $r$ - це многочлени Лежандра; коефіцієнт $r^l$ , який є $P_l(x)$ , є поліномом Лежандра порядку $l$ , і він є поліномом у $x$ включенні членів настільки ж високим, як $x^l$ . $P_{l+1} = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 - 1)^l . \label{1.14.5} \tag{1.14.5}$
11.2: Властивості многочленів Лежандра
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\[\begin{aligned} { \frac{d^{n+1}}{dx^{n+1}}\left[ (x^{2}-1) \frac{d}{dx} (x^{2}-1)^{n} - 2 n x (x^{2}-1)^{n}\right]} &= n(n+1) \frac{d^{n}}{dx^{n}}(x^2-1)^n + 2(n+1) x \frac{d^{n+1}}{dx^{n+1}} (x^2-1...\[\begin{aligned} { \frac{d^{n+1}}{dx^{n+1}}\left[ (x^{2}-1) \frac{d}{dx} (x^{2}-1)^{n} - 2 n x (x^{2}-1)^{n}\right]} &= n(n+1) \frac{d^{n}}{dx^{n}}(x^2-1)^n + 2(n+1) x \frac{d^{n+1}}{dx^{n+1}} (x^2-1)^n+(x^2-1) \frac{d^{n+2}}{dx^{n+2}} (x^2-1)^n \nonumber\\ &-2n(n+1) \frac{d^{n}}{dx^{n}}(x^2-1)^n - 2n x \frac{d^{n+1}}{dx^{n+1}} (x^2-1)^n \nonumber\\ &=-n(n+1) \frac{d^{n}}{dx^{n}}(x^2-1)^n + 2 x \frac{d^{n+1}}{dx^{n+1}} (x^2-1)^n+(x^2-1) \frac{d^{n+2}}{dx^{n+2}} (x^2-1)^n \nonumber\\ &= -\left[…
7.2: Многочлени Лежандра
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$(x-t) \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\sum_{n=0}^{\infty} n P_{n}(x) t^{n-1}-\sum_{n=0}^{\infty} 2 n x P_{n}(x) t^{n}+\sum_{n=0}^{\infty} n P_{n}(x) t^{n+1} . \nonumber$ \[\sum_{n=0}^{\infty} n... $(x-t) \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\sum_{n=0}^{\infty} n P_{n}(x) t^{n-1}-\sum_{n=0}^{\infty} 2 n x P_{n}(x) t^{n}+\sum_{n=0}^{\infty} n P_{n}(x) t^{n+1} . \nonumber$ $\sum_{n=0}^{\infty} n P_{n}(x) t^{n-1}-\sum_{n=0}^{\infty}(2 n+1) x P_{n}(x) t^{n}+\sum_{n=0}^{\infty}(n+1) P_{n}(x) t^{n+1}=0 \label{7.21}$ $\int_{-1}^{1} x^{m} P_{n}(x) d x=\dfrac{1}{2^{n} n !} \int_{-1}^{1} x^{m} \dfrac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n} d x . \nonumber$