# Stark - H400=R40 Y00,410=R41Y10,420=R42Y10,430=R43 Y30 the...

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H 400 = R 40 Y 00 , ψ 410 = R 41 Y 10 , ψ 420 = R 42 Y 10 , ψ 430 = R 43 Y 30, the spherical harmonics are * L SphericalHarmonicY @ 0, 0, θ , ϕ D ; SphericalHarmonicY @ 1, 0, θ , ϕ D ; SphericalHarmonicY @ 2, 0, θ , ϕ D ; H ** L n = 4;l = 0;m = 0; ψ 400 = 2 l + 1 expe - r n * a n - 1 - l r a l H n - 1 - l L ! H n + 1 L ! LaguerreL B n - l - 1, 2 l + 1, 2 r n * a F SphericalHarmonicY @ l, m, θ , φ D ; A = 1 Ÿ 0 2 π Ÿ 0 π Ÿ 0 ψ 400 * ψ 400 * r * r * Sin @ θ D diffd r diffdθ diffdφ ; ψ 400 = FullSimplify B Expand B 2 l + 1 expe - r n * a n - 1 - l r a l H n - 1 - l L ! H n + 1 L ! LaguerreL B n - l - 1, 2 l + 1, 2 r n * a F SphericalHarmonicY @ l, m, θ , φ D * A FF ; n = 4;l = 1;m = 0; ψ 410 = 2 l + 1 expe - r n * a n - 1 - l r a l H n - 1 - l L ! H n + 1 L ! LaguerreL B n - l - 1, 2 l + 1, 2 r n * a F SphericalHarmonicY @ l, m, θ , φ D ; A = 1 Ÿ 0 2 π Ÿ 0 π Ÿ 0 ψ 410 * ψ 410 * r * r * Sin @ θ D diffd r diffdθ diffdφ ; ψ 410 = FullSimplify B Expand B 2 l + 1 expe - r n * a n - 1 - l r a l H n - 1 - l L ! H n + 1 L ! LaguerreL B n - l - 1, 2 l + 1, 2 r n * a F SphericalHarmonicY @ l, m, θ , φ D * A FF ;

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n = 4;l = 2;m = 0; ψ 420 = 2 l + 1 expe - r n * a n - 1 - l r a l H n - 1 - l L ! H n + 1 L ! LaguerreL B n - l - 1, 2 l + 1, 2 r n * a F SphericalHarmonicY @ l, m, θ , φ D ; A = 1 Ÿ 0 2 π Ÿ 0 π Ÿ 0 ψ 420 * ψ 420 * r * r * Sin @ θ D diffd r diffdθ diffdφ ; ψ 420 = FullSimplify B Expand B 2 l + 1 expe - r n * a n - 1 - l r a l H n - 1 - l L ! H n + 1 L ! LaguerreL B n - l - 1, 2 l + 1, 2 r n * a F SphericalHarmonicY @ l, m, θ , φ D * A FF ; n = 4;l = 3;m = 0; ψ 430 = 2 l + 1 expe - r n * a n - 1 - l r a l H n - 1 - l L !
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