Anexo:Funciones Hidrogenoides (No relativistas)

En este artículo se muestran, la definición de función hidrogenoide, sus componentes, las ecuaciones de las que provienen, así como una representación que facilita su obtención. Además este artículo contiene una tabla de las primeras funciones hidrogenoides hasta n=5.

Función hidrogenoide.

Una función $\Psi$ es hidrogenoide si es solución de la ecuación de Schrödinger estacionaria para el caso del átomo hidrogenoide. La cual es:

$-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\Psi (r,\theta ,\phi )+{\frac {Ze}{4\pi \varepsilon _{0}r}}\Psi (r,\theta ,\phi )=E\Psi (r,\theta ,\phi )$

donde:

• $\hbar$ es la constante de Planck reducida,
• $\mu$ es la masa reducida del electrón,
• $e$ es la carga del electrón,
• $Z$ es el número atómico,
• $\varepsilon _{0}$ es la permitividad eléctrica del vacío,
• $r,\theta ,\phi$ son las coordenadas esféricas,
• $E$ es la energía asociada a la función $\Psi$.

De esta ecuación (aplicando separación de variables) se obtiene que existen infinitas funciones que son solución y que pueden ser escritas de la siguiente manera:

$\Psi _{nlm}(r,\theta ,\phi )=R_{nl}(r)Y_{l}^{m}(\theta ,\phi )$

donde $R_{nl}$ son las soluciones radiales , $Y_{l}^{m}$son las soluciones angulares conocidas como armónicos esféricos y $n,l,m$ son constantes de sepración conocidas como números cuánticos.Estos números están relacionados entre sí de la siguiente manera:

$n=1,2,3,...$

$l=0,1,2,...,n-1$

$m=-l,-l+1,...,0,...,l-1,l$

Funciones radiales.

Las funciones radiales cumplen la siguiente ecuación, derivada de la ecuación de Schrödinger:

$\{-r^{2}\hbar ^{2}[{\frac {d^{2}}{dr^{2}}}+{\frac {2}{r}}{\frac {d}{dr}}]+2\mu r^{2}[V(r)-E_{n}]\}R(r)=-\hbar ^{2}l(l+1)R(r)$

Solucionando esta ecuación se obtiene una representación de las funciones radiales caracterizadas por los números cuánticos $n,l$ , la cual es:

$R_{nl}(r)=-({\frac {Z}{a}})^{l+{\frac {3}{2}}}{\sqrt {({\frac {2}{n}})^{2l+3}[{\frac {(n-l-1)!}{2n(\{n+l\}!)^{3}}}]}}e^{-{\frac {Zr}{na}}}r^{l}{\frac {d^{2l+1}}{dx^{2l+1}}}[e^{x}{\frac {d^{n+l}}{dx^{n+l}}}(x^{n+l}e^{-x})]|_{x={\frac {2Zr}{na}}}$

donde:

• $Z$ es el número atómico.
• $a=5.291\times 10^{-11}m$ es una constante conocida como radio de Bohr.

Esta representación es útil para facilitar su obtención.

Ejemplo.

Obtener la función radial $R_{32}(r)$.

$R_{32}(r)=-({\frac {Z}{a}})^{\frac {7}{2}}{\sqrt {({\frac {2}{3}})^{7}[{\frac {1!}{6(5!)^{3}}}]}}e^{-{\frac {Zr}{3a}}}r^{2}{\frac {d^{5}}{dx^{5}}}[e^{x}{\frac {d^{5}}{dx^{5}}}(x^{5}e^{-x})]|_{x={\frac {2Zr}{3a}}}$

$R_{32}(r)={\frac {1}{2430{\sqrt {30}}}}({\frac {Z}{a}})^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}{\frac {d^{5}}{dx^{5}}}[e^{x}(-e^{-x}x^{5}+25e^{-x}x^{4}-200e^{-x}x^{3}+600e^{-x}x^{2}-600e^{-x}x+120e^{-x})]|_{x={\frac {2Zr}{3a}}}$

$R_{32}(r)={\frac {1}{2430{\sqrt {30}}}}({\frac {Z}{a}})^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}{\frac {d^{5}}{dx^{5}}}(-x^{5}+25x^{4}-200x^{3}+600x^{2}-600x+120)|_{x={\frac {2Zr}{3a}}}$

$R_{32}(r)={\frac {1}{2430{\sqrt {30}}}}({\frac {Z}{a}})^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}(120)$

$R_{32}(r)={\frac {4}{81{\sqrt {30}}}}({\frac {Z}{a}})^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}$

Armónicos Esféricos.

Las componentes angulares de la funciones hidrogenoides (conocidas como armónicos esféricos) son soluciones simultáneas de las siguientes ecuaciones.

$-\hbar ^{2}({\frac {\partial ^{2}}{\partial \theta ^{2}}}+cot\theta {\frac {\partial }{\partial \theta }}+{\frac {1}{sin^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}})Y_{l}^{m}(\theta ,\phi )=-\hbar ^{2}l(l+1)Y_{l}^{m}(\theta ,\phi )$

$-i\hbar {\frac {\partial Y_{l}^{m}(\theta ,\phi )}{\partial \phi }}=\hbar mY_{l}^{m}(\theta ,\phi )$

De donde se obtiene que estas funciones admiten la siguiente representación.

$Y_{l}^{m}(\theta ,\phi )={\sqrt {{\frac {2l+1}{4\pi }}{\frac {(l-|m|)!}{(l+|m|)!}}}}e^{im\phi }{\frac {sen\theta ^{|m|}}{2^{l}l!}}{\frac {d^{l+|m|}}{dx^{l+|m|}}}[(x^{2}-1)^{l}]|_{x=cos\theta }$

Esta representación de los armónicos esféricos es útil únicamente para fines de la mecánica cuántica y resulta imprecisa para otros problemas como problemas electrostáticos o de propagación ondas mecánicas.

Ejemplo.

Obtener el armónico esférico $Y_{2}^{-1}$.

$Y_{2}^{-1}(\theta ,\phi )={\sqrt {{\frac {5}{4\pi }}{\frac {1!}{3!}}}}e^{-i\phi }{\frac {sin\theta }{2^{2}2!}}{\frac {d^{3}}{dx^{3}}}[(x^{2}-1)^{2}]|_{x=cos\theta }$

$Y_{2}^{-1}(\theta ,\phi )={\frac {1}{2}}{\sqrt {\frac {5}{6\pi }}}e^{-i\phi }{\frac {sin\theta }{8}}{\frac {d^{3}}{dx^{3}}}(x^{4}-2x^{2}+1)|_{x=cos\theta }$

$Y_{2}^{-1}(\theta ,\phi )={\frac {1}{2}}{\sqrt {\frac {5}{6\pi }}}e^{-i\phi }{\frac {sin\theta }{8}}(24x)|_{x=cos\theta }={\frac {1}{2}}{\sqrt {\frac {5}{6\pi }}}e^{-i\phi }{\frac {sin\theta }{8}}(24cos\theta )$

$Y_{2}^{-1}(\theta ,\phi )={\frac {1}{2}}{\sqrt {\frac {5\cdot 9}{6\pi }}}e^{-i\phi }sin\theta cos\theta$

$Y_{2}^{-1}(\theta ,\phi )={\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}e^{-i\phi }sin\theta cos\theta$

Funciones Hidrogenoides hasta n=5.

En este apartado se muestran todas las funciones hidrogenoides posibles para valores de n entre 1 y 5.

n=1.

$\Psi _{100}(r,\theta ,\phi )={\frac {1}{\sqrt {\pi }}}{\biggl (}{\frac {Z}{a}}{\biggr )}^{\frac {3}{2}}e^{-{\frac {Zr}{a}}}$

n=2.

$\Psi _{200}(r,\theta ,\phi )={\frac {1}{\sqrt {\pi }}}\left({\frac {Z}{2a}}\right)^{\frac {3}{2}}{\biggl (}1-{\frac {Zr}{2a}}{\biggr )}e^{-{\frac {Zr}{2a}}}$

$\Psi _{21-1}(r,\theta ,\phi )={\frac {1}{8{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}re^{-{\frac {Zr}{2a}}}sin\theta e^{-i\phi }$

$\Psi _{210}(r,\theta ,\phi )={\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}re^{-{\frac {Zr}{2a}}}cos\theta$

$\Psi _{211}(r,\theta ,\phi )={\frac {1}{8{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}re^{-{\frac {Zr}{2a}}}sin\theta e^{i\phi }$

n=3.

$\Psi _{300}(r,\theta ,\phi )={\frac {1}{81{\sqrt {3\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {3}{2}}\left[27-18{\frac {Zr}{a}}+2{\frac {Z^{2}r^{2}}{a^{2}}}\right]e^{-{\frac {Zr}{3a}}}$

$\Psi _{31-1}(r,\theta ,\phi )={\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}e^{-{\frac {Zr}{3a}}}r{\biggl (}6-{\frac {Zr}{a}}{\biggr )}e^{-i\phi }sin\theta$

$\Psi _{310}(r,\theta ,\phi )={\frac {1}{81}}{\sqrt {\frac {2}{\pi }}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}e^{-{\frac {Zr}{3a}}}r{\biggl (}6-{\frac {Zr}{a}}{\biggr )}cos\theta$

$\Psi _{311}(r,\theta ,\phi )={\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}e^{-{\frac {Zr}{3a}}}r{\biggl (}6-{\frac {Zr}{a}}{\biggr )}e^{i\phi }sin\theta$

$\Psi _{32-2}(r,\theta ,\phi )={\frac {1}{108{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}e^{-2i\phi }sen^{2}\theta$

$\Psi _{32-1}(r,\theta ,\phi )={\frac {1}{54{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}e^{-i\phi }sen\theta cos\theta$

$\Psi _{320}(r,\theta ,\phi )={\frac {1}{54{\sqrt {6\pi }}}}\left({\frac {Z}{a}}\right)^{-{\frac {7}{2}}}e^{-{\frac {Zr}{3a}}}r^{2}(3cos^{2}\theta -1)$

$\Psi _{321}(r,\theta ,\phi )={\frac {1}{54{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}e^{i\phi }sen\theta cos\theta$

$\Psi _{322}(r,\theta ,\phi )={\frac {1}{108{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{3a}}}r^{2}e^{2i\phi }sen^{2}\theta$

n=4.

$\Psi _{400}(r,\theta ,\phi )={\frac {1}{1536{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {3}{2}}e^{-{\frac {Zr}{4a}}}\left[192-144{\frac {Zr}{a}}+24{\frac {Z^{2}r^{2}}{a^{2}}}-{\frac {Z^{3}r^{3}}{a^{3}}}\right]$

$\Psi _{41-1}={\frac {1}{512{\sqrt {5\pi }}}}e^{-{\frac {Zr}{4a}}}r\left[80-20{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]e^{-i\phi }sin\theta$

$\Psi _{410}(r,\theta ,\phi )={\frac {1}{256{\sqrt {10\pi }}}}e^{-{\frac {Zr}{4a}}}r\left[80-20{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]cos\theta$

$\Psi _{411}={\frac {1}{512{\sqrt {5\pi }}}}e^{-{\frac {Zr}{4a}}}r\left[80-20{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]e^{i\phi }sin\theta$

$\Psi _{42-2}(r,\theta ,\phi )={\frac {1}{1536}}{\sqrt {\frac {3}{2\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{4a}}}r^{2}\left[12-{\frac {Zr}{a}}\right]e^{-2i\phi }sen^{2}\theta$

$\Psi _{42-1}(r,\theta ,\phi )={\frac {1}{768}}{\sqrt {\frac {3}{2\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{4a}}}r^{2}\left[12-{\frac {Zr}{a}}\right]e^{-i\phi }sen\theta cos\theta$

$\Psi _{420}(r,\theta ,\phi )={\frac {1}{1536{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{4a}}}r^{2}\left[12-{\frac {Zr}{a}}\right]{\biggl (}3cos^{2}\theta -1{\biggr )}$

$\Psi _{421}(r,\theta ,\phi )={\frac {1}{768}}{\sqrt {\frac {3}{2\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{4a}}}r^{2}\left[12-{\frac {Zr}{a}}\right]e^{i\phi }sen\theta cos\theta$

$\Psi _{422}(r,\theta ,\phi )={\frac {1}{1536}}{\sqrt {\frac {3}{2\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{4a}}}r^{2}\left[12-{\frac {Zr}{a}}\right]e^{2i\phi }sen^{2}\theta$

$\Psi _{43-3}(r,\theta ,\phi )={\frac {1}{1536{\sqrt {2\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{4a}}}r^{3}e^{-3i\phi }sin^{3}\theta$

$\Psi _{43-2}(r,\theta ,\phi )={\frac {1}{1536}}{\sqrt {\frac {3}{\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{4a}}}r^{3}e^{-2i\phi }sen^{2}\theta cos\theta$

$\Psi _{43-1}(r,\theta ,\phi )={\frac {1}{1536}}{\sqrt {\frac {3}{10\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{4a}}}r^{3}e^{-i\phi }sin\theta {\biggl (}5cos^{2}\theta -1{\biggr )}$

$\Psi _{430}(r,\theta ,\phi )={\frac {1}{768{\sqrt {10\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{4a}}}r^{3}{\biggl (}5cos^{3}\theta -3cos\theta {\biggr )}$

$\Psi _{431}(r,\theta ,\phi )={\frac {1}{1536}}{\sqrt {\frac {3}{10\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{4a}}}r^{3}e^{i\phi }sin\theta {\biggl (}5cos^{2}\theta -1{\biggr )}$

$\Psi _{432}(r,\theta ,\phi )={\frac {1}{1536}}{\sqrt {\frac {3}{\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{4a}}}r^{3}e^{2i\phi }sen^{2}\theta cos\theta$

$\Psi _{433}(r,\theta ,\phi )={\frac {1}{1536{\sqrt {2\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{4a}}}r^{3}e^{3i\phi }sin^{3}\theta$

n=5.

$\Psi _{500}(r,\theta ,\phi )={\frac {1}{46875{\sqrt {5\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {3}{2}}e^{-{\frac {Zr}{5a}}}\left[9375-7500{\frac {Zr}{5a}}+1500{\frac {Z^{2}r^{2}}{a^{2}}}-100{\frac {Z^{3}r^{3}}{a^{3}}}+2{\frac {Z^{4}r^{4}}{a^{4}}}\right]$

$\Psi _{51-1}(r,\theta ,\phi )={\frac {1}{46875{\sqrt {5\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}e^{-{\frac {Zr}{5a}}}r\left[3750-1125{\frac {Zr}{a}}+90{\frac {Z^{2}r^{2}}{a^{2}}}-2{\frac {Z^{3}r^{3}}{a^{3}}}\right]e^{-i\phi }sen\theta$

$\Psi _{510}(r,\theta ,\phi )={\frac {1}{46875}}{\sqrt {\frac {2}{5\pi }}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}e^{-{\frac {Zr}{5a}}}r\left[3750-1125{\frac {Zr}{a}}+90{\frac {Z^{2}r^{2}}{a^{2}}}-2{\frac {Z^{3}r^{3}}{a^{3}}}\right]cos\theta$

$\Psi _{511}(r,\theta ,\phi )={\frac {1}{46875{\sqrt {5\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {5}{2}}e^{-{\frac {Zr}{5a}}}r\left[3750-1125{\frac {Zr}{a}}+90{\frac {Z^{2}r^{2}}{a^{2}}}-2{\frac {Z^{3}r^{3}}{a^{3}}}\right]e^{i\phi }sen\theta$

$\Psi _{52-2}(r,\theta ,\phi )={\frac {1}{187500}}{\sqrt {\frac {3}{7\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{5a}}}r^{2}\left[1050-70{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]e^{-2i\phi }sin^{2}\theta$

$\Psi _{52-1}(r,\theta ,\phi )={\frac {1}{93750}}{\sqrt {\frac {3}{7\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{5a}}}r^{2}\left[1050-70{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]e^{-i\phi }sin\theta cos\theta$

$\Psi _{520}(r,\theta ,\phi )={\frac {1}{93750}}{\sqrt {\frac {1}{14\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{5a}}}r^{2}\left[1050-70{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]{\biggl (}3\cos ^{2}\theta -1{\biggr )}$

$\Psi _{521}(r,\theta ,\phi )={\frac {1}{93750}}{\sqrt {\frac {3}{7\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{5a}}}r^{2}\left[1050-70{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]e^{i\phi }sin\theta cos\theta$

$\Psi _{522}(r,\theta ,\phi )={\frac {1}{187500}}{\sqrt {\frac {3}{7\pi }}}\left({\frac {Z}{a}}\right)^{\frac {7}{2}}e^{-{\frac {Zr}{5a}}}r^{2}\left[1050-70{\frac {Zr}{a}}+{\frac {Z^{2}r^{2}}{a^{2}}}\right]e^{2i\phi }sin^{2}\theta$

$\Psi _{53-3}(r,\theta ,\phi )={\frac {1}{93750{\sqrt {2\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{5a}}}r^{3}{\biggl (}20-{\frac {Zr}{a}}{\biggr )}e^{-3i\phi }sin^{3}\theta$

$\Psi _{53-2}(r,\theta ,\phi )={\frac {1}{93750}}{\sqrt {\frac {3}{\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{5a}}}r^{3}{\biggl (}20-{\frac {Zr}{a}}{\biggr )}e^{-2i\phi }sin^{2}\theta cos\theta$

$\Psi _{53-1}(r,\theta ,\phi )={\frac {1}{93750}}{\sqrt {\frac {3}{10\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{5a}}}r^{3}{\biggl (}20-{\frac {Zr}{a}}{\biggr )}e^{-i\phi }sin\theta {\biggl (}5\cos ^{2}\theta -1{\biggr )}$

$\Psi _{530}(r,\theta ,\phi )={\frac {1}{46875{\sqrt {10\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{5a}}}r^{3}{\biggl (}20-{\frac {Zr}{a}}{\biggr )}{\biggl (}5\cos ^{3}\theta -3cos\theta {\biggr )}$

$\Psi _{531}(r,\theta ,\phi )={\frac {1}{93750}}{\sqrt {\frac {3}{10\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{5a}}}r^{3}{\biggl (}20-{\frac {Zr}{a}}{\biggr )}e^{i\phi }sin\theta {\biggl (}5\cos ^{2}\theta -1{\biggr )}$

$\Psi _{532}(r,\theta ,\phi )={\frac {1}{93750}}{\sqrt {\frac {3}{\pi }}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{5a}}}r^{3}{\biggl (}20-{\frac {Zr}{a}}{\biggr )}e^{2i\phi }sin^{2}\theta cos\theta$

$\Psi _{533}(r,\theta ,\phi )={\frac {1}{93750{\sqrt {2\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {9}{2}}e^{-{\frac {Zr}{5a}}}r^{3}{\biggl (}20-{\frac {Zr}{a}}{\biggr )}e^{3i\phi }sin^{3}\theta$

$\Psi _{54-4}(r,\theta ,\phi )={\frac {1}{375000{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{-4i\phi }sin^{4}\theta$

$\Psi _{54-3}(r,\theta ,\phi )={\frac {1}{93750{\sqrt {2\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{-3i\phi }sin^{3}\theta cos\theta$

$\Psi _{54-2}(r,\theta ,\phi )={\frac {1}{187500{\sqrt {7\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{-2i\phi }sin^{2}\theta {\biggl (}7cos^{2}\theta -1{\biggr )}$

$\Psi _{54-1}(r,\theta ,\phi )={\frac {1}{93750{\sqrt {14\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{-i\phi }sin\theta {\biggl (}7cos^{3}\theta -3cos\theta {\biggr )}$

$\Psi _{540}(r,\theta ,\phi )={\frac {1}{187500{\sqrt {70\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}{\biggl (}35cos^{4}\theta -30cos^{2}\theta +3{\biggr )}$

$\Psi _{541}(r,\theta ,\phi )={\frac {1}{93750{\sqrt {14\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{i\phi }sin\theta {\biggl (}7cos^{3}\theta -3cos\theta {\biggr )}$

$\Psi _{542}(r,\theta ,\phi )={\frac {1}{187500{\sqrt {7\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{2i\phi }sin^{2}\theta {\biggl (}7cos^{2}\theta -1{\biggr )}$

$\Psi _{543}(r,\theta ,\phi )={\frac {1}{93750{\sqrt {2\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{3i\phi }sin^{3}\theta cos\theta$

$\Psi _{544}(r,\theta ,\phi )={\frac {1}{375000{\sqrt {\pi }}}}\left({\frac {Z}{a}}\right)^{\frac {11}{2}}e^{-{\frac {Zr}{5a}}}r^{4}e^{4i\phi }sin^{4}\theta$

Bibliografía.

De la Peña, L.(2006).Introducción a la Mecánica Cuántica. México: Fondo de Cultura Económica.ISBN 9786071601766

Véase también.

Átomo hidrogenoide.

Armónicos esféricos.