Radial Wavefunction of a Hydrogen Atom

Gibson T. Maglasang and John Paul Aseniero

In this article, we outlined the necessary steps in calculating the radial wavefunctions $R_{nl}$ for the Hydrogen atom. Thus, the radial wavefunctions particularly $R_{30$ , $R_{31$ and $R_{32$ are easily obtained without bothering to normalize it.

We use the formula below to find the wavefunction,

$R_{nl}=\frac{1}{r}\rho^{l+1}e^{-\rho}\nu(\rho),$ ( 1)

where

$\nu(\rho)=\sum_{j=0}^{\infty}c_j\rho^j,$ (2)

while $c_j$ is determined by the recursion formula given by,

$c_{j+1}=\frac{2(j+l+1-n)}{(j+1)(j+2l+2)}c_j,$ (3)

and

$\rho=\frac{r}{na}.$ (4)

(i) Now, finding $R_{30}$

Using equation 1, we need to solve first the coefficient $c_j$ from (eqn. 3), with $n=3$ and $l=0$ .

$c_1=\frac{2(0+1-3)}{1(0+0+2)}c_0=-2c_0$

$c_2=\frac{2(1+1-3)}{2(1+0+2)}c_1=-\frac{2}{3}c_0$

$c_3=\frac{2(2+1-3)}{3(2+2)}c_2=0.$

Knowing the value of $c_j$ , (eqn. 2) can now be easily determined,

$\nu(\rho)=c_0\rho^0+c_1\rho^1+c_2\rho^2+c_3\rho^3.$ (5)

Substituting the value of the calculated coefficients to (eqn. 5), we then have

$\nu(\rho)=c_0-2c_0+\frac{2}{3}\rho^2c_0.$ (6)

Thus,

$R_{30}(\nu)=\frac{1}{r}\Big(\frac{r}{3a}\Big)e^{-\rho}[c_0-2c_0+\frac{2}{3}c_0\rho^2].$ (7)

Plugging in (eqn. 4) to (eqn. 7), we finally have

$R_30=\frac{c_0}{3a}\bigg[1-2\Big(\frac{r}{3a}\Big)+\frac{2}{3}\Big(\frac{r}{3a}\Big)\rho^2\bigg]e^{-(r/3a)}.$

Following the same process in (i), the rest of the wavefunctions are just straightforward.

(ii) For $R_{31}$

Determining first the coefficients, with $n=3$ and $l=1$ ,

$c_1=\frac{2(1+1-3)}{1(0+2+2)}c_0=-\frac{1}{2}c_0$

$c_2=\frac{2(1+1+1-3)}{2(1+2+2)}c_0=0$

$c_3=0$

Then,

$\nu(\rho)=c_0-\frac{1}{2}c_0\rho$ .

Thus,

$R_{31}=\Big(\frac{r}{9a^2}\Big)\bigg[1-\frac{r}{6a}\bigg]e^{-r/3a}$

(iii) We have the coefficients for $R_{32}$ with $n=3$ and $l=2$ ,

$c_1=\frac{2(2+1-3)}{1(4+2)}c_0=0$

$c_2=0$

$c_3=0$

Using again (eqn. 1), we have

$R_{32}=\frac{r^2}{27a^3}e^{-r/2a}c_0$

We finally generated the radial wavefunctions ( $R_{30}$ , $R_{31}$ , $R_{32}$ ) for the hydrogen atom which is the main aim of this paper.

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Radial Wavefunction of a Hydrogen Atom

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