- #1
Nacho Verdugo
- 15
- 0
Moved from a technical forum, so homework template missing
I'm trying to prove that the wave function of Hydrogen for the fundamental state is normalized:
$$ \Psi_{1s}(r)=\frac{1}{\sqrt{\pi a^3}}e^{-\frac{r}{a}} $$
What I tried is this:
$$ I= \int_{-\infty}^{\infty} | \Psi^2(x) | dx = 1$$
$$ \int_{-\infty}^{\infty} \frac{1}{\pi a^3}e^{-\frac{r^2}{a^2}} dr $$
Which lead me to two differents results:
a) First, when integrating the next integral
$$ \frac{e^{1/a^2}}{\pi a^3} \int_{-\infty}^{\infty} e^{-r^2}dr$$
and using polar coordinates to integrate ## \int_{-\infty}^{\infty} e^{-r^2}dr ##, I finally obtain:
$$ \frac{e^{1/a^2}}{\pi a^3} \sqrt{\pi} $$
and I don't see a way that this expression equals to one.
b) for my second attempt, I used the next property
$$ \int_{-\infty}^{\infty} e^{-bx^2} dx = \sqrt{\frac{\pi}{b}} $$
So, considering ## b=\frac{1}{a^2} ##, I obtain
$$ I=\frac{a\sqrt{\pi}}{\pi a^2} $$
which as well, doesn't look like one.
did i miss something?
$$ \Psi_{1s}(r)=\frac{1}{\sqrt{\pi a^3}}e^{-\frac{r}{a}} $$
What I tried is this:
$$ I= \int_{-\infty}^{\infty} | \Psi^2(x) | dx = 1$$
$$ \int_{-\infty}^{\infty} \frac{1}{\pi a^3}e^{-\frac{r^2}{a^2}} dr $$
Which lead me to two differents results:
a) First, when integrating the next integral
$$ \frac{e^{1/a^2}}{\pi a^3} \int_{-\infty}^{\infty} e^{-r^2}dr$$
and using polar coordinates to integrate ## \int_{-\infty}^{\infty} e^{-r^2}dr ##, I finally obtain:
$$ \frac{e^{1/a^2}}{\pi a^3} \sqrt{\pi} $$
and I don't see a way that this expression equals to one.
b) for my second attempt, I used the next property
$$ \int_{-\infty}^{\infty} e^{-bx^2} dx = \sqrt{\frac{\pi}{b}} $$
So, considering ## b=\frac{1}{a^2} ##, I obtain
$$ I=\frac{a\sqrt{\pi}}{\pi a^2} $$
which as well, doesn't look like one.
did i miss something?