Homework Help: Normalizing the wave function of the electron in hydrogen

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1. Apr 1, 2017

Cocoleia

1. The problem statement, all variables and given/known data

I am having trouble with part d, where they ask me to prove that the wave function is already normalized

3. The attempt at a solution

But that clearly doesn't give me 1. I tried to use spherical coordinates since it is in 3D? Not really sure how to proceed.
EDIT: I realize that I didn't square the wave function, so

Which still doesn't give me 1

2. Apr 1, 2017

blue_leaf77

The integration limit should be from $0$ to $\infty$.

3. Apr 1, 2017

Cocoleia

Ok. I was thinking 0 to r since the most it could go was to the radius. I guess the logic is wrong

multiplied by 4pi, so the 4's will cancel out, leaving me with (a_0^3)(pi)
Which still doesn't = 1. What am I missing

4. Apr 1, 2017

kuruman

It pays to be methodical
$\psi(r,\theta,\phi) =\frac{1}{\sqrt{\pi}} \left( \frac{1}{a_0} \right )^{3/2} e^{-r/a_0}$
$\psi^*(r,\theta,\phi) \psi(r,\theta,\phi) =\frac{1}{{\pi}} \left( \frac{1}{a_0} \right )^{3} e^{-2r/a_0}$
Now do the integrals.

5. Apr 2, 2017

Cocoleia

If I use the spherical coordinates I still get (a^3)π. I say that 0<Θ<π and 0<Φ<2π

6. Apr 2, 2017

kuruman

Can you show the expression that you integrated to get this result?

7. Apr 2, 2017

Cocoleia

I took out the constant and then used spherical coordinates

8. Apr 2, 2017

Cocoleia

Oh wait. I think I figured it out. I lost my constant along the way.
When I integrate to find the probability of finding the electron in a certain place will I use spherical coordinates again ?

9. Apr 2, 2017

Yes.