Normalizing the wave function of the electron in hydrogen

[Cocoleia]

Homework Statement

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

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

 

[blue_leaf77]

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

 

[Cocoleia]

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

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

 

[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.

 

[Cocoleia]

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.

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

 

[kuruman]

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

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

 

[Cocoleia]

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

I took out the constant and then used spherical coordinates

 

[Cocoleia]

View attachment 117362
I took out the constant and then used spherical coordinates

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 ?

 

[kuruman]

Yes.

 

[rb120134]

Is it true that the probability of finding electron betwee r1= ao/2 and r2= 3a0/2 is 49,65%? For this problem

 

[kuruman]

Is it true that the probability of finding electron betwee r1= ao/2 and r2= 3a0/2 is 49,65%? For this problem

Yes.