# Show that the Hydrogen wave functions are normalized

## Homework Statement

Show that the (1,0,0) and (2,0,0) wave functions are properly normalized.
We know that:

Ψ(1,0,0) = (2/(a0^(3/2))*e^(-r/a0)*(1/sqrt(2))*(1/sqrt(2*pi))

where:
R(r) = (2/(a0^(3/2))*e^(-r/a0)
Θ(θ) = (1/sqrt(2))
Φ(φ) = (1/sqrt(2*pi))

## Homework Equations

(1) ∫|Ψ|^2 dx = 1
(2) |R(r)|^2*|Θ(θ)|^2*|Φ(φ)|^2*r^2*sinθ dr dθ dφ ... (? might be useful)

## The Attempt at a Solution

So I know that in order for a function to be properly normalized, it has to have the absolute value of the sine wave squared equal to 1. I originally integrated from negative inf to positive inf, but that did not give me 1 = 1. I tried looking to see where I went wrong but I found equation (2) in my book, but wasn't sure how to integrate that. What am I doing wrong? I think my limits of integration might be wrong because I was doing:

(3) ∫ ((2/(a0^(3/2))*e^(-r/a0)*(1/sqrt(2))*(1/sqrt(2*pi)))^2 dr = 1 from - inf to + inf

and not worrying about that sin θ thing I mentioned earlier.

Do I need to be using equation (2)? How do I find limits of integration? I know that - infinity doesn't make sense for r, so maybe it is just from 0 to infinity with equation 3 maybe?

Homework Helper
Gold Member
## r ## goes from 0 to ## +\infty ##. ## \theta ## goes from ## 0 ## to ## \pi ##, and ## \phi ## goes from 0 to ## 2 \pi ##.

kneesarethebees
Staff Emeritus
Homework Helper
You need to integrate over all space, so it's going to be a three-dimensional integral. You need to use (2).

kneesarethebees
Okay, so since I was given what R(r), Θ(θ), and Φ(φ) is, I came up with this equation from what vela has suggested and integrating over what Charles said:

∫∫|R(r)|^2*r^2*sinθ dr dθ for r from 0 to inf and theta from 0 to pi

(1/pi*a0^3) ∫∫ e^(-2r/a0)*r^2*sinθ dr dθ

When evaluating the dr integral using a table of integrals, I get the value of 0. Now when evaluating the dθ integral I get -cosθ from 0 to pi, which results in 2. Why is this still not giving me 1?

Thanks guys :)

Staff Emeritus
Homework Helper
Because you’re doing it wrong. You need to show your work. We can’t tell what you did otherwise.

Alright, so the table in my book gives me the ##R(r)##, ##Θ(θ)##, and ##Φ(φ)## for (1,0,0). We know that
$$\int \left | \psi \right| ^2 = 1$$ and
$$\left| R(r) \right| ^2 ~ \left| Θ(θ) \right| ^2 ~ \left| Φ(φ) \right| ^2 ~ r^2 ~ sinθ ~ dr ~ dθ ~ dφ$$ and
$$\psi = \left| R(r) \right| ^2 ~ \left| Θ(θ) \right| ^2 ~ \left| Φ(φ) \right| ^2$$
Therefore, we can say that
$$\int_0^\infty \int_0^\pi \left| \psi \right| ^2 ~ r^2 ~ sinθ ~ dr ~ dθ = 1$$

Okay, so now I just plug in my values for ##\psi## which gave me the integral:
$$\int_0^\infty \int_0^\pi \frac {4} {4 \pi a_0^3} e^{-2r/a_0} r^2 sin \theta ~ dr ~ d\theta$$
Pull the constants out:
$$\frac {1} {\pi a_0^3} \int_0^\infty \int_0^\pi r^2 e^{-2r/a_0} sin \theta ~ dr ~ d\theta$$

Evaluating the ##dr## integral by using a table of integrals:
$$\int_0^\infty r^2 e^{-2r/a_0} ~ dr = \left[ \frac {-e^{-2r/a_0}} {2/a_0} \left( r^2 + \frac {2r} {2/a_0} + \frac {2} {4/a_0^2} \right) \right]_0^\infty$$
##0 - 3.7*10^{-5} = 3.7*10^{-5}## when evaluating

Evaluating ##d\theta##
$$\int_0^\pi sin\theta ~ d\theta = \left[ -cos\theta \right]_0^\pi$$
##1 - \left( -1 \right) = 2##

Using values now:
$$\frac {1} {\pi a_0^3} \left[ 3.7*10^{-4} * 2 \right] = 4.5*10^{-4} \neq 1$$

I took a couple of hours to learn LaTeX hopefully that outlines my steps more clearly? I think I might be integrating the double integral wrong with my two steps which is why I can't get it to equal 1

Homework Helper
Gold Member
If you wrote it correctly, I get ## (1/4 )a_o^3 ## for the ## dr ## integral. Also the ## \phi ## integral gives ## 2 \pi ##. I think the results work.

kneesarethebees
Charles, how are you integrating for the ##\phi## integral? I don't have an equation for that, just a value for Φ(##\phi##) as seen from my response right above yours.

Mentor
Therefore, we can say that
$$\int_0^\infty \int_0^\pi \left| \psi \right| ^2 ~ r^2 ~ sinθ ~ dr ~ dθ = 1$$
That is not correct. First, the order of the integral signs and the integration elements should be consistent. Second, this has to be a three-dimensional integral:
$$\int_0^{2\pi} \int_0^\pi \int_0^\infty \left| \psi \right| ^2 ~ r^2 ~ \sinθ ~ dr ~ dθ ~ d\phi = 1$$

Homework Helper
Gold Member
Charles, how are you integrating for the ##\phi## integral? I don't have an equation for that, just a value for Φ(##\phi##) as seen from my response right above yours.
For ## A ## is a constant, ## \int\limits_{0}^{2 \pi} A \, d \phi=A \, \phi |_{0}^{2 \pi}=A [2 \pi-0]=A (2 \pi) ##. (It's as simple as they come, but you might be somewhat new to calculus, so I spelled it out in detail.)