00:01
Yes, here we're going to find the probability of finding a 1s electron in the hydrogen atom in a certain region around the nucleus.
00:14
So what we'll be using is the probability density, which basically comes from the wave function times its complex conjugate.
00:26
Yes, and here because we are thinking about spherical coordinates, so the hydrogen atom starts with the solution to a differential equation in spherical coordinates, in particular the schrodinger equation.
00:43
So we have to figure out our probability density based on a spherical volume.
00:51
And that is the reason why there is a 4 pi r squared in front of the wave function squared.
00:59
R inside the wave function is the distance from the nucleus, and a0 is the bore radius, a particular number that scales the first orbit around the nucleus in hydrogen.
01:17
A reminder about what a probability density is, though.
01:21
It is pretty much a histogram, a probability histogram, but it's continuous.
01:31
So what our probability density function looks like for the 1s state is it's a fairly rapidly peaking function.
01:43
So that's our probability density as a function of r.
01:47
And it's not a bell curve, but it kind of works the same way that a bell curve would.
01:53
If you find the total area under the entire graph, you should get 100%, meaning you should, should find the electron somewhere in that hydrogen atom.
02:15
The other thing that you can do is integrate from one distance to another, r1 to r2.
02:24
And if you find that area, that's the probability, the total probability, of finding the electron inside a kind of a donut that extends from r1 to r2.
02:44
It's not really a donut.
02:49
It's more like two spherical shells with the space in between them, possibly having the electron in it.
03:00
So we are going to set up the integral and find two different probabilities.
03:07
Our first one is what is the probability? total probability, big b, which is an area, of finding the electron from the region 0 to a knot over just a knot within one more radius.
03:26
And there should be a fairly high probability, but we're basically finding the area under that curve up to the peak actually happens at a knot.
03:39
So we're finding the area under that curve.
03:45
And we just simply set up our integral.
03:48
We can bring the 4 pi out of it.
03:54
And we can also bring the constants that are in front of the wave function squared.
04:02
I can bring those out, pi a not cubed, and then inside we have r squared, e to the minus to r over a0, dr.
04:19
Usually what i like to do at this point is to use a tool to do the integral.
04:26
Polynomials multiplying and exponential are not bad to integrate.
04:32
You can use integration by parts, but you need to do it multiple times and build up an iteration formula.
04:40
So i am going to use an online tool, the wolfram integrate tool...