00:01
Our present -day interpretation of the wave function that solves the schrodinger equation is that if you take the wave function and you square it, then what is true is that wave function gives you the probability of finding the electron within some small volume.
00:28
So it is a probability density.
00:32
And if we just have a function of r, we can write out the volume.
00:38
It gets a little more complicated if there's an angular dependence.
00:42
But we can write out that volume element as basically r squared, dr with a 4 pi in front.
00:54
So it's like you're adding up over various surface areas of spheres.
00:59
And we want to integrate between zero and infinity.
01:08
So the quantity in front of the dr, including the 4 pi, is considered to be a probability density function with a probability of finding the electron between some r and r plus dr.
01:38
Now, typically, what we would expect this function to look like, and it does indeed look like this, which won't prove.
01:46
But it should have some peak, should be going to zero at some point.
01:56
And it's kind of like a gaussian that you would use in statistics.
02:01
That is, if you want to find the probability that a variable takes on a certain range, you simply find the area under the integral.
02:14
And, you know, if there's a finite amount between the two endpoints for your area, yeah, you'll have a finite probability.
02:30
So that's the interpretation of this probability density.
02:35
It is like a statistical probability distribution.
02:40
And what we mean by the absolute value around that wave function is that the way function is that the way function can actually take on complex parts to it.
02:52
And so by taking the absolute value, we're ensuring that the probability density itself is real, real valued, whereas the wave function can be complex.
03:06
And this leads to a very unsettling idea of what the wave function is.
03:12
Where does that wave function exist? it exists in some imaginary dimension.
03:22
But it makes its effects known through this squared valued.
03:29
Anyway, what we would like to do is find the probability density function for the n equals 1 s -state, the hydrogen atom, and find out where it is maximum.
03:58
I will point out that the a parameter in that wave function that occurs both in the exponential and in the denominator is related to the wave equation...