When you want to find where an electron is at any given time in a hydrogen atom, what you’re actually doing is finding how far the electron is from the proton. You can find the expectation value of r, that is,
![image0.png](https://www.dummies.com/wp-content/uploads/396844.image0.png)
the following expression represents the probability that the electron will be found in the spatial element d3r:
![image1.png](https://www.dummies.com/wp-content/uploads/396845.image1.png)
In spherical coordinates,
![image2.png](https://www.dummies.com/wp-content/uploads/396846.image2.png)
So you can write
![image3.png](https://www.dummies.com/wp-content/uploads/396847.image3.png)
as
![image4.png](https://www.dummies.com/wp-content/uploads/396848.image4.png)
The probability that the electron is in a spherical shell of radius r to r + dr is therefore
![image5.png](https://www.dummies.com/wp-content/uploads/396849.image5.png)
And because
![image6.png](https://www.dummies.com/wp-content/uploads/396850.image6.png)
this equation becomes the following:
![image7.png](https://www.dummies.com/wp-content/uploads/396851.image7.png)
The preceding equation is equal to
![image8.png](https://www.dummies.com/wp-content/uploads/396852.image8.png)
(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)
Spherical harmonics are normalized, so this just becomes
![image9.png](https://www.dummies.com/wp-content/uploads/396853.image9.png)
Okay, that’s the probability that the electron is inside the spherical shell from r to r + dr. So the expectation value of r, which is
![image10.png](https://www.dummies.com/wp-content/uploads/396854.image10.png)
which is
![image11.png](https://www.dummies.com/wp-content/uploads/396855.image11.png)
This is where things get more complex, because Rnl(r) involves the Laguerre polynomials. But after a lot of math, here’s what you get:
![image12.png](https://www.dummies.com/wp-content/uploads/396856.image12.png)
where r0 is the Bohr radius:
![image13.png](https://www.dummies.com/wp-content/uploads/396857.image13.png)
The Bohr radius is about
![image14.png](https://www.dummies.com/wp-content/uploads/396858.image14.png)
so the expectation value of the electron’s distance from the proton is
![image15.png](https://www.dummies.com/wp-content/uploads/396859.image15.png)
So, for example, in the 1s state
![image16.png](https://www.dummies.com/wp-content/uploads/396860.image16.png)
the expectation value of r is equal to
![image17.png](https://www.dummies.com/wp-content/uploads/396861.image17.png)
And in the 4p state
![image18.png](https://www.dummies.com/wp-content/uploads/396862.image18.png)