In this post we will be evaluating the expectation value of 1/r : using the Virial Theorem, that we proved and talked about in the last post.
We will be using the expression for Energy of the nth energy level for Hydrogen Atom:
Virial Theorem states that for stationary states,
For Hydrogen,
For 3-dimensions Virial Theorem can be written as:
Plugging the value of in the above equation, we get
We know,
Now plugging the expression of Energy for Hydrogen in the above equation, we get
This is the expectation value of 1/r for Hydrogen.
You can simplify it further by writing in terms of the Bohr’s radius ,
Using the above expression, one can write , as
If you have any questions or doubts regarding the above proof, feel free to post a comment down below.
Thank you Manas, I couldnt find anything about this specific topic in spanish.