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 3-dimensions Virial Theorem can be written as:
Plugging the value of in the above equation, we get
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.