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.
Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.