Expectation Value of 1/r for Hydrogen using Virial Theorem -Quantum Mechanics

In this post we will be evaluating the expectation value of 1/r : \left<\frac{1}{r}\right> 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:

E_n=-\frac{m}{2\hbar^2}\left(\frac{e^2}{4\pi \epsilon_0 }\right)^2\frac{1}{n^2}

Virial Theorem states that for stationary states,
2\left<T\right>=\left<x\frac{\partial V}{\partial x}\right>

For Hydrogen, V=-\frac{e^2}{4\pi \epsilon_0 r}

For 3-dimensions Virial Theorem can be written as:
2\left<T\right>=\left<r\frac{\partial V}{\partial r}\right>

Plugging the value of V in the above equation, we get

\implies 2\left<T\right>= \frac{e^2}{4\pi \epsilon_0 r}

\implies 2\left<T\right>= -\left<V\right>

We know, E_n=\left<T\right>+\left<V\right>

\implies E_n=-\frac{\left<V\right>}{2} + \left<V\right>

\implies E_n=\frac{\left<V\right>}{2}

Now plugging the expression of Energy for Hydrogen in the above equation, we get

\implies -\frac{m}{2\hbar^2}\left(\frac{e^2}{4\pi \epsilon_0 }\right)^2\frac{1}{n^2}=\frac{\left<V\right>}{2}

\implies \left<V\right> = -\frac{m}{\hbar^2}\left(\frac{e^2}{4\pi \epsilon_0 }\right)^2\frac{1}{n^2}

\implies -\frac{e^2}{4\pi \epsilon_0} \left<\frac{1}{r}\right>=  -\frac{m}{\hbar^2}\left(\frac{e^2}{4\pi \epsilon_0 }\right)^2\frac{1}{n^2}

\implies \left<\frac{1}{r}\right>=  \frac{m}{\hbar^2}\frac{e^2}{4\pi \epsilon_0 }\frac{1}{n^2}

This is the expectation value of 1/r for Hydrogen.

You can simplify it further by writing in terms of the Bohr’s radius a_0 ,

a_0= \frac{4\pi \epsilon_0 \hbar^2}{m e^2}

Using the above expression, one can write  \left<\frac{1}{r}\right> , as

 \left<\frac{1}{r}\right> = \frac{1}{n^2 a_0}

If you have any questions or doubts regarding the above proof, feel free to post a comment down below.




PhD 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 softwares from time to time. Can code in most of the popular languages. Like to share my knowledge in Physics and applications using this Blog and a YouTube channel.
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