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.

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Manas Sharma

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.

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