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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4 thoughts on “Expectation Value of 1/r for Hydrogen using Virial Theorem -Quantum Mechanics

  1. Thank you Manas, I couldnt find anything about this specific topic in spanish.

    1. To expectation values of higher powers of 1/r, we have the “Kramer’s relation”.

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