Answer to Question #141739 in Mechanics | Relativity for Afraid

Question

Write down expectation value of 1/r for hydrogen atom.

Answer 1

We will be evaluating the expectation value using the Virial Theorem.

We will be using the expression for Energy of the nth energy level for Hydrogen Atom:

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

For Hydrogen, 

$$V=-\frac{e^2}{4\pi\epsilon_0r}$$

For 3-dimensions Virial Theorem can be written as:

$$2\left\lang T\right\rangle=\left\lang r\frac{dV}{dr}\right\rangle$$

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

$$2\left\lang T\right\rangle=\frac{e^2}{4\pi\epsilon_0r}=-\left\lang V\right\rangle\\E_n=\left\lang T\right\rangle+\left\lang V\right\rangle\to 2E_n=\left\lang V\right\rangle$$

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

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

$$\left\lang \frac{1}{r}\right\rangle=\frac{me^2}{4\pi\epsilon_0\hbar n^2}$$