Calculate the Distance of an Electron from the Proton of a Hydrogen Atom

When you want to find where an electron is at any given time in a hydrogen atom, what you’re actually doing is finding how far the electron is from the proton. You can find the expectation value of r, that is, <r>, to tell you its location. Given that the wave function is

image0.png

the following expression represents the probability that the electron will be found in the spatial element d3r:

image1.png

In spherical coordinates,

image2.png

So you can write

image3.png

as

image4.png

The probability that the electron is in a spherical shell of radius r to r + dr is therefore

image5.png

And because

image6.png

this equation becomes the following:

image7.png

The preceding equation is equal to

image8.png

(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)

Spherical harmonics are normalized, so this just becomes

image9.png

Okay, that’s the probability that the electron is inside the spherical shell from r to r + dr. So the expectation value of r, which is <r>, is

image10.png

which is

image11.png

This is where things get more complex, because Rnl(r) involves the Laguerre polynomials. But after a lot of math, here’s what you get:

image12.png

where r0 is the Bohr radius:

image13.png

The Bohr radius is about

image14.png

so the expectation value of the electron’s distance from the proton is

image15.png

So, for example, in the 1s state

image16.png

the expectation value of r is equal to

image17.png

And in the 4p state

image18.png
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