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

# 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 the following expression represents the probability that the electron will be found in the spatial element d3r: In spherical coordinates, So you can write as The probability that the electron is in a spherical shell of radius r to r + dr is therefore And because this equation becomes the following: The preceding equation is equal to (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 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 which is 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: where r0 is the Bohr radius:  so the expectation value of the electron’s distance from the proton is So, for example, in the 1s state the expectation value of r is equal to And in the 4p state 