Quantum Chemistry 7.7 - Hydrogen Atom Radius

Short lecture on measures for the radius of the hydrogen atom.

The Bohr radius is 0.529 Angstroms, or a_o. The expectation value of the radius of the 1s hydrogen orbital is 1.5 a_0. The most probable value of the radius is 1.0 a_o. Alternatively, we can compute a probability radius, where a certain percent of the electron density falls inside the given radius.

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[EdwardCullensMayo]

nit-picky comment:

the "dr" was accidentally left off of the <r>1s integral.

[nkyu]

I've been thinking... yet i'm not sure why one of the volume element in spherical polar coordinates is rsinθdΦ instead of just rdΦ.
it seems sinθ is there so rsinθdΦ is on the xy plane. but why?

[thomaslassitter]

I've been watching these and following along in McQuarrie and Simon and I noticed that in the text they multiplied R(r)*R(r) by the full area of a sphere, 4*pi*r^2, yet you only multiplied by the r^2 term. I was wondering about the reasoning behind this. Is it because probabilities are proportional so the 4*pi doesn't matter, or is there some other reason I'm missing?

[RishabChawla]

Hi Trent, is the Pnl(r) function supposed to be followed be volume element dr such that the probability of finding the electron between r and r+dr is (4r^2)/(a0^3)exp(-2r/a0)dr ?

[mikusxlr]

The second term in the volume component. Is that rd(theta) or rd(phi)?