[Sophia7X]

Show that the electron distribution is spherically symmetrical for an atom with an electron that occupies each of the 3 p orbitals of a given shell.




 I know that I'm supposed to do something with the angular wavefunctions, but what?

Y_px(θ,Φ) = (3/4π)^0.5sinθcosΦ
Y_py(θ,Φ) = (3/4π)^0.5sinθsinΦ
Y_pz(θ,Φ) = (3/4π)^0.5cosθ

Also, I know that spherically symmetrical = independent of θ and Φ

I would show my attempt at this problem... if I knew how to. Could someone point me in the right direction?

[Jorriss]

Show that the electron distribution is spherically symmetrical for an atom with an electron that occupies each of the 3 p orbitals of a given shell.




 I know that I'm supposed to do something with the angular wavefunctions, but what?

Y_px(θ,Φ) = (3/4π)^0.5sinθcosΦ
Y_py(θ,Φ) = (3/4π)^0.5sinθsinΦ
Y_pz(θ,Φ) = (3/4π)^0.5cosθ

Also, I know that spherically symmetrical = independent of θ and Φ

I would show my attempt at this problem... if I knew how to. Could someone point me in the right direction?

In quantum mechanics, the probability of measuring a particle between points a and b goes like the wave function squared.

So, Y_31m(θ,Φ)^2 goes like the probability. To show it is independent of theta or phi, sum over m and show Y₃₁₁(θ,Φ)^2+Y₃₁₀(θ,Φ)^2+Y_31-1(θ,Φ)^2 has no angular dependence.

That's impressive for High school chemistry.

[Sophia7X]

Yeah, not exactly high school chemistry we're learning in school, lol. I'm just self-studying/furthering my chem studies over the summer.

In quantum mechanics, the probability of measuring a particle between points a and b goes like the wave function squared.

Thanks for reminding me, I keep thinking that probability and wavefunction are the same thing!

OK, here's my work so far after squaring everything...

(3/4π)sin²θcos²Φ + (3/4π)sin²θsin²Φ + (3/4π)cos²θ

(3/4π)(sin²θcos²Φ + sin²θsin²Φ + cos²θ)

I don't know what to do next

[Sophia7X]

Ohh, never mind
It can be factored a little bit more.
And simplified using trig identities, which I never thought I would ever apply to anything else but trig.

(3/4π)(sin²θ(cos²Φ + sin²Φ) + cos²θ)
(3/4π)(sin²θ(1)+ cos²θ)
=(3/4π)

so the distribution is independent of θ and Φ and spherically symmetrical

Thanks for your help

[Jorriss]

There you go.