What does the radial wave function represent?

[Zatman]

I am (attempting to) learn the *basics* of quantum physics in terms of the origin of atomic orbitals from the Schrödinger equation. I understand that the solution for H is split into a product of 2 functions, the radial wave function and the angular wave function.

Then I am being shown plots of the radial wave function for various orbitals. And what seems to be missing, (or perhaps something obvious that I am missing), is what it actually is representing.

I am also aware of the radial distribution function which makes a little more sense to me (probability of an electron being at a distance r from the nucleus).

But I can't seem to see what the radial wave function represents. Any help would be appreciated!

 

[tiny-tim]

Hi Zatman!

… the solution for H is split into a product of 2 functions, the radial wave function and the angular wave function.

Yes, the angular function tells you the general shape of the probability distribution, and the radial function modifies that shape (for different values of the radial quantum number) by telling you how much it depends on distance from the centre.

All you really need to know about the radial function is whether it's zero or non-zero at the centre. ​

 

[Zatman]

So what is represented by the vertical axis? And what does it mean for the radial wave function to go below zero?

 

[tiny-tim]

what vertical axis?

(i'm not sure we're talking about the same thing …

can you link to a diagram that illustrates what you're referring to?)​

 

[Zatman]

This sort of thing:

http://www.google.co.uk/imgres?hl=e...20&ved=1t:429,r:16,s:0,i:117&biw=1366&bih=605

 

[tiny-tim]

And what does it mean for the radial wave function to go below zero?

you mean, as in the graphs in the upper box?

i've never seen those graphs before, I've always seen the bumpy graphs in the lower box, showing the probability, which i think is all you need to know

the upper box shows the amplitude (which of course you still have to multiply by the angular wave function), and amplitudes don't usually have any physical significance (and generally, they're complex, so a negative amplitude is nothing out of the ordinary)

 

[Zatman]

Yes they are what I'm referring to.

(I agree that the radial distribution functions are more important, but I do need to know about radial wave functions too according to my syllabus).

That does make a little more sense, thank you. Just one more thing -- when the amplitude goes negative, is this related to the different phases of different parts of the orbital (it more than likely is, so... how?

 

[tiny-tim]

… when the amplitude goes negative, is this related to the different phases of different parts of the orbital (it more than likely is, so... how?

i don't think it's related to anything

if we were talking about the angular wave function, the complexity (or negativity) of the amplitude at each point wouldn't bother you, would it?

 

[Zatman]

It... wouldn't, no. But are you saying then that it is a coincidence that the number of times the radial wave function goes below zero is exactly correct for the number of changes of phase for each orbital?

 

[tiny-tim]

i'm saying that only the probability matters

(that's for an eigenstate, of course … for a combination of eigenstates, the probability comes from sum of the amplitudes, so i suppose a negative amplitude could cause destructive interference)

 

[Zatman]

Okay, I guess I'll just accept this.

Thanks for your help tiny-tim, much appreciated!