Angular part of the wavefunction

[Chemist20]

Hello,

This question is related to wavefunctions and their radial and angular parts.
I know how to draw the radial part, the RDF but how would you draw the angular part?

Thank you!

 

[ChmDudeCB]

Are you referring to the hydrogen atom?

 

[Chemist20]

Im referring to any of the spherical harmonics, including the one for the hydrogen! ;)

 

[tom.stoer]

google: spherical harmonics
http://www.google.de/search?hl=de&q...urce=og&sa=N&tab=wi&ei=IAsIT82BOMzBtAaexriCDw

 

[Chemist20]

google: spherical harmonics
http://www.google.de/search?hl=de&q...urce=og&sa=N&tab=wi&ei=IAsIT82BOMzBtAaexriCDw

Yes, but those are the wavefunctions themselves, the total wavefunction. not just the angular part.

 

[cgk]

Yes, but those are the wavefunctions themselves, the total wavefunction. not just the angular part.

No, they are not:
1. They are only the angular part (they have only two angles (or alternatively one normalized vector) as parameter!).
2. They are not necessarily wave functions. They are the solutions to the angular part of a Laplace equation with spherically symmetric potential.

There are also "solid harmonics", which actually have an radial part and are three-dimensional functions (unlike the spherical harmonics, which are defined only on the sphere), but normally you don't encounter them.
Why do you think spherical harmonics are total wave functions?

 

[Chemist20]

No, they are not:
1. They are only the angular part (they have only two angles (or alternatively one normalized vector) as parameter!).
2. They are not necessarily wave functions. They are the solutions to the angular part of a Laplace equation with spherically symmetric potential.

There are also "solid harmonics", which actually have an radial part and are three-dimensional functions (unlike the spherical harmonics, which are defined only on the sphere), but normally you don't encounter them.
Why do you think spherical harmonics are total wave functions?

I'm a bit lost now...
Okay, the link takes you to a google page were the shape of different orbitals appear. That's the shape of the orbital itself and so has to be total wavefunction right?

 

[nonequilibrium]

Well, they might remind you of orbital drawings (which is not too silly as they're used in them), but they aren't really orbital pictures when you compare them, are they?

Anyway, those drawings of the amplitude of the spherical harmonics are exact, whereas the orbital drawings of hydrogen are an approximation, right? The latter uses a .9 probability area to draw the wave functions (?) That being said, I'm wondering how orbitals are drawn: what is the method/algorithm? What exactly are we seeing in those drawings of orbitals? I can't seem to find a clear source on it. My quantum physics professor even told me about it the wrong way...

 

[tom.stoer]

the link takes you to a google page were the shape of different orbitals appear.

No, I just googled for "spherical harmonics"

 

[nonequilibrium]

He means that the pictures that show up remind him of the pictures of orbitals.

 

[tom.stoer]

He means that the pictures that show up remind him of the pictures of orbitals.

Yes, I understand that and that's not unreasonable b/c spherical harmonics are part of the wave function -and b/c the usual 'drawings' of orbitals focus on the spherical part, not on the radial one.

 

[nonequilibrium]

Just wondering: do you know how orbitals are drawn? Where and how does the 90% come in?

 

[tom.stoer]

Just wondering: do you know how orbitals are drawn?

An orbital is a region of space (not necessarily connected) which contains the electron with a certain fixed probability. What is drawn are just the outer surfaces. For higher shells this is not correct, the orbitals should look more like onions.

Where and how does the 90% come in?

Don't know; b/c it looks nice?

 

[nonequilibrium]

What is drawn are just the outer surfaces.

Outer surfaces of what? Of for example a region where there's a 90% probability? But it seems there would be an uncountable number of such regions you could pick. Which one is chosen?

 

[tom.stoer]

Define surfaces of constant probability density p; these closed surfaces define several regions (not necessarily connected); the union of all these regions is called 'orbital'. Integrating the probability density over all these regions yields a probability P(p). Now adjust p such that P(p)=90%.

Due to the radial oscillations the structure of the orbital (the different surfaces) can look like an onion; but of course you can only see one such surface, the inner structure is hidden.

 

[nonequilibrium]

Ah interesting. So when you say "Integrating the probability density over all these regions yields a probability P(p)." do you mean integrating the volume enclosed in each of those closed surfaces?

 

[tom.stoer]

Ah interesting. So when you say "Integrating the probability density over all these regions yields a probability P(p)." do you mean integrating the volume enclosed in each of those closed surfaces?

For one region V: integrating the probability density over V yields a probability P

$$\int_{V:\, |\psi|^2 = p\text{ on }\partial V} dV\,|\psi(x)|^2 = P$$

 

[nonequilibrium]

But if you're doing a volume integral restricted to a surface, isn't that zero? After all, if the probability distribution is bounded, you can bound the whole integral by a constant times the volume measure of a surface, which is zero.

 

[tom.stoer]

I do not integrate over a surface but over a volume defined by the surface. It's like saying that "integrating over a disc r² ≤ 1" is defined by "integrating over a two-dim. region defined by a circle r² = 1".

 

[nonequilibrium]

Oh, no problem then, I misunderstood your notation (my fault)!

So you end up with (possibly) different surfaces, each with a (possibly) different (constant) value of probability density on its surface, such that each of these gives a (say) probability of 90% when integrating the volume enclosed by the surface? At least this is what your explicit integral is implying to me.

But how can you then get an onion? If one surface B is enclosing another smaller one A, and the volume integral inside of A is .9, then surely the volume integral inside of B (enclosing the volume inside of A) must be bigger than .9,
so I'm probably again misinterpreting something you've said.

 

[Chemist20]

Oh okay, so if I was asked to draw both the angular part of por example to 2px orbital and the wavefunction itself, what would the difference be? (in the drawing i mean, because mathematically it would be that the wavefunction would be a result of multiplying the spherical harmonic and the lagerre functions, yes?)

Thanks for your answers!

 

[tom.stoer]

But how can you then get an onion? If one surface B is enclosing another smaller one A, and the volume integral inside of A is .9, then surely the volume integral inside of B (enclosing the volume inside of A) must be bigger than .9,
so I'm probably again misinterpreting something you've said.

The difficulty arises if you take into account the radial oscillations. Then you may have several spherical shells 0 < r_a < r_b < r_c < r_d < ... where the probability for [0, r_a] is small, the probability in [r_a, r_b] is large, the probability in [r_b, r_c] is small ...

That means that the orbital may not consist of one ball but of several spherical shells, e.g. [r_a, r_b], [r_c, r_d], ... From the outside it may look like one ball, but in reality it's something like an onion.

Unfortunately this is rarely explained when orbitals are introduced. One always get's the impression of a solid ball. That's why my recommendation is to look at the wave function or the probability density directly.

http://sevencolors.org/images/photo/hydrogen_density_plots.jpg (e.g. 411)
http://c2h2.ifa.hawaii.edu/Tutorial/graphics/Fundamentals/electron_orbitals1.jpg