Angular part of the wavefunction

In summary, the conversation discusses the concept of spherical harmonics and their relation to wavefunctions and orbitals. It is clarified that spherical harmonics are only the angular part of the wavefunction and are not necessarily wave functions themselves. The conversation also touches on the method of drawing orbitals and the significance of the 90% probability in orbital drawings. It is explained that orbitals are drawn as regions of space with a fixed probability of containing the electron, and the 90% probability is determined by integrating the probability density over these regions. The conversation also briefly mentions the layered structure of orbitals and how this can be represented as an onion-like shape.
  • #1
Chemist20
86
0
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!
 
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  • #2
Are you referring to the hydrogen atom?
 
  • #3
Im referring to any of the spherical harmonics, including the one for the hydrogen! ;)
 
  • #6
Chemist20 said:
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?
 
  • #7
cgk said:
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?
 
  • #8
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...
 
  • #9
Chemist20 said:
the link takes you to a google page were the shape of different orbitals appear.
No, I just googled for "spherical harmonics"
 
  • #10
He means that the pictures that show up remind him of the pictures of orbitals.
 
  • #11
mr. vodka said:
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.
 
  • #12
Just wondering: do you know how orbitals are drawn? Where and how does the 90% come in?
 
  • #13
mr. vodka said:
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.

mr. vodka said:
Where and how does the 90% come in?
Don't know; b/c it looks nice?
 
  • #14
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?
 
  • #15
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.
 
  • #16
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?
 
  • #17
mr. vodka said:
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

[tex]\int_{V:\, |\psi|^2 = p\text{ on }\partial V} dV\,|\psi(x)|^2 = P[/tex]
 
  • #18
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.
 
  • #19
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".
 
  • #20
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.
 
  • #21
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!
 
  • #22
mr. vodka said:
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 < ra < rb < rc < rd < ... where the probability for [0, ra] is small, the probability in [ra, rb] is large, the probability in [rb, rc] is small ...

That means that the orbital may not consist of one ball but of several spherical shells, e.g. [ra, rb], [rc, rd], ... 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
 
Last edited by a moderator:

1. What is the angular part of the wavefunction?

The angular part of the wavefunction is a component of the overall quantum wavefunction that describes the spatial orientation and shape of a particle's wave. It is responsible for determining the probability of finding a particle at a particular location in space.

2. How is the angular part of the wavefunction related to the radial part?

The angular part of the wavefunction is related to the radial part through the separation of variables technique, which allows us to solve the Schrödinger equation for a multi-dimensional system by separating it into simpler one-dimensional problems. The angular part is dependent on the radial part, but they are solved separately and then combined to form the complete wavefunction.

3. What is the significance of the angular part of the wavefunction?

The angular part of the wavefunction is significant because it contains information about the angular momentum of a particle. This includes its orbital angular momentum, which determines its shape and orientation, as well as its spin angular momentum, which describes its intrinsic angular momentum. The total angular momentum of a particle is a conserved quantity in quantum mechanics and plays a crucial role in many physical phenomena.

4. How is the angular part of the wavefunction affected by external forces?

The angular part of the wavefunction can be affected by external forces, such as electric or magnetic fields, which can alter the shape and orientation of the particle's wave. This is because these forces can change the energy and momentum of the particle, which are related to its angular momentum. The resulting changes in the particle's wavefunction can then be observed in its behavior and interactions with other particles.

5. Can the angular part of the wavefunction be measured directly?

No, the angular part of the wavefunction cannot be measured directly. It is a mathematical description of the quantum state of a particle and does not have a physical representation. However, its effects can be observed through experiments and measurements of other physical quantities, such as the particle's energy or position, which are related to the angular part of the wavefunction.

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