Spherical symmetri of eigenfunctions?

In summary: Thanks for any help you can give!In summary, the four eigenfunctions of the hydrogen atoms are: psi_{100}(r, \theta, \phi), psi_{200}(r, \theta, \phi), psi_{210}(r, \theta, \phi), and psi_{21\pm 1}(r, \theta, \phi). These functions are sphericaly symmetric within each shell, and the superposition of these functions has the same value in a arbitrary r and -r.
  • #1
Azael
257
1
I have been given these 4 eigenfunctions of the hydrogen atoms first 2 n-shells.

[tex] \psi_{100}(r, \theta, \phi )=\frac{1}{\sqrt{\pi a^3_0}}e^{-r/a_0} [/tex]

[tex] \psi_{200}(r, \theta, \phi )=\frac{1}{\sqrt{8\pi a^3_0}}(1-\frac{r}{2a_0})e^{-r/2a_0} [/tex]

[tex] \psi_{210}(r, \theta, \phi )=\frac{1}{4\sqrt{2\pi a^3_0}}(\frac{r}{a_0})e^{-r/2a_0} cos\theta[/tex]

[tex] \psi_{21\pm 1}(r, \theta, \phi )=\pm\frac{1}{8\sqrt{\pi a^3_0}}(\frac{r}{a_0})e^{-\frac{r}{2a_0}} sin\theta e^{\pm i\phi}[/tex]

Where [tex] a_0 [/tex] is the Bohr radius.

I am suposed to show that the superposition
[tex] |\psi_{nlm}(r, \theta, \phi )|^2 [/tex] is sphericaly symmetric within each shell.

Now what I don't know is how do I show spherical symmetri(not even generaly and not just in this particular case). Any hints?:confused:
 
Physics news on Phys.org
  • #2
do I basicly just show that the superposition has the same value in a arbitrary r and -r? and the same for [tex] \phi [/tex] and [tex] \theta [/tex]
 
  • #3
I guess I should rephrase.

How do I show that any function is sphericaly symetric? Right now my brain has frozen even though I know it must be ridicilously simple.

Ignore the -r brainfart in the previous post btw.

What I mean is that should I show the function is the same if I hold r, theta constant and replace phi with -phi and then if I hold r and phi constant and replace theta with -theta?
 
Last edited:
  • #4
the first one
[tex] |\psi_{100}|^2 = \frac{1}{\pi a_0^3}e^{-\frac{2r}{a_0}} [/tex]
this is obviously sphericaly symmetric since it only depend on the radius. Its a sphere pure and simple. But is that proof enough?

The second one

[tex] |\psi_{200}|^2 = \frac{1}{8\pi a_0^3}[1-\frac{r}{2a_0}]^2 e^{-\frac{r}{a_0}} [/tex]

same as above? only radialy dependant=the function gest weaker in the radial direction, obviously spericaly symetric.

third one

[tex] |\psi_{210}|^2=\frac{r}{32\pi a^3_0}e^{-\frac{r}{a_0}} (cos\theta)^2[/tex]

Now this one. If I hold r constant and replace theta with -theta I get the same answere. So does that mean sphericaly symetric? Do I just have to say that to prove it?

In the fourth one at the end

The [tex] sin\theta e^{\pm i\phi} [/tex] should I read that as [tex] sin(\theta e^{\pm i\phi}) [/tex] or [tex] sin(\theta) e^{\pm i\phi} [/tex] ??
 
Last edited:

1. What is spherical symmetry in eigenfunctions?

Spherical symmetry in eigenfunctions refers to the property where the eigenfunctions of a system exhibit the same behavior under rotation around a fixed point. This means that the shape and orientation of the eigenfunctions remain unchanged when the system is rotated.

2. How is spherical symmetry related to the shape of the potential energy surface?

In quantum mechanics, the potential energy surface is a representation of the energy of a system as a function of its coordinates. The shape of this surface can provide information about the symmetry of the system. If the potential energy surface exhibits spherical symmetry, it means that the system has a spherical symmetry and its eigenfunctions will also exhibit this symmetry.

3. Can eigenfunctions have other types of symmetries besides spherical symmetry?

Yes, eigenfunctions can exhibit other types of symmetries such as axial symmetry, planar symmetry, or no symmetry at all. The type of symmetry of the eigenfunctions depends on the type of potential energy surface and the physical properties of the system.

4. How does spherical symmetry affect the energy levels of a system?

In systems with spherical symmetry, the energy levels are typically degenerate, meaning that they have the same energy value. This is because the eigenfunctions with different orientations have the same energy. However, if the symmetry is broken, the degeneracy is lifted and the energy levels become non-degenerate.

5. What are some real-life examples of systems with spherical symmetry?

One example is the hydrogen atom, where the electron is confined by a spherical potential created by the positively charged nucleus. The eigenfunctions of the hydrogen atom exhibit spherical symmetry. Another example is a water molecule, where the molecule has a spherical symmetry and its eigenfunctions also exhibit this symmetry.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
1K
Replies
6
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
874
  • Advanced Physics Homework Help
Replies
7
Views
927
  • Advanced Physics Homework Help
Replies
1
Views
250
  • Advanced Physics Homework Help
Replies
2
Views
1K
Replies
1
Views
641
  • Advanced Physics Homework Help
Replies
0
Views
602
  • Advanced Physics Homework Help
Replies
1
Views
799
Back
Top