Schrödinger wave function: How to use it to get 3-D atomic orbitals?

[Tymothee Waldner]

Hi, I am 16 year old and I am very interested in Physics.

This summer I solved Schrödinger equation using griffiths' introduction to quantum physics and other sources. I achieved to get an exact solution of the wave function but I would like to plot it in a programm in order to get the 3d atomic orbital. My function for 1s energy level is the following:

(e^(-r/a))/(sqrt(pi*a))= psi

Where a is the bohr radius. It is in spherical coordonates and according to griffiths' book it is correct. What should I do to obtain 3d orbital ? Can I find some program able to convert it into a 3d graph ? Should I programm it myself ? In this last case could you give me some hints please ?

Thank you very much for your help !

 

[Frabjous]

Try wolfram alpha www.wolframalpha.com

 

[Tymothee Waldner]

Try wolfram alpha (wolframalpha.com)

I already tried and it does not work, I suppose it is because my function is not expressed in cartesian coordonates however I don't know how to pass it back to this system...

But thank you for your help !

 

[Frabjous]

Sorry. You want to plot a value that is a function of 3 variables xyz; i.e. you want to plot 4 things. Since we live in a 3d world, you are going to have to make artistic decisions on what you plot. If you look at the images you get if you google atomic orbitals, the artistic decision was to emphasize the directionality of the orbitals. The 1s orbital is spherically symmetric, so it is going to look the same from any direction. It has a maximum value at r =0 and undergoes exponential decay in the radial direction from the origin.

 

[Tymothee Waldner]

Sorry. You want to plot a value that is a function of 3 variables xyz; i.e. you want to plot 4 things. Since we live in a 3d world, you are going to have to make artistic decisions on what you plot. If you look at the images you get if you google atomic orbitals, the artistic decision was to emphasize the directionality of the orbitals. The 1s orbital is spherically symmetric, so it is going to look the same from any direction. It has a maximum value at r =0 and undergoes exponential decay in the radial direction from the origin.

Okay I think I understoof what you said... So I should pass my actual function into a cartesian system and then plot it into walfram alpha. Should I only convert the r of my function into its cartesian coordonates equivalence (sqrt (x²+y²+z²)), or do I have to do more works ? And finally what do you call "artistic decisions" ?

I am very grateful for your help (sorry for my writing errors I am french...)

 

[Frabjous]

As an example, you will want to find something that can plot e^{-√(x2+y2+z2)}=0.5
If it can handle this, it should be able to handle more complicated orbitals (I couldn’t get wolfram alpha to do this)
You could solve this yourself and then just plot the solution points.
Artistic in this case is picking 0.5 (i.e., making the decision on what brings out what you want).

Sorry I couldn‘t be of more help.

 

[Frabjous]

This page does orbitals in mathematica
https://community.wolfram.com/groups/-/m/t/840468

 

[Tymothee Waldner]

As an example, you will want to find something that can plot e^{-√(x2+y2+z2)}=0.5
If it can handle this, it should be able to handle more complicated orbitals (I couldn’t get wolfram alpha to do this)
You could solve this yourself and then just plot the solution points.
Artistic in this case is picking 0.5 (i.e., making the decision on what brings out what you want).

Sorry I couldn‘t be of more help.

Oh okay I see what you mean. I am going to try all of this today, and if it does not work I will keep investigating.

Thank you very much !

 

[aaroman]

One way would be to do a volumetric visualization of the probability density.

Sort of like this:

 

[Tymothee Waldner]

One way would be to do a volumetric visualization of the probability density.

Sort of like this:

Hi, thank you for your contribution, I am going to look more in deep on this way. I've been investigating and I am now thinking that orbitals are isosurfaces of the squared of my function (as psi²= probability of finding the electron). Do you know some program able to show isosurfaces of a function (wolfram alpha does not do it) ?

 

[aaroman]

I used VTK https://vtk.org/ for that implementation. In that case, it's volumetric visualization, not isosurfaces, but those are possible as well with VTK (see for example https://www.evl.uic.edu/aspale/cs526/final/3-5-2-0.htm ). Other packages/libraries have also that possibility.

PS
The code is available here https://github.com/aromanro/DFTQuantumDot for those that want to take a look. I also have a python repository https://github.com/aromanro/PythonCompphys, the Density Functional Theory notebook covers the same things that are in the C++ project.

 

[Tymothee Waldner]

I used VTK https://vtk.org/ for that implementation. In that case, it's volumetric visualization, not isosurfaces, but those are possible as well with VTK (see for example https://www.evl.uic.edu/aspale/cs526/final/3-5-2-0.htm ). Other packages/libraries have also that possibility.

PS
The code is available here https://github.com/aromanro/DFTQuantumDot for those that want to take a look. I also have a python repository https://github.com/aromanro/PythonCompphys, the Density Functional Theory notebook covers the same things that are in the C++ project.

Thank you so much for your help ! So I will use this way to obtain my orbitals. Did you do all that stuff yourselves ? It is very interesting