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Hydrogen atom

Three analytically solvable problems in quantum physics are: 
1. Hydrogen atom
 
2. Particle in a potential well 
3.
Quantum oscillator
 
For other quantum problems can be solved by starting from one of the problems above


Theory of the hydrogen atom

The hydrogen atom consists of an electron interacting with a proton through the Coloumb force.

The state of the electron is described by its wave function,

The magnitude of the wave function squared gives us the probability to find the electron in a region around the nucleus of the atom.


The wave function is normalized, the probability to find the electron in the all space is 1.

The wave function is the solation of the Schrodinger Equation:

http://phelafel.technion.ac.il/~joeyfox/Hydrogen/Sch_Eq.PNG

The normalized position wave functions, given in spherical coordinates are:

 \psi_{n\ell m}(r,\vartheta,\varphi) = \sqrt {{\left (  \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n(n+\ell)!} } e^{- \rho / 2} \rho^{\ell} L_{n-\ell-1}^{2\ell+1}(\rho) Y_{\ell}^{m}(\vartheta, \varphi )

where

-The coordinate transformations are:

http://phelafel.technion.ac.il/~joeyfox/Hydrogen/sph_coord.png

-         is Bohr radius 



The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state.

 

There are three quantum numbers:

-         n is an integer resulting from the quantization of energy.

-         l is an integer resulting from quantization of angular momentum squared.

-         m is an integer resulting from quantization of angular momentum.

 

After solving the Schrodinger's Equation we get a boundary condition for n, l and m:


Clarification: m can be a negative number; its meaning is the projection of the angular momentum on axis z.

The probability is a function of n, l and |m|

For m=1 and m=-1 the probability is identical:

 

http://phycomp.technion.ac.il/~webteach/phys3/ph114053/adler/atom/hatom3.gif

 

 

The equation is solved by making a separation of variables

-         is the generalized Laguerre polynomial .

Its' the solution for the part that depends on r (equals to r).

 

L_n^{(\alpha)}(x)=

{x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right).

 

-          is the spherical harmonic.

Its' the solution for the part that depends on and.

 

 Y_\ell^m( \theta , \varphi ) = \sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!\over (\ell+m)!}}  \, P_\ell^m ( \cos{\theta} ) \, e^{i m \varphi }


Hydrogen atom visualizations

Sixty years ago



Today


1s m=0
2p m=1
2p m=0
3d m=2
3d m=1
3d m=0
4f m=3
4f m=2
4f m=1
4f m=0
2s m=0
3p m=1

After putting on your 3D glasses , click on the one of the images in the table above and observe.
For more information on the 3D stereo see Dan Peled's website.


 

How to create the Hydrogen Atom Visualizations?

The visualizations were creating by Aviz, which uses with .xyz files, in a LINUX system.

(More information about Aviz can be found here: http://phycomp.technion.ac.il/~newaviz)

The original visualizations of the hydrogen atom wave function were made by Joey Fox,

see: http://phelafel.technion.ac.il/~joeyfox/Hydrogen/Hydrogen.html.

In this project I have added stereo capability and much clearer instructions for creating the visualizations.

Create a file:

For creating the .xyz file we use a MATHEMATICA code:

http://phelafel.technion.ac.il/~joeyfox/Hydrogen/Hydrogen2.nb

In this code the electron density points in 3D space is calculated.

To view this, we draw a volume box around each point and place a number of dots (proportional to the density value) at random points of the box.

We also use a different color for different densities, but the color change is not continuous.

These dot points at locations (x,y,z) are the basis for the visualization.

For every energy state (n,l,m) we control 6 parameters:

multFactor - factor which multiples the probability function when we use at the function Ruond (function to calculate number of random points to create within a certain box) in the code.

mini/maxcoordinate- number for the max/min coordinate that of the x, y and z values.

rangeOne/Two/Three/Four- ranges that determine which colors to use in the code.

After setting the values and running the notebook, we get a file with the name "DataOrbital_n_l_m" .

On the first line we have the six parameters in the same order as described above.

The remainder of this file contains a number of points in space, each point has a color that represents the probability of the electron to be at that point.

For using the file in Aviz we must add the number of lines containing x,y,z coordinates at the top and change the format of the file to .xyz by the order: mv file name file name.xyz.

 

 

Creating visualization:

Open Aviz and go to file->open->open xyz file

Choose the file you want to work with.

Then go to elements->Atoms->Atoms/Molecules a, and pick the color you want for the atoms.

You can work with the color code or other colors of your choice.

Hence the resulting image (example):

Button 1- black background.

Button 2- white background.

Button 3 - obtaining a three dimensional stereo image.

Button 4/5- select the distance between the red and blue color in the 3D image.

Recommendation: for 3D image choose colors so that the red and blue is not absorbed in.

 

Button Auto - the box begins to turn, for selecting the direction of the turn we have the arrows on the left side of the screen.

Every second the program creates an image and saves it in the folder where the .xyz file is.

For creating a movie go to File -> Animation, select the folder and press the button "Create Animation".

Be aware that the program takes all the images in the folder for the movie, so before you begin the animation delete all unnecessary images.

Additionally, the images are numbered (you can see it at the name of the file image, e.g. for n,l,m=1,0,0 the ninth file is -> Orbital_Data_1_0_0.0009.png) take care that the all images are arranged in ascending order, if it not the movie will not come out in a continuous manner.

To add captions to an image go to Elements->Annotation.

To crop the image go to View->Slicing.

 

 

 

 

 

These visualizations are helpful for students in Modern Physics (Physics 3 and 3H) and Quantum Mechanics I at the Technion.

Back to the Computational Physics Class index page
Updated: April 2013