Plots of Quantum Probability Density Functions in the Hydrogen Atom

The main goal of this Demonstration is to plot 3D density clouds of the position of the electron in the hydrogen atom in states defined by the three quantum numbers (principal), (azimuthal), and (magnetic). Each dot of the cloud represents a possible result of a measurement of the position of the electron in an individual atom. By imagining that the measurement is repeated many times in different atoms at the same quantum state, you can get a plot representing the probability density function associated with that state. A 2D view can also be obtained by a plane slice containing the axis. You can select the number of position measurements to be simulated, the quantum numbers , , and (or all values of combined in a single plot), and the type of view (3D or 2D). In the 2D slice view you can choose the slice orientation (four possibilities) or all four slices combined for better statistics. The length unit is set equal to the Bohr radius.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


In spherical coordinates, the wavefunction associated with the quantum state (, , ) is , where is a spherical harmonic and , where is a normalization constant and is a generalized Laguerre polynomial. The probability density function,, is independent of and of the sign of .
[1] R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, New York: Wiley, 1985.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+