Atomic Quantum Mechanics:

Visualizing the Wave Function

Introduction: A fundamental principle of quantum mechanics (the Heisenberg uncertainty principle) tells us that it is not possible to simultaneously know the exact position and velocity of  a particle.  As a result, we can never describe the state of an electron as being at a particular location at a certain time.  The best we can do is to specify the "wave function" for the electron.

The meaning of the wave function, , is that the absolute value squared of the wave function, , gives the probability of finding the electron at position (x,y,z).  The absolute value squared is always positive or zero, as probabilities must be.

The wave function itself can be positive or negative (in fact, it can also equal a complex number), so that two wave functions can add constructively or destructively.  If wave functions on neighboring atoms have the same sign, they add constructively (the sum is bigger than either of the individual wave functions) and this makes a bonding orbital.  If the wave functions have opposite signs, they combine destructively (the sum is smaller than the individual wave functions) and this makes an anti-bonding orbital.  We will learn more about how the combination of wave functions gives rise to chemical bonding interactions in the next chapter.

  • You can run an interactive JAVA applet, which allows you to choose an atomic configuration and calculate the wave functions for each electron in the atom.
  • You can visit topic pages to help you master the concepts of atomic quantum mechanics.
  • Each page poses various questions to enhance your understanding.  Many of these questions prompt you to run the JAVA applet to help you answer questions about atomic quantum mechanics.

Objectives:  The main goal of this module is to enrich your
understanding of this important scientific principle through explanation,
illustration, and exploration.  As a result of using this module, you will be
able to achieve these objectives:

  • to explain what a wave function is by defining it in your own words
  • to explain what a radial node is by defining it in your own words
  • to explain what an angular node is by defining it in your own words
  • to explain Heisenberg uncertainty by defining it in your own words
  • to illustrate how wave functions change when atoms are ionized by computing wave functions for various electronic configurations and describing each result
  • to illustrate how wave functions with different principal and angular momentum quantum numbers differ by graphing various wave functions and describing the results
  • to analyze how wave functions for different atoms differ by solving for wave functions of different atoms and comparing the result
  • to analyze how excited and ionized atomic configurations affect the wave functions by solving for wave functions in various configurations and comparing the results
  • to synthesize the results of your explorations by writing an essay of your own description of atomic quantum mechanics, including your analysis of the effects of ionization, excitation, and change of element on the electronic wave functions.
Getting Started:  You can select your own way to use this
module.  Visit the topic pages by using the outline below or the arrow
buttons above.  (Topic pages will be online soon!)  Explore freely by running the applet on your own, or use the applet to help you answer some of the questions on the topic pages.  It's up to you!


Mr. Eric J. Walter   &  Prof. Andrew M. Rappe
Prof. Charlotte R. Zales
Department of Chemistry
Department of Education
University of Pennsylvania
Immaculata College
231 South 34th Street
Loyola Hall
Philadelphia, PA 19104-6323
Immaculata, PA 19345
ewalter  (at)
czales2  (at)
rappe  (at)

© Andrew M. Rappe