Probability Density for a Classical Harmonic Oscillator

A classical harmonic oscillator with mass and spring constant has a total energy , dependent on its amplitude . We determine the probability density as the position varies between and , making use of its oscillation frequency (or period ). Thus we find the probability density function where representing the probability that the mass would be found in the infinitesial interval to .


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


Given a classical harmonic oscillator with particle mass , much can be calculated using the principle of energy conservation. We have
where is the oscillating amplitude and is the displacement. We need to find the result of integration for the period ,
Therefore we obtain the probability density function as a function of its total energy and displacement
The result of classical harmonic oscillator mechanical behavior analyzed by the probability density function shows that the characteristic of this density function with six kinds of spring constants, where varies from to .
Snapshots 1, 2, 3, and 4 describe the behavior of the density function for six different spring constants moving simultaneously from its equilibrium point with the given total energy. The density function is plotted against its change of particle position. Graphs show that the lowest spring constant has the furthest displacement. The sharp vertical lines on the graph indicate the furthest displacement from each particle position. These lines are a vertical asymptote at the furthest displacement limit of each particle position showing an infinite value of this function. For the change of particle position , it increases the total energy indicating the decrease of density function towards zero. In fact, dynamically, the difference of the spring constant factor causes the change of the oscillation amplitudes . As a result of this function, a group of the classical harmonic oscillators doing the vibration motion from the equilibrium position to the largest displacement simultaneously may result in harmonic and no harmonic motions. The smallest shows that the value of this function remains constant longer than the larger with increasing the total energy. This can be studied from the results of demonstration for the four harmonic graphs with determination of the certain position from the equilibrium point .
[1] D. A. B. Miller, Quantum Mechanics for Scientists and Engineers, Cambridge: Cambridge University Press, 2007.
[2] Y.-K. Lim, Problems and Solutions on Thermodynamics and Statistical Mechanics, Singapore: World Scientific, 1990.
    • 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+