Probability Density for a Classical Harmonic Oscillator

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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 .

Contributed by: Muhamad Hamdi and Walana Yumna Aribah (January 2012)
Open content licensed under CC BY-NC-SA



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.

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