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
Snapshots
Details
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
.
References
[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.
Permanent Citation