Visualizing the Probability Density Function of a Classical Harmonic Oscillator

In classical mechanics, the solution of equations of motion of a physical system usually gives well-defined trajectories. With the help of these trajectories, the future progress of the system can be predicted.¹ Therefore, a probability density function (PDF) is not required for such systems and is rarely discussed in classical mechanics. The foundation of quantum mechanics is based on the concept of probability, and it is represented by the square of the wave functions that are solutions to the Schrödinger equation for a given system.^2,3 In addition, the textbooks refer to the simple harmonic oscillator as an example to compare the quantum and classical probability distribution functions and further explain Bohr's correspondence principle. Therefore, it becomes important to discuss the PDF for a simple harmonic oscillator in a classical framework before introducing the quantum harmonic oscillator to undergraduate students. However, the PDF makes sense in the classical context...