   Next: Quantum Harmonic Oscillator Up: ho Previous: Introduction

# Classical Harmonic Oscillator

Consider two masses and at positions and , connnected by a spring with spring constant . If the rest length of the spring is , then the two equations governing the motion of the masses are   (1)   (2)

It will be more convenient to work in terms of , which is the amount by which the spring is stretched or compressed from its natural length. Denoting this quantity simply as , the equations reduce to   (3)   (4)

Subtracting the second equation from the first gives   (5)

It will be convenient to introduce the total mass and center of mass coordinates, defined as   (6)   (7)

so that we now have   (8)

This means that the center of mass is not accelerating or decelerating, but is either at rest or moves with constant velocity; of course this simply reflects the fact that there is no external force acting on the two masses.

Now dividing (3) by and (4) by and subtracting the second equation from the first, we can obtain   (9)

Since , we have . If we also introduce the reduced mass   (10)

we obtain   (11)

which is a second-order differential equation describing the displacement from the rest length as a function of time. This can be solved to yield   (12)

where is defined as   (13)

and represents the frequency of oscillation (in rad s ) of the oscillator. One could also define a frequency in Hertz ( ) through .   Next: Quantum Harmonic Oscillator Up: ho Previous: Introduction
David Sherrill 2002-10-16