Harmonic Oscillator as an Effective Theory
The concepts of Effective Theories are illustrated allegorically within the context of one of the most ubiquitous models of oscillating physical phenomena—the harmonic oscillator.
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Harmonic Oscillator as an Effective Theory
Abstract The concepts of Effective Theories are illustrated allegorically within the context of one of the most ubiquitous models of oscillating physical phenomena—the harmonic oscillator.
2.1 Basics of the Harmonic Oscillator The concepts and issues related to effective theories can be illustrated quite nicely by the harmonic oscillator problem. The harmonic oscillator is one of the most ubiquitous mathematical models of physics phenomena. It is present in almost every system with a restoring force, which includes the galaxy, solar system, springs, atoms, molecules, and innumerable other configurations. The main point I would like to illustrate is that the lowest order effective potential for the harmonic oscillator is an excellent approximation to the motion of a system over a wide range of amplitudes. However, at some point it breaks down when the amplitude is large enough, and then control over the system is lost unless a deeper theory is understood. We shall not go into the construction of deeper theories in this chapter, but rather focus on the domain of applicability of the harmonic oscillator effective theory, and show how small corrections can be anticipated and then measured by precise experiments to start building a more complete picture of the potential governing the system. To keep the illustration simple, we will restrict ourselves to one-dimensional harmonic motion of a particle subject to the restoring potential V (x) = kx 2 /2. The Lagrangian of the system is then L=
2 x2 x˙ −k . dt m 2 2
J. D. Wells, Effective Theories in Physics, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-34892-1_2, © The Author(s) 2012
(2.1)
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2 Harmonic Oscillator as an Effective Theory
From the principle of least action the equation of motion gives Newton’s second law of motion F = ma the form m x¨ = −kx =⇒ m x¨ + kx = 0.
(2.2)
Defining ω2 = k/m, we can rewrite this as x¨ + ω2 x = 0
(2.3)
x(t) = A sin(ωt)
(2.4)
which has the solution where A is the amplitude, and the boundary condition of x = 0 at t = 0 is enforced. Let us review a few basic facts about the harmonic oscillator solution. The period is T period
2π = 2π = ω
m . k
(2.5)
The amplitude A of motion is related to the initial velocity by equating full potential energy at maximum amplitude to the full kinetic energy at maximum velocity: 1 2 1 mv = k A2 =⇒ A = vmax 2 max 2
vmax T period vmax m = = . k ω 2π
(2.6)
It should also be noted that the period of the harmonic motion is not dependent on the amplitude of the motion. This is clear from Eq. 2.5 where it is shown that the period only depends on the input parameters m and k. The amplitude and √ maximum velocity conspire with each other such that vmax /A is always equal to k/m.
2.2 Ubiquity of the Harmonic Oscillator The harmonic oscillator problem is ubiquitous in physics, describing small motions of an object attached to a string, molecules vibrating in crystals, electrical circuit response, etc. There is a straightforward reason w
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