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PHYS195 NotesChapter6

# PHYS195 NotesChapter6 - Chapter 6 Special Classical...

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Chapter 6 Special Classical Physical Systems 6.1 Introduction In order to understand the ideas of modern physics, it is essential to under- stand the operations of some special classical systems. Not only do these provide a physical intuition but also a vocabulary. In the previous chapter, Chapter 5, we dealt in some detail with two important physical systems, the free particle and the particle moving in a constant force. These were dealt with there to illustrate the principles and uses of symmetry and action. They obviously belong to the category of “Special Classical Physical Sys- tems” but since they were treated there will not be treated here. Instead we will deal with the harmonic oscillator as an example of a more complicated but still simple system and the string as an example of a field system. 6.2 The Harmonic Oscillator 6.2.1 Importance After the free particle, the harmonic oscillator is the most important me- chanical system. Harmonic oscillators or systems that are almost harmonic oscillators are ubiquitous in nature. These are basically objects that when disturbed slightly return to there starting position but because of inertia overshoot and jiggle. The simplest example is the simple spring with a mass on the end. The general definition is that the system is a harmonic oscillator if the 189

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190 CHAPTER 6. SPECIAL CLASSICAL PHYSICAL SYSTEMS Figure 6.1: A mass and Hook’s Law Spring A mass, m, on the end of an ideal spring is an example of a harmonic oscillator. An ideal spring or Hook’s Law spring,is one in which the force at the end of the spring is proportional to the stretch of the spring, F = k ( x - x 0 ). force on the system that emerges from movement from equilibrium is pro- portional to the amount of movement from equilibrium and is directed to remove the displacement from equilibrium. Defined this way, harmonic oscil- lators come in lots of forms. A mass on the end of a string suspended above the earth, if displaced to the side by a small amount is a harmonic oscillator. A shallow pan filled with water sloshes back and forth when disturbed and can be analyzed as a harmonic oscillator. We will discuss these examples in Section #6.2.3 In a very real sense, any object that is held in place but still moves a little about that fixed point is generally well approximated by the harmonic oscillator system. Even more important to our purposes, we will find that the harmonic oscillator is essential to the modern interpretation of the nature of particles. The quantum harmonic oscillator is the only system that can provide a framework for creating a quantum field theory satisfies the requirements of having a particle interpretation. 6.2.2 Dynamics In the most general case, for a mass that can move freely in space, since acceleration and force are vector quantities F = ma , a harmonic oscillator is a system which obeys:
6.2. THE HARMONIC OSCILLATOR 191 ma = - k ( x - x 0 ) , (6.1) where a is the acceleration of the position of the block and x and x 0 are the position and neutral position of the mass. k is called the spring constant.

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PHYS195 NotesChapter6 - Chapter 6 Special Classical...

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