Harmonic oscillator

"If one begins by considering a kind of state or condition for Bose particles which do not interact with each other (we have assumed that the photons do not interact with each other), and then considers that into this state there can be put either zero, or one, or two, ... up to any number n of particles, one finds that this system behaves for all quantum mechanical purposes exactly like a harmonic oscillator. By such an oscillator we mean a dynamic system like a weight on a spring or a standing wave in a resonant cavity. And that is why it is possible to represent the electromagnetic field by photon particles. From one point of view, we can analyze the electromagnetic field in a box or cavity in terms of a lot of harmonic oscillators, treating each mode of oscillation according to quantum mechanics as a harmonic oscillator. From a different point of view, we can analyze the same physics in terms of identical Bose particles. And the results of both ways of working are always in exact agreement. There is no way to make up your mind whether the electromagnetic field is really to be described as a quantized harmonic oscillator or by giving how many photons there are in each condition. The two views turn out to be mathematically identical. So in the future we can speak either about the number of photons in a particular state in a box or the number of the energy level associated with a particular mode of oscillation of the electromagnetic field. They are two ways of saying the same thing. The same is true of photons in free space. They are equivalent to oscillations of a cavity whose walls have receded to infinity."

"Many of the mechanical elements constituting a musical instrument behave approximately as linear systems. By this we mean that the acoustic output is a linear function of the mechanical input, so that the output obtained from two inputs applied simultaneousl is just the sum of the outputs that would be obtained if they were applied separately. For this statement to be true for the instrument as a whole, it must also be true for all of its parts, so that deflections must be proportional to applied forces, flows to applied pressures, and so on. Mathematically, this property is reflected in the requirement that the differential equations describing the behavior of the system are also linear, in the sense that the dependent variable occurs only to the first power. An example is the equation for the displacement y of a simple harmonic oscillator under the action of an applied force F(t): m \frac{\mathrm{d}^2y}{\mathrm{d}t^2} + R\frac{\mathrm{d}y}{\mathrm{d}t} + Ky = F(t), ... where m, R, and K are, respectively, the mass, damping coefficeint, and spring coefficent, all of which are taken to be constants. ... A little consideration shows, of course, that this description must be an over-simplification ..."

"The simple mechanical system of the classical harmonic oscillator underlies important areas of modern physiccal theory. ... The concept of degeneracy arises in the two-dimensional oscillation of a square plate or diaphragm. Three-dimensional harmonic oscillation relates to oscillatory modes in the Rayleigh-Jeans equation ... Vibration of a macroscopic three-dimensional crystal is treated by Debye's theory ... Harmonic oscillator theory is important when it succeeds and also when if fails, as we shall see in the motivation to find a theory of radiation that we now call the quantum theory ..."