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The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal ...
Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency.
The Hooke's atom is a simple model of the helium atom using the quantum harmonic oscillator. Modelling phonons, as discussed above. A charge with mass in a uniform magnetic field is an example of a one-dimensional quantum harmonic oscillator: Landau quantization.
The harmonic oscillator and the systems it models have a single degree of freedom. More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of the others.
Starting with the example used in the derivation above, the simple harmonic oscillator has the potential energy function = =, where k is the spring constant of the oscillator and ω = 2π/T is the natural angular frequency of the oscillator.
The fundamentals of vibration analysis can be understood by studying the simple Mass-spring-damper model. Indeed, even a complex structure such as an automobile body can be modeled as a "summation" of simple mass–spring–damper models. The mass–spring–damper model is an example of a simple harmonic oscillator.
To see an example where Liouville's theorem does not apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping. Consider again the system of N {\displaystyle N} particles each in a 3 {\displaystyle 3} -dimensional isotropic harmonic potential, the Hamiltonian for which is ...
A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1).