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The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics.
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 ...
The result of Mehler can also be linked to probability. For this, the variables should be rescaled as x → x/ √ 2, y → y/ √ 2, so as to change from the 'physicist's' Hermite polynomials H (.) (with weight function exp(− x 2)) to "probabilist's" Hermite polynomials He (.) (with weight function exp(− x 2 /2)).
The Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the ...
In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an isotropic oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions.
[clarification needed] For each parameter there is a set of ladder operators; these are then a standardized way to navigate one dimension of the root system and root lattice. [2] The ladder operators of the quantum harmonic oscillator or the "number representation" of second quantization are just special cases of this
The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems. [2] For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook [ 3 ] ).
Harmonic oscillator; ... From the formula for = (,) and the coordinate-based definition of ... is an infinitesimal arc length. From the above formulation, one can ...