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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.
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in ...
For example, a quantum harmonic oscillator may be in a state |ψ for which the expectation value of the momentum, | ^ | , oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector | ψ , the momentum operator p ^ {\displaystyle {\hat {p}}} , or both.
Such an oscillator, when quantized, is described by the mathematics of a quantum harmonic oscillator. [1] Quantum oscillators are described using creation and annihilation operators ^ † and ^. Physical quantities, such as the electric field strength, then become quantum operators.
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 ...
(Sections I to IV of this article provide an overview over the Wigner–Weyl transform, the Wigner quasiprobability distribution, the phase space formulation of quantum mechanics and the example of the quantum harmonic oscillator.) "Weyl quantization", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Terence Tao's 2012 notes on Weyl ordering
Some expanded derivations and an example of the harmonic oscillator in the Heisenberg picture ; The original Heisenberg paper translated (although difficult to read, it contains an example for the anharmonic oscillator): Sources of Quantum mechanics B.L. Van Der Waerden
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues.Examples are position and momentum of a particle, and the (dimension-less) electric field in the amplitude (phase 0) and in the mode (phase 90°) of a light wave (the wave's quadratures).