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A simple harmonic oscillator is an oscillator that is neither driven nor damped.It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point x = 0 and depends only on the position x of the mass and a constant k.
Since the force acting on the oscillator is conservative and the motion occurs over a finite domain, the motion will be cyclic with some period which will be denoted T. Since the probability of the oscillator being at any possible position between the minimum possible x-value and the maximum possible x-value must sum to 1, the normalization
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 Fradkin tensor is orthogonal to the angular momentum =: = contracting the Fradkin tensor with the displacement vector gives the relationship =. The 5 independent components of the Fradkin tensor and the 3 components of angular momentum give the 8 generators of (), the three-dimensional special unitary group in 3 dimensions, with the relationships
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 ...
For the harmonic oscillator, x and p enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L 2. [nb 5]
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 [2]
The zero-point energy E = ħω / 2 causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed. The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle.