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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.
Mathematically, =, where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m −1), and x is the displacement from the equilibrium position (in metres). For any simple mechanical harmonic oscillator:
where v(x) is the speed of the oscillator as a function of its position. (Note that because speed is a scalar, v(x) is the same for both half periods.) At this point, all that is needed is to provide a function v(x) to obtain P(x). For systems subject to conservative forces, this is done by relating speed to energy.
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 potential energy U of such a system can be determined through the spring constant k and its displacement x: [14] = The kinetic energy K of an object in simple harmonic motion can be found using the mass of the attached object m and the velocity at which the object oscillates v: [14]
(E.g. 1 mm diameter wire is ~18 AWG, 2 mm diameter wire is ~12 AWG, and 4 mm diameter wire is ~6 AWG). This quadruples the cross-sectional area and conductance. A decrease of ten gauge numbers (E.g. from 12 AWG to 2 AWG) multiplies the area and weight by approximately 10, and reduces the electrical resistance (and increases the conductance ) by ...
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]
The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. [2] Van der Pol found stable oscillations, [3] which he subsequently called relaxation-oscillations [4] and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes.