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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.
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.
This force becomes so strong that it becomes the dominant force between uncharged conductors at submicron scales. In fact, at separations of 10 nm – about 100 times the typical size of an atom – the Casimir effect produces the equivalent of about 1 atmosphere of pressure (the precise value depends on surface geometry and other factors). [17]
The electromagnetic tensor has an electromagnetic four-potential field possessing gauge symmetry. The strong (color) interaction is mediated by gluons, which can have eight color charges. There are eight gluon field strength tensors with corresponding gluon four potentials field, each possessing gauge symmetry.
(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 ...
Without the presence of an electric field, the electrons have no net velocity. When a DC voltage is applied, the electron drift velocity will increase in speed proportionally to the strength of the electric field. The drift velocity in a 2 mm diameter copper wire in 1 ampere current is approximately 8 cm per hour. AC voltages cause no net movement.
As an isolated system the pendulum is a harmonic oscillator with a frequency of /. The potential energy of a spring is 1 2 k x 2 {\displaystyle {\tfrac {1}{2}}kx^{2}} where k is the spring constant and x is the displacement.
In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules (e.g. the partial derivative of L with respect to the z velocity component of particle 2, defined by v z,2 = dz 2 /dt, is ...