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An oscillator is a physical system characterized by periodic motion, such as a pendulum, tuning fork, or vibrating diatomic molecule.Mathematically speaking, the essential feature of an oscillator is that for some coordinate x of the system, a force whose magnitude depends on x will push x away from extreme values and back toward some central value x 0, causing x to oscillate between extremes.
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 The computations for the hydrogen atom in the Heisenberg representation originally from a paper of Pauli [3]
The equation for describing the period: = shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. The above equation is also valid in the case when an additional constant force is being applied on the mass, i.e. the additional constant force cannot change the period of oscillation.
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)).
Quantity (common name/s) (Common) symbol/s SI units Dimension Number of wave cycles N: dimensionless dimensionless (Oscillatory) displacement Symbol of any quantity which varies periodically, such as h, x, y (mechanical waves), x, s, η (longitudinal waves) I, V, E, B, H, D (electromagnetism), u, U (luminal waves), ψ, Ψ, Φ (quantum mechanics).
The forced Duffing oscillator with cubic nonlinearity is described by the following ordinary differential equation: ¨ + ˙ + + = (). The frequency response of this oscillator describes the amplitude z {\displaystyle z} of steady state response of the equation (i.e. x ( t ) {\displaystyle x(t)} ) at a given frequency of excitation ω ...
where is the oscillation amplitude and is a constant defined by the anharmonic coefficients. Second, the shape of the resonance curve is distorted ( foldover effect ). When the amplitude of the (sinusoidal) external force F {\displaystyle F} reaches a critical value F c r i t {\displaystyle F_{\mathrm {crit} }} instabilities appear.
Defining equation SI unit Dimension Wavefunction: ψ, Ψ To solve from the Schrödinger equation: varies with situation and number of particles Wavefunction probability density: ρ = | | = m −3 [L] −3: Wavefunction probability current: j: Non-relativistic, no external field: