5 The Harmonic Oscillator
... has magnitude proportional to x, with proportionality constant k. (For example, a spring that obeys Hooke's law) At x = 0, F = 0, so x = 0 is a position of stable equilibrium. The force can be represented analytically as F = −kx. The negative sign shows that the direction of F is always toward the p ...
... has magnitude proportional to x, with proportionality constant k. (For example, a spring that obeys Hooke's law) At x = 0, F = 0, so x = 0 is a position of stable equilibrium. The force can be represented analytically as F = −kx. The negative sign shows that the direction of F is always toward the p ...
4, 7, 9, 13, 15 / 2, 6, 17, 18, 24, 29, 41, 48, 51, 54, 74
... The time required for a particle in simple harmonic motion to travel through one complete cycle (the period) is independent of the amplitude of the motion, even though at larger amplitudes the particle travels further. This is possible because, at larger amplitudes, the maximum speed of the particle ...
... The time required for a particle in simple harmonic motion to travel through one complete cycle (the period) is independent of the amplitude of the motion, even though at larger amplitudes the particle travels further. This is possible because, at larger amplitudes, the maximum speed of the particle ...
STOCHASTIC DYNAMICS OF LONG SUPPLY CHAINS WITH
... • For a large number of parts the function min{µ, V ρ} is, under the expectation, replaced by the function µ[1−exp(− Vµρ )] which has the same limiting behavior for large and small densities (the limits ρ → 0 and ρ → ∞). • The effect of the random on / off switches can be incorporated into the model ...
... • For a large number of parts the function min{µ, V ρ} is, under the expectation, replaced by the function µ[1−exp(− Vµρ )] which has the same limiting behavior for large and small densities (the limits ρ → 0 and ρ → ∞). • The effect of the random on / off switches can be incorporated into the model ...
CHAPTER 11: Through the Looking Glass
... A troubling inconsistency had escaped the attention of most classical physicists: physics described Nature as “schizophrenic.” Newtonian mechanics dealt with particles. Maxwellian electromagnetics dealt with waves. But particles and waves are mutually exclusive. Whereas particles are localized in s ...
... A troubling inconsistency had escaped the attention of most classical physicists: physics described Nature as “schizophrenic.” Newtonian mechanics dealt with particles. Maxwellian electromagnetics dealt with waves. But particles and waves are mutually exclusive. Whereas particles are localized in s ...
Coherent whistler emissions in the magnetosphere
... at which the phase speed of the waves equals their group speed. Periodic momentum exchange between the electrons and protons, mediated by Maxwell stresses, gives rise to the wave packet structures which are a kind of solitary Gendrin wave packets. The layout of the paper is as follows. Section 2 con ...
... at which the phase speed of the waves equals their group speed. Periodic momentum exchange between the electrons and protons, mediated by Maxwell stresses, gives rise to the wave packet structures which are a kind of solitary Gendrin wave packets. The layout of the paper is as follows. Section 2 con ...
... Region of validity of geometrical optics: features of interest should be much bigger than the wavelength – Problem: geometrical point objects/images are infinitesimally small, definitely smaller than !!! – So light focusing at a single point is an artifact of the geometric approximations – Moreov ...
Large-scale numerical simulations of ion beam instabilities in
... level and continue to drive them throughout the simulation; second, the parameters of our particle distributions are such that waves propagating in only one direction are subject to strong linear instability; finally, we use an electromagnetic PIC model, with three velocity dimensions, rather than a ...
... level and continue to drive them throughout the simulation; second, the parameters of our particle distributions are such that waves propagating in only one direction are subject to strong linear instability; finally, we use an electromagnetic PIC model, with three velocity dimensions, rather than a ...
Wave packet
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In physics, a wave packet (or wave train) is a short ""burst"" or ""envelope"" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating.Quantum mechanics ascribes a special significance to the wave packet; it is interpreted as a probability amplitude, its norm squared describing the probability density that a particle or particles in a particular state will be measured to have a given position or momentum. The wave equation is in this case the Schrödinger equation. It is possible to deduce the time evolution of a quantum mechanical system, similar to the process of the Hamiltonian formalism in classical mechanics. The dispersive character of solutions of the Schrödinger equation has played an important role in rejecting Schrödinger's original interpretation, and accepting the Born rule.In the coordinate representation of the wave (such as the Cartesian coordinate system), the position of the physical object's localized probability is specified by the position of the packet solution. Moreover, the narrower the spatial wave packet, and therefore the better localized the position of the wave packet, the larger the spread in the momentum of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg uncertainty principle,and will be illustrated below.