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# Topic: Harmonic oscillator

 The simple harmonic oscillator   (Site not responding. Last check: 2007-10-16) Oscillations are central to the study of sound (and almost everything else in physics). The mass oscillating on a spring is the simplest example of the simplest kind of oscillation, known as simple harmonic oscillation. For simple harmonic motion, the oscillations are measured with respect to the position where the mass can remain at rest when it is not oscillating (this is called the equilibrium position). carini.physics.indiana.edu /P105S98/Simple-harmonic-oscillator.html   (1017 words)

 Harmonic oscillator - Wikipedia, the free encyclopedia If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude). In such situation, the frequency of the oscillations is smaller than in the non-damped case, and the amplitude of the oscillations decreases with time. In summary: at a steady state the frequency of the oscillation is the same as that of the driving force, but the oscillation is phase-offset and scaled by amounts that depend on the frequency of the driving force in relation to the preferred (resonant) frequency of the oscillating system. en.wikipedia.org /wiki/Harmonic_oscillator   (1810 words)

 More on Harmonic Oscillator A harmonic oscillator is either a mechanical system in which there exists a returning force F directly proportional to the displacement x, i.e. The motion of a simple harmonic oscillator, called simple harmonic motion, is essentially a sine function oscillating about the equilibrium displacement, x = 0, at which the returning force is zero. In summary: at steady state the frequency of oscillation is the same as the driving force, but the oscillation is phase offset and scaled by amounts that depend on the frequency of the driving force in relation to the preferred (resonant) frequency of the oscillating system. www.artilifes.com /harmonic-oscillator.htm   (967 words)

 4.1 Harmonic Oscillator The simple harmonic oscillator (SHO) is a mass connected to some elastic object of negligible mass that is fixed at the other end and constrained so that it may only move in one dimension. The amplitude that the oscillator eventually acquires depends on the relation of the driving frequency to the natural frequency of the oscillator and on the damping factor. In the damped harmonic oscillator we saw exponential decay to an equilibrium position with natural periodicity as a limiting case. hypertextbook.com /chaos/41.shtml   (1136 words)

 Harmonic Oscillator The harmonic oscillator is a canonical system discussed in every freshman course of physics. Nearly all oscillators and oscillations in physics are modeled by this equation of motion, at least in a first approximation, because it can be solved analytically. Another look at the dynamics of the damped and driven harmonic oscillator is the following one: Instead of discussing the solution as a function of time, discuss it as a function of the initial conditions. monet.physik.unibas.ch /~elmer/pendulum/harmosc.htm   (894 words)

 More on Quantum Harmonic Oscillator In the one-dimensional harmonic oscillator problem, a particle of mass m is subject to a potential V(x) = (1/2)mω2 x2. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3,... www.artilifes.com /quantum-harmonic-oscillator.htm   (2215 words)

 Simple Harmonic Motion A simple harmonic oscillator moves back and forth between the two positions of maximum displacement, at x = A and x = - A. Examples of simple harmonic oscillators include: a mass attached to a spring, a molecule inside a solid, a car stuck in a ditch being ``rocked out'' and a pendulum. The motion of any simple harmonic oscillator is completely characterized by two quantities: the amplitude, and the period (or frequency). theory.uwinnipeg.ca /physics/shm/node2.html   (298 words)

 The Forced Harmonic Oscillator   (Site not responding. Last check: 2007-10-16) After the transient motion decays and the oscillator settles into steady state motion, the displacement is in phase with force. Notice that the frequency of the steady state motion of the mass is the driving (forcing) frequency, not the natural frequency of the mass-spring system. Since the oscillator is being driven near resonance the amplitude quickly grows to a maximum. www.kettering.edu /~drussell/Demos/SHO/mass-force.html   (439 words)

 Harmonic Oscillator   (Site not responding. Last check: 2007-10-16) x is the displacement of the oscillator from equilibrium, x0, v is the velocity of the oscillator, and m is its mass. A C programwhich implements the Verlet Algorithm to step through the dynamics of a harmonic oscillator is available for browsing. To see the program in action, click here The results are displayed graphically using the relative displacement of the oscillator from equilibrium as the ordinate, and the fraction of the total energy of the oscillator which is stored as potential energy (or any one of four other options) as the abscissa. web.mit.edu /10.491-md/www/ICEMat_HO.html   (416 words)

 A Simple Harmonic Oscillator in a Thermal Bath If a simple harmonic oscillator is immersed in a thermal bath, then impacts with neighboring atoms change the phase and energy in an irregular way. Since the oscillator is surrounded by a huge thermal bath and impacts from the bath are not predictable, the changes in motion of the oscillator are probabilistic. In a heat bath the oscillator can exchange energy with the surrounding medium and the distribution is more spread out, according to the Rayleigh distribution. www.lecb.ncifcrf.gov /~toms/paper/ccmm/latex/node6.html   (485 words)

 harmonic I presume you're referring to the fact that when you quantize the harmonic oscillator, the spectrum of the Hamiltonian is discrete. In other words, if the momentum and position of a harmonic oscillator starts out at (p,q), after time t it will be (p cos t - q sin t, p sin t + q cos t), at least if the frequency of the oscillator is chosen right. This funny extra 1/2 in the eigenvalues of the harmonic oscillator Hamiltonian can be thought of as the "zero-point energy" or "vacuum energy" due to the uncertainty principle. math.ucr.edu /home/baez/harmonic.html   (1838 words)

 The Harmonic Quantum Oscillator Each oscillator can only absorb or emit energy in portions – quanta – which are proportional to f, which means that the following must be valid: E = h · f, where E is the value of the absorbed or emitted portion of energy, and h is a proportionality constant, the so called Planck's Constant. of a harmonically oscillating particle with the mass m and the amplitude A of the oscillation is, derived according to classical mechanics, given by: Very interestingly we note that the energy content of a specific harmonical oscillator – which is a local subsystem in the Universe – is equal to a rational fraction of the total content of energy of the Universe, which is M www.rostra.dk /louis/quant_15.html   (1028 words)

 Quantum Harmonic Oscillator   (Site not responding. Last check: 2007-10-16) The energy levels of the quantum harmonic oscillator are This form of the frequency is the same as that for the classical simple harmonic oscillator. The quantum harmonic oscillator has implications far beyond the simple diatomic molecule. hyperphysics.phy-astr.gsu.edu /hbase/quantum/hosc.html   (138 words)

 THE HARMONIC OSCILLATOR IN PHYSICS - AND THEN SOME The oscillator exhibits a varying reponsiveness to the driving frequency w that depends on ((w_0)Â² - wÂ²), which measures the closeness of w to the natural frequency in the same analytical way that the square of a standard deviation in statistical theory, and uncertainty in quantum mechanics are measured. The energy or intensity of an oscillation is measured by the square of its amplitude, and oscillators will always have natural resonances at certain frequencies where their responses are more sensitive than at other frequencies. These are unnormalized transition amplitudes where the full oscillator energy motivates the transitions, and the T(n) eigenvectors represent phases (pointer positions of a digital clock) of the oscillator "as a Q clock". graham.main.nc.us /~bhammel/PHYS/sho.html   (13359 words)

 2 The Ideal Driven Harmonic Oscillator The retarded Green's function is the solution where the oscillator is at rest at the origin before the impulsive force is applied, and therefore after the force, the oscillator will be vibrating. The advanced Green's function tells you the amplitude and phase the oscillator would need initially so that it would end up at rest at the origin after the impulsive force is applied. That is, the retarded Green's function is the solution of the adjoint equation that has the oscillator at rest at the origin after the impulse, while the advanced Green's function is the solution of the adjoint equation that has the oscillator at rest at the origin before the impulse. fermi.la.asu.edu /PHY531/hogreen/node2.html   (674 words)

 Glauber States: Coherent states of Quantum Harmonic Oscillator   (Site not responding. Last check: 2007-10-16) The probability distribution of the coherent state behaves as the n=0 state whose shape moves as a classical oscillator with the frequency omega. In the toy below about 25 first states of harmonic oscillator are used when in the coherent state mode, i.e. The oscillator starts with amplitudes corresponding to a coherent state with the shown energy (average energy). web.ift.uib.no /AMOS/MOV/HO   (392 words)

 HARMONIC OSCILATOR by Harry Lythall - SM0VPO For the higher crystal frequencies you will need to be certain that the crystal is not a harmonic crystal, as this oscillator will only function at the fundamental frequency of the crystal, but it is just a suggestion. The oscillator modules also have a FM input that will shift the carrier a few KHz, which, at the 5th harmonic, gives us some 300KHz of deviation (looks like an ideal start for a WBFM transmitter!!). The output should be about 1mW at the 5th harmonic although it is about the same region for all harmonics, up to about the 10th. w1.859.telia.com /~u85920178/conv/xtal-0.htm   (1247 words)

 Harmonic Oscillator Model: The harmonic oscillator model is important because not only does it demonstrate quantization of energy, but it also shows the phenomenon called quantum tunnelling, in which an electron can pass into a finite potential barrier, which is not permitted by classical mechanics. is the quantum number for this system; it determines the specific state which the harmonic oscillator electron occupies. This effect does not mean that classical mechanics is necessarily incorrect, rather it helps one to understand that classical mechanics concerns itself with average values pertaining to objects which possess large amounts of energy, relative to electrons. user.mc.net /~buckeroo/HARM.html   (735 words)

 SIMPLE HARMONIC OSCILLATOR It represents the relative motion of atoms in a diatomic molecule or the simultaneous motion of atoms in a polyatomic molecule along an "normal mode" of vibration. Here x is the displacement of the oscillating particle from its equilibrium position (x=0), t is the time, k is the Hooke's law force constant, and m is the particle mass. Derive expressions for the integration constants A and B of the classical oscillator in terms of the total energy and the initial displacement. www.sci.wsu.edu /idea/quantum/SHO.htm   (912 words)

 Physics: Fourier Transform In order for an object to oscillate, there must be a force acting on it. The amplitude tells you how strong the oscillation is, the phase tells you at what stage of oscillation the system was at time t = 0. Even if you were successful in doing it, a slowly oscillating small object is very inefficient in converting vibrational energy to sound. www.relisoft.com /Science/Physics/oscillator.html   (738 words)

 4.1 Harmonic Oscillator The importance of this problem, however, lies in the fact that equations of a similar form arise when a particle moves through any region whose potential has one or more local minima: planetary and satellite motion, the classical description of an electron in orbit around a nucleus, pendulums, etc. behavior, oscillating at the damped frequency with an amplitude that decays exponentially. portion, which has the same solution as the damped harmonic oscillator, dies out exponentially and depends on the initial conditions. www.story.com /math/cfd.4.1.html   (1164 words)

 Damped Harmonic Oscillator   (Site not responding. Last check: 2007-10-16) When a damped oscillator is subject to a damping force which is linearly dependent upon the velocity, such as viscous damping, the oscillation will have exponential decay terms which depend upon a damping coefficient. This will seem logical when you note that the damping force is proportional to c, but its influence inversely proportional to the mass of the oscillator. When a damped oscillator is underdamped, it approaches zero faster than in the case of critical damping, but oscillates about that zero. hyperphysics.phy-astr.gsu.edu /hbase/oscda.html   (109 words)

 The Simple Harmonic Oscillator   (Site not responding. Last check: 2007-10-16) The simplest example of an oscillating system is a mass connected to a rigid foundation by way of a spring. is the amplitude of the oscillation, and φ is the phase constant of the oscillation. The movie at right (25 KB Quicktime movie) shows how the total mechanical energy in a simple undamped mass-spring oscillator is traded between kinetic and potential energies while the total energy remains constant. www.kettering.edu /~drussell/Demos/SHO/mass.html   (381 words)

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