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Topic: Quantum harmonic oscillator

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Perturbation theory (quantum mechanics)

Harmonic oscillator

Schrodinger equation

Mathematical formulation of quantum mechanics

Particle in a box

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Oscillation

Vacuum state

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Phonon

	Quantum Harmonic Oscillator (Site not responding. Last check: 2007-10-09)
		This form of the frequency is the same as that for the classical simple harmonic oscillator.
		The most surprising difference for the quantum case is the so-called zero-point vibration" of the n=0 ground state.
		The quantum harmonic oscillator has implications far beyond the simple diatomic molecule.
	hyperphysics.phy-astr.gsu.edu /hbase/quantum/hosc.html (138 words)

	Quantum harmonic oscillator
		The quantum harmonic oscillator is a quantum mechanical analogue of the classical harmonic oscillator.
		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 quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem.
	www.ebroadcast.com.au /lookup/encyclopedia/la/Ladder_operator.html (1756 words)

	Quantum Harmonic Oscillator
		If you examine the ground state of the quantum harmonic oscillator, the correspondence principle seems far-fetched, since the classical and quantum predictions for the most probable location are in total contradiction.
		If the equilibrium position for the oscillator is taken to be x=0, then the quantum oscillator predicts that for the ground state, the oscillator will spend most of its time near that center point.
		The harmonic oscillator is an important problem in both the quantum and classical realm.
	hyperphysics.phy-astr.gsu.edu /hbase/quantum/hosc6.html (708 words)

	Reference.com/Encyclopedia/Harmonic oscillator
		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.
		In general, the pulsation-also known as angular frequency, of a straight-line simple harmonic motion is the angular speed of the corresponding circular motion.
	www.reference.com /browse/wiki/Harmonic_oscillator (2226 words)

	Quantum Harmonic Oscillator - Top Quantum Harmonic Oscillator Sites, Reviews and Information (Site not responding. Last check: 2007-10-09)
		Quantum harmonic of definition oscillator Quantum harmonic The quantum harmonic oscillator is the quantum mechanical analogue of resonance the classical harmonic oscillator.
		Harmonic Quantum Oscillator The energy levels of the quantum harmonic oscillator are capacitor has quantum harmonic oscillator The implications far beyond oscillation the simple diatomic molecule.
		Quantum Mechanics: Harmonic Dimensional 2 Oscillator Applet This java applet is a quantum mechanics simulation that function shows the behavior a a particle in of two analysis dimensional harmonic oscillator.
	www.justoscillators.com /listings83.html (568 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		For the quantum mechanical case the probability of finding the oscillator in an interval Dx is the square of the wavefunction, and that is very different for the lower energy states.
		Two particular systems related to the Gaussian function via limiting procedures are considered, the harmonic oscillator perturbed by a Dirac distribution in the strong coupling limit and the 1D hydrogen atom.
		Index Key: PHY031 Author: Patton Subject: Harmonic oscillator energy levels Text: The quantum energy levels for the kinetic energy of a particle in a box are obtained as the eigenvalues for the wave equation.
	www.lycos.com /info/harmonic-oscillator.html (533 words)

	Quantum Harmonic Oscillator (Site not responding. Last check: 2007-10-09)
		The Schrodinger equation for a harmonic oscillator may be solved to give the wavefunctions illustrated below.
		But as the quantum number increases, the probabability distribution becomes more like that of the classical oscillator - this tendency to approach the classical behavior for high quantum numbers is called the correspondence principle.
		The wavefunctions for the quantum harmonic oscillator contain the Gaussian form which allows them to satisfy the necessary boundary conditions at infinity.
	hyperphysics.phy-astr.gsu.edu /hbase/quantum/hosc5.html (254 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.
		In quantum field theory, a and a† are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.
		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)

	- Quantum Harmonic Oscillator
		He was the first one to consider harmonic oscillator wave functions normalizable in the time variable.
		In 1963, Dirac used coupled harmonic oscillators to construct a representation of the O(3,2) de Sitter group which is the basic scientific language for two-mode squeezed states.
		He proposed harmonic oscillators for relativistic extended particles five years before Hofstadter observed that protons are not point particles in 1955.
	worldcrossing.com /WebX?230@@.1de1408d (370 words)

	Quantum field theory
		Quantum field theory originated in the problem of computing the power radiated by an atom when it dropped from one quantum state to another of lower energy.
		As described in the article on identical particles, quantum mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged.
		The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles.
	www.zamandayolculuk.com /cetinbal/quantumalankuram.htm (3742 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.
		See the article quantum harmonic oscillator for a discussion of the harmonic oscillator in quantum mechanics.
	www.artilifes.com /harmonic-oscillator.htm (967 words)

	Interactive Extras: The Quantum Harmonic Oscillator
		The quantum mechanical harmonic oscillator is among our most important model systems.
		Below, you can change the v quantum number (keep it less than 11) and watch the wavefunction and its associated energy change.
		In the graph on the bottom, the wavefunction and its square are shown superimposed on a plot of the potential energy function with the first 11 (v = 0 to 10) allowed energies drawn as lines connecting the classical turning points at each energy.
	www.dartmouth.edu /~pchem/62/thps/quantumho.html (237 words)

	The Harmonic Quantum Oscillator
		The transformation from a classical and continuous physical equation to a quantum physical expression is reduced to inserting the quantized quantities given by (3), (4) and (5).
		of a harmonically oscillating particle with the mass m and the amplitude A of the oscillation is, derived according to classical mechanics, given by:
		We see that the cosmic evolution quantum number is the physical quantity uniting the smallest and the greatest physical quantities of the Universe.
	www.rostra.dk /louis/quant_15.html (1028 words)

	[No title]
		Quantum Harmonic Oscillator The plots of the quantum mechanical harmonic oscillator wavefunctions in the text show that the oscillator can be in regions forbidden to a classical harmonic oscillator.
		Consider a 35Cl2 molecule to be a quantum mechanical harmonic oscillator.
		Calculate the probability that the quantum mechanical harmonic oscillator is in the classically forbidden region.
	www.carthage.edu /dept/cvl/documents/pchem2.doc (1327 words)

	Glauber States: Coherent states of Quantum Harmonic Oscillator (Site not responding. Last check: 2007-10-09)
		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).
	www-troja.fjfi.cvut.cz /~ladi/HO (392 words)

	The Harmonic Oscillator and the Quantum Field (Site not responding. Last check: 2007-10-09)
		This equality is the basis of all the theory of the quantum field.
		In a generic harmonic oscillator, these two operators are merely a way to raise and lower the state of the system.
		One of the fundamental results of quantum theory is that at n=0 a harmonic oscillator has a non--zero energy.
	webphysics.davidson.edu /Projects/AnAntonelli/node38.html (575 words)

	[No title]
		The program uses the Cartesian harmonic oscillator basis to expand single-particle or single-quasiparticle wave functions of neutrons and protons interacting by means of the Skyrme effective interaction and zero-range pairing interaction.
		The expansion coefficients are determined by the iterative diagonalization of the mean field Hamiltonians or Routhians which depend non-linearly on the local neutron and proton densities.
		present, the harmonic oscillator is described as damped.
	www.lycos.com /info/harmonic-oscillator--miscellaneous.html (394 words)

	Forced Oscillator (Site not responding. Last check: 2007-10-09)
		Solving for the ratio of the oscillation amplitude of the mass to the amplitude of the wiggling motion,
		, the motion of the mass is in phase with the wiggling motion and the amplitude of the mass oscillation is greater than the amplitude of the wiggling.
		As the forcing frequency approaches the natural frequency of the oscillator, the response of the mass grows in amplitude.
	www.physics.nmt.edu /~raymond/classes/ph13xbook/node123.html (485 words)

	QRHO
		The construction of a proper model for a quantum relativistic oscillator is of general interest in physics, as a first approximation to periodic dynamics with positive energy for systems in which the velocity of constituents is not small with respect to c.
		The extension of this observation to the quantum mechanical terms was developed in the paper by V.
		Group-theoretical construction of the quantum relativistic harmonic oscillator.
	www.elp.uji.es /juan_home/research/quantum_harmonic_oscillator.htm (364 words)

	Eigenstates of the Quantum Harmonic Oscillator (Site not responding. Last check: 2007-10-09)
		All the eigenvectors of the harmonic oscillator hamiltonian can be created by repeatedly acting on the ground state with the raising operator, a^dag.
		One finds that the position and momentum expectation values of a general wave function oscillate sinusoidally as for the classical harmonic oscillator.
		For the nth eigenvector, the expectation value for the position is zero, and the root mean square position, Delta x, is the classical maximum amplitude divided by the square root of 2.
	www.phys.ufl.edu /~selman/teaching/fall01/lecture23.html (148 words)

	Quantum Harmonic Oscillator (Site not responding. Last check: 2007-10-09)
		The ground state energy for the quantum harmonic oscillator can be shown to be the minimum energy allowed by the uncertainty principle.
		The energy of the quantum harmonic oscillator must be at least
		This is a very significant physical result because it tells us that the energy of a system described by a harmonic oscillator potential cannot have zero energy.
	hyperphysics.phy-astr.gsu.edu /hbase/quantum/hosc4.html (178 words)

	Visualizing Quantum Dynamics (Site not responding. Last check: 2007-10-09)
		The basic tools used are the Bloch Sphere, for two-level systems (such as a two-level atom, a spin 1/2 system or a Cooper-pair box), and the Q-function (sometimes called the Husimi function) for a quantum harmonic oscillator, such as a single mode of the electromagnetic field, or a nanoresonator.
		The evolution of the two-level system is plotted on the Bloch sphere, and simultaneously a 3-D plot of the evolving Q function of the field is displayed.
		The initial state of the atom is arbitrary; the initial state of the oscillator is restricted to be a coherent state.
	comp.uark.edu /~jgeabana/blochapps (440 words)

	Details for 2D Quantum Harmonic Oscillator Applet
		This simulation shows time-dependent 2D quantum bound state wavefunctions for a harmonic oscillator potential.
		Multiple-energy-eigenstate wavefunctions can be created through changes in the amplitude and phase of the basis states using spinors, or through the creation of Gaussian, elliptical, or square wavefunctions with the mouse.
		The quantum numbers of the states are shown.
	www.compadre.org /Quantum/items/detail.cfm?ID=1491 (157 words)

	Harmonic oscillator - Wikipedia, the free encyclopedia
		In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x according to Hooke's law:
		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).
		Period, the time for one complete oscillation (time for the bob to return to its starting position), is given by 2π divided by whatever is multiplying the time in the argument of the cosine
	en.wikipedia.org /wiki/Harmonic_oscillator (1828 words)

	Quantum Harmonic Oscillator State (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		From this starting point, we can put the atom into various quantum states of motion by application of optical and rf electric fields.
		Some of these states resemble classical states (the coherent states), while others are intrinsically quantum, such as number states or squeezed states.
		2 Note on the forced and damped oscillator in quantum mechanic..
	citeseer.ist.psu.edu /624993.html (554 words)

	7 Quantum Harmonic Oscillator
		Having shown an interconnection between the mathematics of classical mechanics and electromagnetism, let's look at the driven quantum harmonic oscillator too.
		Let's start with a one-dimensional quantum harmonic oscillator in its ground state at time t = 0, and apply a force F(t).
		Let's assume that the force is weak, and let's calculate the expectation value of the operator x as a function of time.
	fermi.la.asu.edu /PHY531/hogreen/node7.html (146 words)

	CHP - Harmonic Oscillator
		We can model the bond in a molecule as a spring connecting two atoms and use the harmonic oscillator expression to describe the potential energy for the periodic vibration of the atoms.
		A more realistic model of the potential well of a diatomic molecule is the Morse potential, which does model the dissociation energy.
		The simple harmonic oscillator provides a good fit to energies for the lowest energy levels, but fails at higher energies.
	www.chem.vt.edu /chem-ed/quantum/harmonic-oscillator.html (488 words)

	Units
		Absorption or emission of infrared light can cause transitions between energy levels in the harmonic oscillator.
		to refer to fundamental (actually observed) frequencies, which differ from the harmonic (model) frequencies because the potential wells in diatomic molecules are not strictly harmonic but contain an anharmonic contribution.
		, but these are not really natural units for the very small energies and distances involved in a quantum oscillator.
	vergil.chemistry.gatech.edu /notes/ho/node4.html (108 words)

	Quantum Mechanical Harmonic Oscillator (Site not responding. Last check: 2007-10-09)
		The quantum mechanical harmonic oscillator shares the characteristic of other quantum mechanical bound state problems in that the total energy can take on only discrete values.
		The energies accessible to a quantum mechanical mass-spring system are given by the formula
		In other words, the energy difference between successive quantum mechanical energy levels in this case is constant and equals the classical resonant frequency for the oscillator,
	www.physics.nmt.edu /~raymond/classes/ph13xbook/node124.html (88 words)

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