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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.
A degree of freedom corresponds to one quantity that changes the configuration of the system, for example the angle of a pendulum, or the arc length traversed by a bead along a wire. If it is possible to find from the constraints as many independent variables as there are degrees of freedom, these can be used as generalized coordinates. [ 5 ]
A passive isolation system, such as a shock mount, in general contains mass, spring, and damping elements and moves as a harmonic oscillator. The mass and spring stiffness dictate a natural frequency of the system. Damping causes energy dissipation and has a secondary effect on natural frequency. Passive Vibration Isolation
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.
Since the force acting on the oscillator is conservative and the motion occurs over a finite domain, the motion will be cyclic with some period which will be denoted T. Since the probability of the oscillator being at any possible position between the minimum possible x-value and the maximum possible x-value must sum to 1, the normalization
The dynamical symmetry group of the n dimensional quantum harmonic oscillator is the special unitary group SU(n). As an example, the number of infinitesimal generators of the corresponding Lie algebras of SU(2) and SU(3) are three and eight respectively.
This gives the harmonic nuclear motion Hamiltonian. Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one-dimensional harmonic oscillator Hamiltonians. The one-dimensional harmonic oscillator is one of the few systems that allows an exact solution of the Schrödinger equation.
A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10 13 Hz to approximately 10 14 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm −1 and wavelengths of approximately 30 to 3 μm.