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Glossary
Free Vibrations
See http://www.sharetechnote.com/html/DE_Modeling_Example_SpringMass.html#SingleSpring_SimpleHarmonic http://www.sharetechnote.com/html/DE_Modeling_Example_SpringMass.html#SingleSpring_SimpleHarMonic_Horiz_NoDamp https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III__Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.06%3A_The_Quantum_Harmonic_Oscillator
A simple harmonic oscillator is a particle or system that undergoes harmonic motion about an equilibrium position, such as an object with mass vibrating on a spring. In this section, we consider oscillations in onedimension only. Suppose a mass moves backandforth along the xdirection about the equilibrium position, x = 0. In classical mechanics, the particle moves in response to a linear restoring force given by F_{x} = −kx, where x is the displacement of the particle from its equilibrium position. The motion takes place between two turning points, x±A, where A denotes the amplitude of the motion. The position of the object varies periodically in time with angular frequency ω=k/m−−−−√, which depends on the mass m of the oscillator and on the force constant k of the net force, and can be written as


The potential energy well of a classical harmonic oscillator.  Mathematica code. 
In this plot, the motion of a classical oscillator is confined to the region where its kinetic energy is nonnegative, which is what the energy relation Equation 7.6.2 says. Physically, it means that a classical oscillator can never be found beyond its turning points, and its energy depends only on how far the turning points are from its equilibrium position. The energy of a classical oscillator changes in a continuous way. The lowest energy that a classical oscillator may have is zero, which corresponds to a situation where an object is at rest at its equilibrium position. The zeroenergy state of a classical oscillator simply means no oscillations and no motion at all (a classical particle sitting at the bottom of the potential well in Figure. When an object oscillates, no matter how big or small its energy may be, it spends the longest time near the turning points, because this is where it slows down and reverses its direction of motion. Therefore, the probability of finding a classical oscillator between the turning points is highest near the turning points and lowest at the equilibrium position. (Note that this is not a statement of preference of the object to go to lower energy. It is a statement about how quickly the object moves through various regions.)
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