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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.
A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing .
The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems. [2] For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook [ 3 ] ).
The presence of degenerate energy levels is studied in the cases of Particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems. Particle in a rectangular plane
For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency (in the harmonic oscillator approximation). See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be
The zero-point energy E = ħω / 2 causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed. The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle.
A harmonic oscillator in classical mechanics (A–B) and quantum mechanics (C–H). In (A–B), a ball, attached to a spring, oscillates back and forth.(C–H) are six solutions to the Schrödinger equation for this situation.
The old quantum theory yields a recipe for the quantization of the energy levels of the harmonic oscillator, which, when combined with the Boltzmann probability distribution of thermodynamics, yields the correct expression for the stored energy and specific heat of a quantum oscillator both at low and at ordinary temperatures.