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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.
The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal ...
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 an axially symmetric shape with the axis of symmetry being the z axis, the Hamiltonian is = + (+) ( ). Here m is the mass of the nucleon, N is the total number of harmonic oscillator quanta in the spherical basis, is the orbital angular momentum operator, is its square (with eigenvalues (+)), = (/) (+) is the average value of over the N shell, and s is the intrinsic spin.
Harmonic analysis – Study of superpositions in mathematics; Harmonics (electrical power) – Sinusoidal wave whose frequency is an integer multiple; Harmonic generation – Nonlinear optical process; Harmonic oscillator – Physical system that responds to a restoring force inversely proportional to displacement; Harmony – Aspect of music
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.
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 ] ).
Elementary examples that show mathematically how energy levels come about are the particle in a box and the quantum harmonic oscillator. Any superposition ( linear combination ) of energy states is also a quantum state, but such states change with time and do not have well-defined energies.