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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 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 coherent state describes a state ...
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.
Such an oscillator, when quantized, is described by the mathematics of a quantum harmonic oscillator. [1] Quantum oscillators are described using creation and annihilation operators ^ † and ^. Physical quantities, such as the electric field strength, then become quantum operators.
For example, a quantum harmonic oscillator may be in a state |ψ for which the expectation value of the momentum, | ^ | , oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector | ψ , the momentum operator p ^ {\displaystyle {\hat {p}}} , or both.
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum.
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.
See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one, = but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because ...