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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 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.
Further, in contrast to the energy eigenstates of the system, the time evolution of a coherent state is concentrated along the classical trajectories. The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems.
Considering the ion's motion along the direction of the static trapping potential of an ion trap (the axial motion in -direction), the trap potential can be validly approximated as quadratic around the equilibrium position and the ion's motion locally be considered as that of [1] a quantum harmonic oscillator with quantum harmonic oscillator eigenstates | .
These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows. Assuming that is an eigenstate of the Hamiltonian ^ =. Using these commutation relations, it follows that [6] ^ = ().
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.
Then the wave function factors into a product of momentum eigenstates in the direction and harmonic oscillator eigenstates | shifted by an amount in the direction: (,,) = (+) where = /. In sum, the state of the electron is characterized by the quantum numbers, n {\displaystyle n} , k y {\displaystyle k_{y}} and k z {\displaystyle k_{z}} .
The ladder operators of the quantum harmonic oscillator or the "number representation" of second quantization are just special cases of this fact. Ladder operators then become ubiquitous in quantum mechanics from the angular momentum operator , to coherent states and to discrete magnetic translation operators.