Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary 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. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

The Hamiltonian of the particle is:

where m is the particle's mass, ω is the angular frequency of the oscillator, ∧x is the position operator (given by x), and ∧p is the momentum operator, given by ${\displaystyle {\hat {p}}=-i\hbar {\partial \over \partial x}\,.}$ The first term in the Hamiltonian represents the possible kinetic energy states of the particle, and the second term represents its corresponding possible potential energy states.

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