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In classical mechanics, a harmonic oscillator is a system that, ... The general solution is a sum of a transient solution that depends on initial conditions, ...
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; The quantum harmonic oscillator with an applied uniform field [1] The Inverse square root potential [2] The periodic potential The particle in a lattice; The particle in a lattice of finite length [3] The Pöschl–Teller potential; The quantum pendulum; The three-dimensional potentials The rotating system The ...
In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel.It provides the fundamental solution [3] φ(x,t) to
It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. [1] The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems. [2]
The general solution is a superposition of the normal modes where c 1, c 2, φ 1, and φ 2 are determined by the initial conditions of the problem. The process demonstrated here can be generalized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics .
The Kepler problem and the simple harmonic oscillator problem are the two most fundamental problems in classical mechanics. They are the only two problems that have closed orbits for every possible set of initial conditions, i.e., return to their starting point with the same velocity (Bertrand's theorem). [1]: 92
The method deals with differential equations in the form + = + (,) for a smooth function f along with appropriate initial conditions. The parameter ε is assumed to satisfy <. If ε = 0 then the equation becomes that of the simple harmonic oscillator with constant forcing, and the general solution is