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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.
The tuning application, for instance, is an example of band-pass filtering. The RLC filter is described as a second-order circuit, meaning that any voltage or current in the circuit can be described by a second-order differential equation in circuit analysis. The three circuit elements, R, L and C, can be combined in a number of different ...
The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill [1] and David M. Fradkin, [2] is a conservation law used in the treatment of the isotropic multidimensional harmonic oscillator in classical 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 ...
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.
In such cases, the solution provided by the use of the retarded Green's function depends only on the past sources and is causal whereas the solution provided by the use of the advanced Green's function depends only on the future sources and is acausal. In these problems, it is often the case that the causal solution is the physically important one.
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
This plot corresponds to solutions of the complete Langevin equation for a lightly damped harmonic oscillator, obtained using the Euler–Maruyama method. The left panel shows the time evolution of the phase portrait at different temperatures. The right panel captures the corresponding equilibrium probability distributions.