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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. They are the source of virtually all sinusoidal ...
This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method. A wavefunction different from the exact one is obtained by use of the method described above. [citation needed]
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.
which is in the form of the harmonic oscillator Hamiltonian. The generating function F for this transformation is of the third kind, = (,). To find F explicitly, use the equation for its derivative from the table above, =,
Since the force acting on the oscillator is conservative and the motion occurs over a finite domain, the motion will be cyclic with some period which will be denoted T. Since the probability of the oscillator being at any possible position between the minimum possible x-value and the maximum possible x-value must sum to 1, the normalization
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
The harmonic oscillator is an important case. Finding the matrices is easier than determining the general conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillator , with Hamiltonian H = 1 2 P 2 + 1 2 X 2 + ε X 3 . {\displaystyle H={\tfrac {1}{2}}P^{2}+{\tfrac {1}{2}}X^{2}+\varepsilon X^{3}~.}
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.