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Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position.
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.
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]
For the harmonic oscillator, x and p enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L 2. [nb 5]
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 ...
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. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction.
In the context of the quantum harmonic oscillator, one reinterprets the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system. Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin).