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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.
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 solid plot represents the quantum mechanical probability density, while the dotted line represents the classical probability density. The dashed vertical lines indicate the classical turning points of the system. Starting with the example used in the derivation above, the simple harmonic oscillator has the potential energy function
The Hamiltonian of a simple harmonic oscillator is given by = + where is the linear momentum and the position coordinate. The action variable is given by = where we have used that = is the energy and that the closed trajectory is 4 times the trajectory from 0 to the turning point = /.
Download as PDF; Printable version; In other projects ... The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator ...
In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum .
The second quantized treatment of the one-dimensional quantum harmonic oscillator is a well-known topic in quantum mechanical courses. We digress and say a few words about it. The harmonic oscillator Hamiltonian has the form = († +)