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The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is ...
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. Balance of forces (Newton's second law) for the system is.
Creation and annihilation operators. Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. [1] An annihilation operator (usually denoted ) lowers the number of particles in a given state by one.
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={1 \over 2}P^{2}+{1 \over 2}X^{2}+\varepsilon X^{3}~.}
Ladder operator. In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the ...
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in ...
Wigner function for the harmonic oscillator ground state, displaced from the origin of phase space, i.e., a coherent state. Note the rigid rotation, identical to classical motion: this is a special feature of the SHO, illustrating the correspondence principle. From the general pedagogy web-site. [23]
Quantum field theory. In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum ...