The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually...

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional...

physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state...

operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation...

(helioseismology), stars (asteroseismology) and Neutron-star oscillations. Quantum harmonic oscillator Playground swing String instruments Torsional vibration Tuning...

Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. There is a relationship...

the uncertainty principle. (Refs ) Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators...

discrete, rather than continuous, values. These are known as quantum harmonic oscillators. This process of restricting energies to discrete values is called...

Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator is ( − ℏ 2 2 m d 2 d x 2 + 1 2 m ω 2 x 2 ) ψ ( x ) = E ψ ( x...

dependent on the length of the box. The wave functions for the quantum harmonic oscillator can be expressed in terms of Hermite polynomials Hn, they are...

freedom of molecules towards its thermodynamic variables. A quantum harmonic oscillator has an energy spectrum characterized by: E j , n = ℏ ω j ( n...

zero-point energy of a quantum harmonic oscillator. An exact amount of energy ħω must be supplied to the harmonic oscillator lattice to push it to the...

of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version...

Schrödinger equation for the one-dimensional quantum harmonic oscillator (a quantum particle in the oscillator potential), [ − ℏ 2 2 m ∂ 2 ∂ x 2 + 1 2 m...

the quantum harmonic oscillator can be used to look at the equilibrium situation for a quantum ideal gas in a harmonic trap, which is a harmonic potential...

network of interacting quantum harmonic oscillators and injecting the input by periodical state resets of a subset of the oscillators. With a suitable choice...

approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking,...

particle in a spherically symmetric potential The quantum harmonic oscillator The quantum harmonic oscillator with an applied uniform field The hydrogen atom...

qubits to this intermediate. A quantum bus is often implemented as a microwave cavity modeled by a quantum harmonic oscillator. Coupled qubits may be brought...

Quantum mechanics was combined with the theory of relativity in the formulation of Paul Dirac. Other developments include quantum statistics, quantum...

common and useful example of quantum mechanics applied to a continuous degree of freedom is the quantum harmonic oscillator.: 24  This system is defined...

operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. Another type of operator in quantum field theory...

a complex-valued function found to be the propagator of the quantum harmonic oscillator. Mehler (1866) defined a function E ( x , y ) = 1 1 − ρ 2 exp...

two-dimensional isotropic quantum harmonic oscillator has symmetry group SU(2), while the symmetry algebra of the rational anisotropic oscillator is a nonlinear...

quantum harmonic oscillator described by a single quantum number, this means the quantum number will remain unchanged. Figure 1 shows how a harmonic oscillator...

John S. (2012). "Chapter 7: The One-Dimensional Harmonic Oscillator". A Modern Approach to Quantum Mechanics. University Science Books. pp. 247–250,...

|^{2n}}{\pi n!}}|\langle 0|\alpha \rangle |^{2}~.} Consider the damped quantum harmonic oscillator with the following master equation, d ρ ^ d t = i ω 0 [ ρ ^ ...

ion's motion locally be considered as that of a quantum harmonic oscillator with quantum harmonic oscillator eigenstates | n ⟩ {\displaystyle |n\rangle }...

harmonic oscillator potential does not affect the energies. The energies of this system are thus identical to those of the standard quantum harmonic oscillator...

of light modes, which are harmonic oscillators. Examples of Wigner-function time evolutions in a quantum harmonic oscillator Wigner quasiprobability distribution...