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...

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...

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...

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

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

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

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...

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

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

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

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

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

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...

field at each point in space is a simple harmonic oscillator, its quantization places a quantum harmonic oscillator at each point. Excitations of the field...

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...

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...

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...

triangular potential The quadratic potentials The quantum harmonic oscillator The quantum harmonic oscillator with an applied uniform field The Inverse square...

that contains a large number of independent three-dimensional quantum harmonic oscillators of the same frequency. The independence assumption is relaxed...

are the ground state | 0 ⟩ {\displaystyle |0\rangle } of the quantum harmonic oscillator and the family of coherent states | α ⟩ {\displaystyle |\alpha...

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

particle in a ring particle in a spherically symmetric potential quantum harmonic oscillator hydrogen atom ring wave guide particle in a one-dimensional lattice...

system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can...

a complex-valued function found to be the propagator of the quantum harmonic oscillator. It was first discovered by Mehler in 1866, and since then, as...

which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe...

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

of a quantum mechanical object (potential and kinetic) is described by its Hamiltonian which also describes the system as a harmonic oscillator, or wave...

bosonic modes.[clarification needed] Unlike a two-level system, a quantum harmonic oscillator has infinitely many energy levels in a single physical system...