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

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

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

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

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

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

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...

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...

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...

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

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

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

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...

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

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...

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

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

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

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

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...

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

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

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

connections between the quantum system under study and the classical limit. Consider the example of a simple harmonic oscillator initially at rest with...

quadrature; physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when...

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

a QDO uses a quantum harmonic oscillator, in the form of a pseudoelectron connected to an oppositely charged pseudonucleus by a harmonic spring. A QDO...

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