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Quantum Harmonic Oscillator
Published in Zbigniew Ficek, Quantum Physics for Beginners, 2017
Suppose that the state |ϕ0〉 of energy E is the lowest (ground) state of the harmonic oscillator. Thus, the energy spectrum (eigenvalues), shown in Fig. 15.2, forms a ladder of equally spaced levels separated by ħω, which one ascends by the action of ↠and descends by the action of â. The quantum harmonic oscillator, therefore, has a discrete energy spectrum.
Applications of the Formalism-II
Published in Shabnam Siddiqui, Quantum Mechanics, 2018
In the previous equation, it is clear that the energies of a quantum harmonic oscillator are discrete and have half integral values. The lowest energy of the harmonic oscillator is 1/2 ℏω.
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Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[atomic, nuclear] Vibration and propagation in a two-dimensional system. The vibration will occur in one direction only with propagation in the orthogonal direction. The quantum harmonic oscillator is the quantum–mechanical equivalent of the classical linear oscillator. In quantum mechanics the existence of an arbitrary potential can in most cases be approximated by a harmonic potential oscillating around a stable equilibrium point. The quantum harmonic oscillator provides an important tool in modeling in quantum mechanical systems. In quantum mechanics the wave-equation is replaced by an operator function equation; the Hamiltonian (t), expressed for a particle in a potential well as H⌢, where H⌢=(p^2/2m)+(1/2)mω2x^2the angular velocity of the oscillation,ω the mass of the particle (e.g., electron) m, is the quantum operator for momentum with p^=−iℏ(∂/∂x), where ℏ=h/2πPlanck’s constant, for velocity h = 6.62606957 × 10−34m2kg/s and v the location operator. The solutions contain eigenvalue that are linked to the Schrödinger equation with quantum number x^ confined solutions. This process also applied to phonons (also seeharmonic oscillator) (see Figure L.109).
Controllability of bilinear quantum systems in explicit times via explicit control fields
Published in International Journal of Control, 2021
In the current section, we briefly propose a possible application of Theorem 1.1. Let us consider an electron trapped in a one-dimensional guide of length m and represented by the quantum state ψ. We suppose that the electron is subjected to an external time-depending electromagnetic field with and a positive time. Let kg be the mass of the electron and with ℏ the reduced Planck constant. The dynamics of ψ is modelled by the Schrödinger equation We substitute , and Now, are dimensionless (without unit of measurement) and (20) corresponds to If the potential is equal to , then we obtain the (BSE) We point out that the last equation can be used to model the dynamics of an electron subjected to two external fields. The first one forces its behaviour to a quantum harmonic oscillator with time dependent intensity. The second field instead traps the electron in a potential well.
Casimir-Lifshitz quantum state of superhydrophobic black-silicon surfaces manufactured by a metal-assisted hierarchical nano-microtexturing process
Published in Philosophical Magazine, 2019
Bhaskar Parida, Sel Gi Ryu, Keunjoo Kim
From the canonical variables and , we have an equation of harmonic motion . From wave equation we also have , and so the Hamiltonian is . This is the simple harmonic oscillator with the dielectric constant of optical inertia instead of mass m. For the quantisation condition, , we define the wave solution of time-dependent vector potential as Hermitian operators of the annihilation and creation operators, and , respectively. The number operator is also defined as and shows . Therefore, the Hamiltonian operator shows the quantum harmonic oscillator of .
Atomic swap gate, driven by position fluctuations, in dispersive cavity optomechanics
Published in Journal of Modern Optics, 2019
Anil Kumar Chauhan, Asoka Biswas
In all the existing proposals, the interaction between the qubits is often simulated by the aid of an auxiliary quantum system. For example, a controlled-phase gate between two optical pulses can be obtained by an atom trapped inside a high-quality factor cavity (7). On the other hand, the quantum controlled-NOT gate between two atoms, trapped inside a cavity, is obtained by their interaction with the common cavity mode and exchange of virtual photons only (8). Several other protocols towards realisation of quantum gates in such cavity QED systems can be found in (9–16). In all these cases, the cavity modes are modelled as a quantum harmonic oscillator, confined to its lowest eigenstates. On the contrary, in this paper, we show that the interaction between two qubits, and thereby quantum logic gates can be mediated by the motion of a mesoscopic mechanical oscillator, at large temperature limit. This further paves the way for controlling the quantum dynamics by mesoscopic systems. In this paper, we choose a cavity optomechanical system to demonstrate this main idea.