By using a test-function method, we construct $n$ exact solutions of a quantum harmonic oscillator with a time-dependent "spring constant". Any $n$-th solution describes a wave-packet train consisting of $n+1$ packets. Its center oscillates like the classically harmonic oscillator with variable frequency, and width and highness of each packet change simultaneously. When the deformation is small, it behaves like a soliton train, and the large deformation is identified with collapse and revival of the wave-packet train.
In this paper we review some known results on the motion of Bloch Oscillators in the crystal momentum representation. We emphasize that the acceleration theorem, as usually stated by most of the authors, is incomplete, but in the case of pure Bloch states. In fact, the exact version of the acceleration theorem should contain a phase factor depending on both the time and crystal momentum variables. As we show this phase factor plays an essential role in order to understand the motion of Bloch Oscillators in the position variable.
The su(1,1) dynamical algebra from the Schrödinger ladder operators for N-dimensional systems: hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator
Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential
Normally, the half-harmonic oscillator is active when $x>0$ and absent when $x<0$. From a canonical quantization perspective, this leads to odd eigenfunctions being present while even eigenfunctions are absent. In that case, only the usual odd eigenfunctions will appear if the wall slides to negative infinity. However, if an affine quantization is used, sliding the wall away shows that all the odd and even eigenfunctions are encountered, exactly like any full-harmonic oscillator. We provide numerical support for this.
We treat the quantum dynamics of a harmonic oscillator as well as its inverted counterpart in the Schrödinger picture. Generally in the most papers of the literature, the inverted harmonic oscillator is formally obtained from the harmonic oscillator by the replacement of ω to iω, this leads to unbounded eigenvectors. This explicitly demonstrates that there are some unclear points involved in redefining the variables in the harmonic oscillator inversion. To remedy this situation, we introduce a scaling operator (Dyson transformation) by connecting the inverted harmonic oscillator to an anti-PT-…
We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$, when $\hbar / m \ll 1$. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the resu…
We discuss the diagonalization of a general Hamiltonian operator for a set of coupled harmonic oscillators and determine the conditions for the existence of bound states. We consider the particular cases of two and three oscillators studied previously and show the conditions for bound states in the latter example that have been omitted in an earlier treatment of this model.
We consider a quantum space with rotationally invariant noncommutative algebra of coordinates and momenta. The algebra contains tensors of noncommutativity constructed involving additional coordinates and momenta. In the rotationally invariant noncommutative phase space harmonic oscillator chain is studied. We obtain that noncommutativity affects on the frequencies of the system. In the case of a chain of particles with harmonic oscillator interaction we conclude that because of momentum noncommutativity the spectrum of the center-of-mass of the system is discrete and corresponds to the spectr…
By using the complete solution of the Milburn equation (beyond the Lindblad form that it is generally used) that describes intrinsic decoherence, we study the decaying dynamics of a displaced harmonic oscillator. We calculate the expectation values of position quadrature, and the number operator in initial coherent and squeezed states.
We use the Fourier operator to transform a time dependent mass quantum harmonic oscillator into a frequency dependent one. Then we use Lewis-Ermakov invariants to solve the Schrödinger equation by using squeeze operators. Finally we give two examples of time dependencies: quadratically and hyperbolically growing masses.
As known all physical properties of solids are described well by the system of quantum linear harmonic oscillators. It is shown in the present paper that the system consisting of classical linear harmonic oscillators having temperature dependent masses or (and) frequencies has the same partition function as the system consisting of quantum linear harmonic oscillators having temperature independent masses and frequencies while the means of the square displacements of the positions of the oscillators from their mean positions for the system consisting of classical linear harmonic oscillators hav…
We discuss the positional fluctuations of a quantum harmonic oscillator in a heat bath. Analytic expressions are given for the probability distribution functions of the oscillator position in general and limiting (classical and ground state) cases.
The ladder operators in harmonic oscillator are a well-known strong tool for various problems in physics. In the same sense, it is sometimes expected to handle the problems of repulsive harmonic oscillator in a similar way to the ladder operators in harmonic oscillators, though their analytic solutions are well known. In this paper, we discuss a simple algebraic way to introduce the ladder operators of the repulsive harmonic oscillators, which can reproduce well-known analytic solutions. Applying this formalism, we discuss the charged particles in a constant electric field in relation to the S…
We investigate the Shortcuts To Adiabaticity (STA) of a quantum harmonic oscillator under time-dependent frictional force, using invariant based inverse engineering method with a class of invariants characterized by a time-dependent frictional coefficient. We discuss the implementation of shortcut protocol in a generalized framework for any arbitrary time-dependent frictional interaction and study a class of such interaction from which the STA for the quantum harmonic oscillator can be obtained. We also discuss the application of the above protocol for the harmonic oscillator with time-varying…
