In this chapter we will study the features of one of the most important potentials in physics, it's the harmonic oscillator potential which is included now in the Hamiltonian V (x) = mω 2 2 x 2. (5.1) There are two possible ways to solve the corresponding time independent Schrödinger equation, the algebraic method, which will lead us to new important concepts, and the analytic method, which is the straightforward solving of a differential equation. 5.1.1 Algebraic Method We start again by using the time independent Schrödinger equation, into which we insert the Hamiltonian containing the harmonic oscillator potential (5.1) H ψ = − 2 2m d 2 dx 2 + mω 2 2 x 2 ψ = E ψ. (5.2) We rewrite Eq. (5.2) by defining the new operatorx := mωx H ψ = 1 2m i d dx 2 + (mωx) 2 ψ = 1 2m p 2 +x 2 ψ = E ψ. (5.3) We will now try to express this equation as the square of some (yet unknown) operator p 2 +x 2 → (x + ip)(x − ip) = p 2 +x 2 + i(px −xp) , (5.4) but since x and p do not commute (remember Theorem 2.3), we only will succeed by taking the x − p commutator into account. Eq. (5.4) suggests to factorize our Hamiltonian by defining new operators a and a † as: 95

Physical Review E, 2004

We study the occurrence of physically observable phase locked states between chaotic oscillators and rotors in which the frequencies of the coupled systems are irrationally related. For two chaotic oscillators, the phenomenon occurs as a result of a coupling term which breaks the 2 invariance in the phase equations. In the case of rotors, a coupling term in the angular velocities results in very long times during which the coupled systems exhibit alternatively irrational phase synchronization and random phase diffusion. The range of parameters for which the phenomenon occurs contains an open set, and is thus physically observable.

2003

A general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented. These coherent states, by construction are not potential specific and rely on the properties of the orthogonal polynomials, for their derivation. The information about a given quantum mechanical potential enters into these states, through the orthogonal polynomials associated with it and also through its ground state wave function. The time evolution of some of these states exhibit fractional revivals, having relevance to the factorization problem.

Physics Letters A, 1991

The phase properties ofthe fractional coherent states are discussed from the point ofview ofthe Pegg-Barnett Hermitianphase formalism. Exact analytical formulas for the phase variance are obtained and illustrated graphically. The results can serve as a test for the range of validity of the scaling law for the phase variance.

Mathematical and Computer Modelling, 2011

In a recent paper [A. Yıldırım, Z. Saadatnia, H. Askari, "Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms", Mathematical and Computer Modelling 54 (2011) 697-703] the so-called Hamiltonian approach (HA) was applied to obtain analytical approximate solutions for conservative nonlinear oscillators with certain elastic terms. In this paper we demonstrate that the approach proposed is equivalent to the well known harmonic balance method (HBM) and that the equations they obtained using the HA can be easily derived from the well known first-order HBM applied to conservative nonlinear oscillators with odd nonlinear elastic terms. This implies that the approximate frequency and periodic solution obtained using the HA with the trial function proposed in that paper are the same as those obtained using the first-order HBM. We think the comments presented here could be useful for people working in approximate analytical methods for nonlinear oscillators and they would have to be taken into account in the developing and application of some approximate techniques.

Journal of Physics A: Mathematical and Theoretical, 2011

2013

In this paper we construct rational orthogonal systems with respect to the normalized area measure on the unit disc. The generating system is a collection of so called elementary rational functions. In the one dimensional case an explicit formula exists for the corresponding Malmquist–Takenaka functions involving the Blaschke functions. Unfortunately, this formula has no generalization for the case of the unit disc, which justifies our investigations. We focus our attention for such special pole combinations, when an explicit numerical process can be given. In [4] we showed, among others, that if the poles of the elementary rational functions are of order one, then the orthogonalization is naturally related with an interpolation problem. Here we take systems of poles which are of order both one and two. We show that this case leads to an Hermite–Fejér type interpolation problem in a subspace of rational functions. The orthogonal projection onto this subspace is calculated and also t...

2016

The uncertainty ∆ pp , the g (2) and the average number of photons ⟨n⟩ as a function of θ α (in rads), for the coherent states |2, 2; θ α , 0⟩. .. .. xii List of Figures 4.1 H(α, β; 0, 0)(Weyl function) for the state of Eq. (4.2) with α = 2 and β = 0. The arrows indicate the autoparts (A) and cross-parts (C). .. 95 4.2 H(α, β; π 2 , π 2)(Wigner function) for the state of Eq. (4.2) with α = 2 and β = 0. The arrows indicate the autoparts (A) and cross-parts (C). 96 4.3 |H(α, β; π 4 , π 4)| for the state of Eq. (4.2) with α = 1.8 and β = 0.. .. . 97 4.4 H(α, β; π 4 , π 4) for the state of Eq. (4.2) with α = 2 and β = 0.. .. .. . 98 4.5 H(α, β; π 2 , π 4) for the state of Eq. (4.2) with α = 2 and β = 0.. .. .. . 99 4.6 H(α, β; π 4 , π 2) for the state of Eq. (4.2) with α = 2 and β = 0.. .. .. . 100 4.7 H(α, β; 0, π 4) for the state of Eq. (4.2) with α = 2 and β = 0.. .. .. . 101 4.8 H(α, β; π 4 , 0) for the state of Eq. (4.2) with α = 2 and β = 0.. .. .. . 102 4.9 The Q-function Q (α, β; 0, 0) for the state of Eq. (4.

Theoretical and Mathematical Physics, 2012

Advances in Mathematical Physics, 2020

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequency ω . The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component r o of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented tech...