Overview on the Hilbert Space Harmonics Oscillator (original) (raw)
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The harmonic oscillator as a tutorial introduction to quantum mechanics
2018
Stemming from the similar linearities of the Schrödinger equation in quantum mechanics on the one hand and of the harmonic oscillations in classical mechanics on the other hand, the idea that any N-degree-of-freedom harmonic oscillator (HON) is formally equivalent to a N-level quantum system is put forward. It is shown that the complex dynamic variables α introduced by R. J. Glauber can be regarded as the components of a state vector belonging to some N-dimension complex Hilbert space, and whose time-evolution is ruled by a Schrödinger-like equation. In case the classical HON is parametrically excited, the unitarity of the time-evolution of the associated quantum system is related to the Ehrenfest adiabaticity of the parametric excitation.
Journal of Mathematics Research
This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr¨odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.
The Quantum Harmonic Oscillator
UNITEXT for Physics, 2018
Think of a sliding block, constrained to move along one direction on an idealized frictionless surface, attached to an idealized spring. The block has mass and the spring has spring constant. This is an example of a classical one-dimensional harmonic oscillator.
One dimensional (1-D) harmonic oscillators revisited
Leonardo Journal of Sciences, 2018
The concept of harmonic oscillator particularly one dimensional (1-D) is mentioned in literature repeatedly and is explained in more complex manner by using various methods. This creates difficulties in understanding the description of the concept for new learners. The purpose of this article was to enlighten different methods to formulate harmonic oscillator in improving knowledge about detailed steps to derive eigen energy values in more comprehensible manner for the beginners. The energy values are derived by using classical method, quantum mechanically, Schrodinger time independent equation, perturbation technique, variation method, WKB approximation etc. A coherent way of derivation of eigen values using various approaches makes this article as unique.
Journal of Physics A: Mathematical and General, 2003
An extension of the classical orthogonal functions invariant to the quantum domain is presented. This invariant is expressed in terms of the Hamiltonian. Unitary transformations which involve the auxiliary function of this quantum invariant are used to solve the time-dependent Schrödinger equation for a harmonic oscillator with time-dependent parameter. The solution thus obtained is in agreement with the results derived using other methods which invoke the Lewis invariant in their procedures.
Europhysics Letters (EPL), 2006
The amplitude and phase are shown to be variables that are not uniquely determined for the one-dimensional time-dependent harmonic-oscillator equation. Even when the parameter is time independent, it is shown that the amplitude and frequency may be time dependent and the phase nonlinear with respect to time. The consequences of this indeterminacy in the energy and the orthogonal-functions invariant are evaluated. The measurement of these quantities is discussed.
A geodesical approach for the harmonic oscillator
Revista Mexicana de Física E, 2020
The harmonic oscillator (HO) is present in all contemporary physics, from elementary classical mechanicsto quantum field theory. It is useful in general to exemplify techniques in theoretical physics. In this work,we use a method for solving classical mechanic problems by first transforming them to a free particle formand using the new canonical coordinates to reparametrize its phase space. This technique has been used tosolve the one-dimensional hydrogen atom and also to solve for the motion of a particle in a dipolar potential.Using canonical transformations we convert the HO Hamiltonian to a free particle form which becomestrivial to solve. Our approach may be helpful to exemplify how canonical transformations may be used inmechanics. Besides, we expect it will help students to grasp what they mean when it is said that a problemhas been transformed into another completely different one. As, for example, when the Kepler problem istransformed into free (geodesic) motion on a spheri...
Constants of Motion of the Harmonic Oscillator
Mathematical Physics, Analysis and Geometry, 2020
We prove that Weyl quantization preserves constant of motion of the Harmonic Oscillator. We also prove that if f is a classical constant of motion and Op(f) is the corresponding operator, then Op(f) maps the Schwartz class into itself and it defines an essentially selfadjoint operator on L 2 (R n). As a consequence, we provide detailed spectral information of Op(f). A complete characterization of the classical constants of motion of the Harmonic Oscillator is given and we also show that they form an algebra with the Moyal product. We give some interesting examples and we analize Weinstein average method within our framework.