Equation of motion for simple harmonic oscillators 8. Thus, for a collection of n point masses, free to move in three dimensions, one would. Isotropic harmonic oscillator 5 since each of the roots, including the three zero roots, satis es p i 0, it follows that p. Find the expected values of position and impulse and check the ehrenfest theorem. Quantum mechanics qm the branch of physics that describes how particles act at very small scales, and is a fundamental part of our best understanding of how the universe works. Pdf anisotropic harmonic oscillator, noncommutative. Isotropic harmonic oscillator weizmann institute of science. Paper open access canonical quantization of anisotropic. Anisotropic harmonic oscillator, noncommutative landau problem and exotic newtonhooke symmetry. Exact propagator of a two dimensional anisotropic harmonic. The quantum harmonic oscillator stephen webb the importance of the harmonic oscillator the quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems that can really be solved in closed form, and is a very generally useful solution, both in approximations and in exact solutions of various. Expand the initial wave function by eigenstates of the anisotropic harmonic oscillator, and determine the time evolution of the system.
It is shown that, in the magnetic field, the levels continue to be equidistant. We do because we know how to solve it exactly, and it is a very good approximation for many, many systems. Find the energy and the angular momentum as a functions dependent of time and compare them with initial values. Several configurations of the electromagnetic field are considered.
Coherent states for the isotropic and anisotropic 2d. Homework statement a twodimensional isotropic harmonic oscillator of mass. This piece is simultaneously a physical representation of data, as well as an exploration of the extent and limitations of human knowledge. Qm the most successful theory in human history, having successfully passed. Phase space in classical physics quantum mechanically, we can actually count the number of microstates consistent with a given macrostate, specified for example by the total energy.
Chapter 8 the simple harmonic oscillator a winter rose. The derivative has an advantage for the negativepower cases, but the harmonic oscillator receives no bene. The anisotropic harmonic oscillator in a magnetic field. Pdf relativistic chaos in the anisotropic harmonic. The anisotropic harmonic oscillator wave equation is solved in cylindrical coordinates and the subsequent wave functions used to find the desired matrix element expressions.
In quantum physics, when you are working in one dimension, the general particle harmonic oscillator looks like the figure shown here, where the particle is under the influence of a restoring force in this example, illustrated as a spring. Pdf in this paper, we investigate a two dimensional isotropic harmonic oscillator on a timedependent spherical background. The symmetry algebra of the twodimensional anisotropic quantum harmonic oscillator with rational ratio of frequencies, which is characterizing. In an earlier paper, a group, g, associated with the n. It includes the exotic newtonhooke particle and the noncommutative landau problem as special, isotropic and maximally anisotropic, cases. The isotropic harmonic oscillator in an angular momentum basis. Relativistic chaos in the anisotropic harmonic oscillator. Pdf shells in a simple anisotropic harmonic oscillator. The isotropic harmonic oscillator in an angular momemtum basis.
A nonrelativistic charged particle moving in an anisotropic harmonic oscillator potential plus a homogeneous static electromagnetic field is studied. The timeindependent schrodinger equation for a 2d harmonic oscillator with commensurate frequencies can generally given by. Wolframalpha brings expertlevel knowledge and capabilities to the broadest possible range of peoplespanning all professions and education levels. The anisotropic harmonic oscillator has therefore been widely used in unified model calculations 2, particularly in calculations of the moments of inertia of rotational states although the oscillator system can be easily solved and the states completely specified in terms of particle oscillator quantum numbers, certain other operators are of. He begins with qualitative discussion on bound state solutions and then moves on to the quantitative treatment of harmonic. In this paper we will examine the quantum twodimen sional anisotropic harmonic. In section 6 we discuss about the big bang naked singularity of the expanding universe which how can it is removed in the quantum perspective of the system. An harmonic oscillator is a particle subject to a restoring force that is proportional to the displacement of the particle. Pdf we investigate the planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with noncommutative coordinates find.
The rain and the cold have worn at the petals but the beauty is eternal regardless. Anisotropic harmonic oscillator in the presence of a magnetic field jose m. For the latter we derive closedform analytical expressions. Qionggui lin department of physics, sun yatsen university, guangzhou 510275, peoples republic of china and china center of advanced science and technology world laboratory, p. Pdf two dimensional isotropic harmonic oscillator on a time. A particular representation r v of l, when restricted to g, was proved to be unitarily equivalent to. Coherent states for the isotropic and anisotropic 2d harmonic. Charged anisotropic harmonic oscillator and the hydrogen atom in.
We find transformation matrices allowing to express noncommutative three dimensional harmonic oscillator in terms of an isotropic commutative oscillator. An accurate formula for the density of states for 87rb is obtained from fitting the. The harmonic oscillator is an essential tool, widely used in all branches of physics in order to understand more realistic systems, from classical to quantum and relativistic regimes. Quantum harmonic oscillator part i mit opencourseware. Threedimensional isotropic harmonic oscillator wolfram. In classical physics this means f mam 2 x aaaaaaaaaaaaa t2 kx. By downloading this thing, you agree to abide by the license. It is proposed a new approach of three dimensional optical instrument using spreadsheet excell by which ray trajectories or anisotropic mechanical harmonic oscillator form lissajous curves. We know that the harmonic oscillator is integrable in newtonian mechanics, whether forced, damped or multidimensional. We investigate the planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with noncommutative. Havent seen it as an example before, so i am posting this here.
Maximizing riesz means of anisotropic harmonic oscillators. New isotropic vs anisotropic phase of noncommutative 2d. Amazing but true, there it is, a yellow winter rose. In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2d harmonic oscillator. The problem of the energy levels of a threedimensional anisotropic harmonic oscillator in a uniform magnetic field with an arbitrary strength and orientation is exactly solved. If we ignore the mass of the springs and the box, this one works. Anisotropic threedimensional harmonic oscillator in a threedimensional problem, consider a particle of mass m and of potential energy. Anisotropic harmonic oscillator in a static electromagnetic field. Time dependent entropy and decoherence in a modified. An important property of this curve is that a threedimensional region of space can be portrayed stigmatically with perspective projection both azimuthal rotation. Anisotropic quantum harmonic oscillator by lnemzer is licensed under the creative commons attribution license.
The energy levels and wave functions are obtained explicitly. The wavefunction is separable in cartesian coordinates, giving a product of three onedimensional oscillators with total energies. Pdf accurate density of states for condensed 87rb in. New isotropic vs anisotropic phase of noncommutative 2dharmonic oscillator anais smailagicb, euro spalluccia adipartimento di fisica teorica, universit a di trieste and infn, sezione di trieste binfn, sezione di trieste we prove existence of a new representation of the 2d non commutative harmonic oscillator in an isotropic commutative. Pdf symmetry algebra of the planar anisotropic quantum harmonic. Perturbation of a degenerate isotropic 2d harmonic oscillator. Since these expressions are in a form conducive to rapid machine computation. Harmonic oscillator node theorem still holds many symmetries present evenlyspaced discrete energy spectrum is very special.
Anisotropic quantum harmonic oscillator by lnemzer. We define a new set of ladder operators for the 2d system as a linear combination of the x and y ladder operators and construct the s u 2 coherent states, where these are. Homework equations the energy operator hamiltonian. She needed a physical example of a 2d anisotropic harmonic oscillator where x and y have different frequencies. Rotational properties of the twodimensional anisotropic.
Symmetry algebra of the planar anisotropic quantum harmonic. What are its energies and eigenkets to first order. Shells in a simple anisotropic harmonic oscillator. The twodimensional anisotropic harmonic oscillator with rational ratio of frequencies is a well known example of a classical superintegrable system. The anisotropic harmonic oscillator potential is of physical interest in quantum mechanics, since it may describe the motion of, say, an electron in an anisotropic. An exact solution of the quantummechanical problem of the energy spectrum and wave functions of the stationary states of an anisotropic oscillator in a onedimensional magnetic field has been obtained by using the commutation relations for the components of the operator of the velocity and the operators of the coordinates of the particle. We applied a generalized scalarvectortensor brans dicke gravity model to study canonical quantization of an anisotropic bianchi i cosmological model. The quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. Anisotropic mechanical harmonic oscillator in lissajous. In this paper we introduce a new method for constructing coherent states for 2d harmonic oscillators. Threedimensional anisotropic harmonic oscillator in a. Pdf anisotropic harmonic oscillator, noncommutative landau. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. In section 5 we discuss about the quantization of the adm energy of the system.