The energy formula of the two dimensional harmonic oscillator in cylindrical coordinates is found by numerical integration of schrodinger equation. Exact propagator of a two dimensional anisotropic harmonic oscillator in the presence of a magnetic field. Many potentials look like a harmonic oscillator near their minimum. Selecting this option will search all publications across the scitation platform selecting this option will search. Introduction to quantum mechanics with applications to chemistry dover books on physics by linus pauling, e. The eigenvalues of n and the corresponding eigenstates may be displayed in the form of a ladder fig. The komaba solution is presented for the twodimensional harmonic oscillator in the polar coordinate representation, in contrast to the pasadena solution presented by dr. The threedimensional, quantummechanical, anisotropic.
In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2d harmonic oscillator. Quantum reports free fulltext coherent states for the. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are determined using algebraic methods of general applicability to quantum. Deformed u2 algebra as the symmetry algebra of the. On the degeneracy of the twodimensional harmonic oscillator. Wigner function of two dimensional isotropic harmonic oscillator.
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. 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. Find the expected values of position and impulse and check the ehrenfest theorem. The 1d harmonic oscillator the harmonic oscillator is an extremely important physics problem. The harmonic oscillator has only discrete energy states as is true of the onedimensional particle in a box problem. Find the energy and the angular momentum as a functions dependent of time and compare them with initial values. By regarding the hamiltonian as a linear operator acting through the poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear hamiltonians, it is shown how all possible rational constants of. The hamiltonian for this seems quite complicated, but i imagine there is some trick like the one dimensional case which simplifies the problem a lot. This may come a bit elemental, what i was working on a direct way to find the eigenfunctions and eigenvalues of the isotropic two dimensional quantum harmonic oscillator but using polar coordinates. For a fluctuating background, transition probabilities per unit time are obtained. Coincidence of quantumclassical orbits for periodically driven twodimensional anisotropic oscillatorberry phase and hannay angle junli xin1,2 and jq liang1 1institute of theoretical physics, shanxi university, taiyuan 030006, people. Wigner function of twodimensional isotropic harmonic. New isotropic vs anisotropic phase of noncommutative 2d.
In complement byir, we shall study the isotropic three dimensional. In this paper we will examine the quantum twodimen sional anisotropic harmonic oscillator with a 2. This system, though particular, is in teresting in two respects. Although the system treated here is two dimensional, many of the results obtained can be applied directly to the three dimensional anisotropic oscillator. The twodimensional anisotropic harmonic oscillator with rational ratio of frequencies is a well known example of a classical superintegrable system. 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. A isotropic twodimensional harmonic oscillator for classical mechanics a, univ. In order to give one possible answer, ill just take the isotropic harmonic oscillator in 2d and do a finitedifference calculation by discretizing the xy plane with constant spacing a. In this thesis, we examine perturbationinduced scarring in a two dimensional anisotropic elliptic oscillator characterized by the anisotropy parameter, i. This may come a bit elemental, what i was working on a direct way to find the eigenfunctions and eigenvalues of the isotropic twodimensional quantum harmonic oscillator but using polar coordinates. Here is the construction of the resulting matrix for the hamiltonian, h. Pdf two dimensional isotropic harmonic oscillator on a time.
Anisotropic three dimensional harmonic oscillator in a three dimensional problem, consider a particle of mass wand of energy. Pdf two dimensional isotropic harmonic oscillator on a. Coincidence of quantumclassical orbits for periodically. Coherent states for the isotropic and anisotropic 2d. The hamiltonian 1 is separable into 2a independent harmonic oscillators corresponding to the x and y coordinates of each particle. Quantum scarring in a twodimensional elliptical harmonic. Because the creation and destruction operators each commute with themselves, multi quantum states are unchanged under exchange of quanta, which therefore behave as bosons. 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 problems. The symmetry algebra of the twodimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a nonlinear extension of the u2. Pdf exact propagator of a two dimensional anisotropic harmonic. Threedimensional anisotropic harmonic oscillator in a. Let a mass point m be bound to the origin by an isotropic harmonicforce. Anisotropic threedimensional harmonic oscillator in a threedimensional problem, consider a particle of mass m and of potential energy. The harmonic oscillator is characterized by the hamiltonian.
The periodic orbits pos of the two dimensional 2d anisotropic elliptic ho were first investigated by bowditch 2 and later in more detail by lissajous 3. The harmonic oscillator ho is a linchpin in various fields of physics 1. Exact propagator of a two dimensional anisotropic harmonic oscillator in the presence of a magnetic field jose m. 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. The symmetry algebra of the two dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a nonlinear extension of the u2. Consider a diatomic molecule ab separated by a distance with an equilbrium bond length. Although the system treated here is twodimensional, many of the results obtained can be applied directly to the three dimensional anisotropic oscillator. In order to give one possible answer, ill just take the isotropic harmonic oscillator in 2d and do a finitedifference calculation by discretizing the xy plane with constant spacing a here is the construction of the resulting matrix for the hamiltonian, h. The result is useful for the treatment of quasi two dimensional samples in the field of magnetic effects in nanostructures and quantum optics.
Qm the most successful theory in human history, having successfully passed. The coherent states of the twodimensional isotropic harmonic. How can we treat the three dimensional anisotropic harmonic oscilator in quantum mechanics without complicating the problem a lot. The anisotropic oscillator in the twodimensional case both in classical and quantum mechanics was discussed by jauch and hill 16. In more than one dimension, there are several different types of hookes law forces that can arise. Working with threedimensional harmonic oscillators dummies.
Anisotropic three dimensional harmonic oscillator in a three dimensional problem, consider a particle of mass m and of potential energy. It is shown that, in the magnetic field, the levels continue to be equidistant. Abstract a further study of the degeneracy of the two dimensional harmonic oscillator is made, both in the isotropic and anisotropic cases. Is there an easy way to treat the anisotropic harmonic. Pdf quantum monodromy in the isotropic 3dimensional. The presented solution, after minor extensions, can also be used for motion in three. In this paper we introduce a new method for constructing coherent states for 2d harmonic oscillators.
The e ect of noncommutativity in the new phase, induces energy levels splitting of the isotropic quantum. Threedimensional isotropic harmonic oscillator youtube. The restoring force has the form fx kxx in one dimension. The coherent states of the twodimensional isotropic. The two dimensional isotropic harmonic oscillator is defined by the hamiltonian more in units where the mass, the angular frequency, and plancks constant equal one. Spain abstract in this paper we solve exactly the problem of the spectrum and feyn. The symmetry algebra of the twodimensional anisotropic quantum harmonic oscillator with rational ratio of frequencies is identified as a deformation of the u2 algebra. Exact propagator of a two dimensional anisotropic harmonic.
Anisotropic quantum harmonic oscillator by lnemzer. The quantum harmonic oscillator is the quantum mechanical analog of the classical harmonic oscillator. Classification of the stationary states by the quantum numbers n, and n. A further study of the degeneracy of the two dimensional harmonic oscillator is made, both in the isotropic and anisotropic cases. These lissajous orbits are sensitive on the frequency ratio of the confinement. 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. Schrodinger eigenvalue problem in two dimensions harmonic.
We then prove that a twodimensional timeindependent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states in terms of the spin values j. Equilibration of quantum gases terry farrellydiscretetime reservoir engineering with entangled bath and stabilising squeezed states zibo miao and alain sarletterecent citations coherent states in magnetized anisotropic 2d dirac materials e diazbautista et altwodimensional nonhermitian harmonic oscillator. Jan 10, 2011 expand the initial wave function by eigenstates of the anisotropic harmonic oscillator, and determine the time evolution of the system. The equation for these states is derived in section 1. This piece is simultaneously a physical representation of data, as well as an exploration of the extent and limitations of human knowledge. Anisotropic quantum harmonic oscillator by lnemzer thingiverse. This option allows users to search by publication, volume and page selecting this option will search the current publication in context. Rotational properties of the twodimensional anisotropic. Path integrals in quantum mechanics, statistics, polymer physics, and. Symmetry algebra of the planar anisotropic quantum. If we consider the bond between them to be approximately harmonic, then there is a hookes law force between. The twodimensional harmonic oscillator l z 0 f rom the one parameter family of twodimensional harmonic oscilla tors with l z 0 we will consider the one in the x, z plane.
The twodimensional isotropic harmonic oscillator is defined by the hamiltonian, in units where the mass, the angular frequency, and plancks constant equal one. In this paper we will examine the quantum twodimen sional anisotropic harmonic. The effect of the background can be represented as a minimally coupled field to the oscillator s hamiltonian. Symmetry algebra of the planar anisotropic quantum harmonic.
Jun 27, 2012 the problem of the energy levels of a three dimensional anisotropic harmonic oscillator in a uniform magnetic field with an arbitrary strength and orientation is exactly solved. Two dimensional isotropic harmonic oscillator on a timedependent sphere article pdf available in journal of physics a mathematical and theoretical 4546 october 2012 with 585 reads. First, the timeindependent problem is known to separate in two coordinate sys tems, cartesian and parabolic and, second, its energy. The two dimensional harmonic oscillator l z 0 f rom the one parameter family of two dimensional harmonic oscilla tors with l z 0 we will consider the one in the x, z plane. Furthermore, it is one of the few quantum mechanical systems for which an exact. The komaba solution is presented for the twodimensional harmonic oscillator in the polar coordinate representation, in contrast to the pasadena solution. By regarding the hamiltonian as a linear operator acting through the poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear hamiltonians, it is shown how all possible rational constants of the motion may be generated. Two dimensional harmonic oscillator in cylindrical coordinates. By regarding the hamiltonian as a linear operator acting through the poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear hamiltonians, it is shown how all possible rational constants of the motion may be. Oct 20, 2012 two dimensional isotropic harmonic oscillator on a timedependent sphere article pdf available in journal of physics a mathematical and theoretical 4546 october 2012 with 585 reads.