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College Physics

Discussion on rotating trap of two-dimensional harmonic oscillator

Yin Xiang-yu

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

To study the rotating trap which is related with BEC,two-dimensional harmonic oscillator is ana- lyzed and Hamihonian is provided.Wave function and energy eigenvalue are given.A coincidence is found that the rotating trap of two-dimensional harmonic oscillator has the similar form as the potential of charged particle in mag- netic field.

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Abstract We study the dynamics of a quantum or classical particle in a two-dimensional rotating anisotropic harmonic potential. By a sequence of symplectic transformations for constant rotation velocity we find uncoupled normal generalized coordinates and conjugate momenta in which the Hamiltonian takes the form of two independent harmonic oscillators. The decomposition into normal-mode dynamics enables us to design fast trap-rotation processes to produce a rotated version of an arbitrary initial state, when the two normal frequencies are commensurate.
The quantum harmonic oscillator is a basic model for many complex ones.After integration,the Wigner function of quantum harmonic oscillator can be expressed by a simple form,which can be used to solve many related problems.This paper has placed the harmonic oscillator in non-commutative phase space,and studied the Wigner function of two-dimensional charged harmonic oscillator in electric field.The results show that when the non-commutative parameter is zero,the Wigner function of two-dimensional charged harmonic oscillator is equivalent to its form in commutative space.
We investigate the angular momentum of a two-dimensional non-interacting Fermi gas confined in a rotating harmonic trap. By employing a quantum statistical approach, analytical expressions for the angular momentum are derived and analyzed as functions of particle number and rotation frequency. The rotational motion introduces a shell structure in the energy spectrum, giving rise to oscillatory behavior analogous to the de Haas–van Alphen oscillations observed in electronic systems subjected to magnetic fields. Numerical calculations reveal that these oscillations are most pronounced for weak rotation and small particle numbers, while they become progressively damped as the particle number increases. The results emphasize the close analogy between ultracold rotating gases and charged particles in magnetic fields and provide further insight into quantum rotational effects in confined Fermi systems.
Complete description of the classical and quantum dynamics of a particle in an anisotropic, rotating, harmonic trap is given. The problem is studied in three dimensions and no restrictions on the geometry are imposed. In the generic case, for an arbitrary orientation of the rotation axis, there are two regions of instability with different characteristics. The analysis of the quantum-mechanical problem is made simple due to a direct connection between the classical mode vectors and the quantum-mechanical wave functions. This connection is obtained via the matrix Riccati equation that governs the time evolution of squeezed states of the harmonic oscillator. It is also shown that the inclusion of gravity leads to a resonant behavior -- the particles are expelled from the trap.