Collective motions of a quantum gas confined in a harmonic trap

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Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency νc is invariant under a squeeze-type unitary transformation, which gives collective radial breathing motion of frequency 2νc to the gas. The amplitudes of the oscillating and breathing motions from the exact invariances could be arbitrarily large. For a Fermi system, appearance of 2νc mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.