Coherent states of quantum non-linear systems

Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 h̄ . In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z >= z | z > (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position x̂ and momentum p̂ operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z >= |exp(−iωt)z > (2) where ω is the oscillator′s frequency. Then, because the state | z > is concentrated around x = √ 2h̄ ω Rez it follows that Ut|z > has its support on the classical trajectory x(t) = √ 2h̄ ω Re(exp(−iωt)z). The simple classical evolution z → exp(−iωt)z is a consequence of the simple representation of √ωx+ i p √ ω = √ Iexp(iθ) in terms of the action-angle variables (I, θ) for the harmonic oscillator. We are going to generalize the construction of coherent states to integrable non-linear systems. It is known that the quantum non-linear dynamics cannot be transformed into a coherent state dynamics exactly (see the discussion in [2][3];these authors define generalized coherent states, which do not coincide with ours). However, wave packets approximately localized around the classical trajectory of an electron in the hydrogen atom have been constructed [4][5] . For this purpose a large dynamical symmetry relating the hydrogen atom to the fourdimensional oscillator has been utilized. In this Letter we do not assume any symmetry, but the integrability of a classical Hamiltonian system. Our construction of wave packets can be applied to multidimensional anharmonic