On the exact revival of Morse oscillator wave packets

M.L. Strekalov

Alvason Zhenhua Li, William G. Harter

Alvason Zhenhua Li, William G. Harter

Robert G. Littlejohn

Zhenhua Li

Analytical eigenfunctions and eigenvalues for the Morse oscillator were applied to investigate the quantum resonant beats and revivals of wave packet propagation. A concise way for exact prediction of the complete revival period of the Morse oscillator was given for the first time. It was suggested that any complete period was made of integer numbers of the minimum or fundamental period. Within the fundamental period, the anharmonicity of this oscillator appeared to cause interesting space-time phenomena that include relatively simple Farey-sum revival structures. In addition, a simple sum of two Morse oscillators led to a double-Morse well whose geometric symmetry provided analytical eigenfunctions and eigenvalues for certain low-lying energy levels. The quantum tunneling between the double-Morse well significantly affected the resonant beats and revivals local to each well, and gave rise to interesting tsunami-like waves in the middle of the double well. Furthermore, quantum rotor wave functions based upon Wigner-D matrix were applied to investigate the quantum resonant beats and revivals that occur in experimentally accessible spin systems. Interesting physical effects in quantum rotors between half-integer spin and integer spin systems were observed to show effects of symmetry. Essentially, the quantum revivals in these quantum systems exhibited number-information aspects of surprisingly simple Farey-sum and Ford circles geometry. Such quantum dynamics will provide a physical insight to further develop matter wave packet technology, and might have applications for quantum information processing and quantum computing.

Kalbe Razi Naqvi, S. Waldenstrøm

If the probability density P ( x , t ) = |ψ ( x , t )| 2 of a non-stationary state happens to meet the condition P ( x , t 0 ) = P ( x , t 0 + Υ), the wave packet is said to experience exact revivals at intervals of Υ. The definition implies that a harmonic oscillator wave packet will experience revivals with a period Υ = 2π/ω, where ω is the angular frequency of the corresponding classical system. A harmonic oscillator wave packet of Gaussian form, centered initially ( t = 0) at x = x 0 , is shown to spread almost everywhere, if its initial width is sufficiently narrow, in a time very much shorter than the period of the corresponding classical oscillator. However, the packet reassembles itself twice during a single period: (i) it experiences a mirror revival, defined by the relation P ( x , ½Υ) = P (- x , 0), and (ii) an exact revival, P ( x , Υ) = P ( x , 0). These results are used to counter a recent claim (which apparently gives voice to a commonly held belief) that a harmonic oscillator wave packet ψ( x , t ) does not collapse and has no revivals. Plots of the recurrence probability |⟨ψ( x , 0)|ψ( x , t )⟩| 2 are presented and the need for formulating unambiguous definitions of the two concepts, revival and collapse, is stressed.