The motion of circular wave packets for one electron in a central Coulomb field with a high atomic number Z is calculated. The wave packet is defined in terms of solutions of the Dirac equation in order to take into account all possible relevant effects, in particular the spin-orbit potential. A time scale is defined within which spin dynamics must be taken into account, mainly in atoms with high Z. Within this time scale there exists a mechanism of collapses and revivals of the spin, already shown by the authors for harmonic oscillator potential and called the spin-orbit pendulum. Though this effect does not have the exact periodicity of the simpler model, the wave packet spatial motion is nevertheless quite similar.
Localized quantum wave packets can be produced in a variety of physical systems and are the subject of much current research in atomic, molecular, chemical, and condensed-matter physics. They are particularly well suited for studying the classical limit of a quantum-mechanical system. The motion of a localized quantum wave packet initially follows the corresponding classical motion. However, in most cases the quantum wave packet spreads and undergoes a series of collapses and revivals. We present a generic treatment of wave-packet evolution, and we provide conditions under which various types of revivals occur in ideal form. The discussion is at a level appropriate for an advanced undergraduate or first-year graduate course in quantum mechanics. Explicit examples of different types of revival structure are provided, and physical applications are discussed.
We have calculated the time evolution of the density matrix reduced to the spin degree of freedom for a particle in a spherical harmonic oscillator with a constant spin-orbit interaction. We have studied two classes of spatial wave functions in the initial state: pure orbital states \ensuremath{\Vert}lm〉 and coherent states. When the initial spin is a pure state, its evolution in time produces a mixed state during a time called the collapse time. This behavior of the spin is similar to that observed in the Jaynes-Cummings model. A similar dynamics is produced for a broad range of geometrical initial conditions for the spin as well as for the wave packet. The system is exactly recurrent but we have also found the existence of approximate recurrences corresponding to a reversal of the spin direction. We have tentatively applied our result to mimic the evolution of the spin in the H atom. In this case only a few spin-orbit partners play a role and, after a long time, produce a state where the admixture is not complete. The average quantum behavior is always very different from that of a classical particle having a spin as a vector of constant length. The increase of the number 2S+1 of spin-orbit partners changes the time scale of the collapse by the factor \ensuremath{\surd}S , and a strong mixing is always observed.
Time evolution of radial wave packets built from the eigenstates of Dirac equation for a hydrogenic system is considered. Radial wave packets are constructed from the states of different n quantum numbers and the same lowest angular momentum. In general they exhibit a kind of breathing motion with dispersion and (partial) revivals. Calculations show that for some particular preparations of the wave packet one can observe interesting effects in spin motion, coming from inherent entanglement of spin and orbital degrees of freedom. These effects manifest themselves through some oscillations in the mean values of spin operators and through changes of spatial probability density carried by upper and lower components of the wave function. It is also shown that the characteristic time scale of predicted effects (called T ls ) is much smaller for radial wave packets than in other cases, reaching values comparable to (or even less than) the time scale for the wave packet revival.
After a Rydberg wave packet forms, it is known to undergo a series of collapses and revivals within a time period called the revival time
trev, at the end of which it resembles its original shape. We study the behavior of Rydberg wave packets on time scales much greater than
trev. We find that after a few revival cycles the wave packet ceases to reform at multiples of the revival time. Instead, a new series of collapses and revivals commences, culminating after a time period
tsr≫trev with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time
trev. Furthermore, at times that are rational fractions of
tsr, we show that the motion of the wave packet is periodic with periodicities that can be expressed as fractions of the revival time
trev. These periodicities indicate a new type of fractional revival, occurring for times much greater than
trev. We also examine the effects of quantum defects and laser detunings on the revival structure of Rydberg wave packets for alkali-metal atoms.
Quantum mechanical wave packet states can approach the classical ideal of a spatially localized particle traveling along a well-defined trajectory. These states offer a system in which the classical limit of quantum atomic theory can be studied. In this work, a wave packet that is radially localized and whose radial motion occurs at the classical orbital period is studied. Despite these strong classical features, the long-term evolution displays the quantum nature of the state. As the wave packet evolves, it becomes dispersed, and the radial motion becomes irregular. However, the wave packet does not remain dispersed, and classical motion returns with the return of the spatial localization. Both the collapse and revival of the wave packet occur because of the quantized nature of the atom. In this experiment, the wave packet was excited to the Rydberg slates by a short, ultraviolet pulse from the ground state of potassium. The probe pulse delay was scanned, and the resulting photo-ionization signal displayed the evolution of the wave packet. The collapse and revival of the radial wave packet were observed.1