Particle Wave Packet Research Articles

The exchange potential in path integral studies: Analytical justification

We present analytical justification for our previously described exchange pseudopotential. We show how the fermi quantum partition function can be constructed from the Boltzmann (distinguishable particle) wave functions if the states that correspond to like-spin electrons occupying the same quantum state are excluded. A class of weighting functions that satisfy this constraint approximately is discussed. Our previous pseudopotential falls under this class. Essentially, our pseudopotential forces the unwanted states to have high energy and, hence, to make negligible contribution to the partition function. Exchange potentials of the form discussed in this article should be useful for studying systems where the (allowed) correlated Boltzmann wave functions have negligible amplitude for like-spin fermion–fermion distances less than the diameter of the individual particle wave packets. For example, in the case of two spin-up (or spin-down) fermions, if one fermion is located at r, then ‖Ψ(r,q)‖2 is negligible if q≂r. This should be the case for systems where a tight binding model is appropriate or for systems with strong interparticle repulsions.

Time evolution of a non-gaussian wave packet

The behavior of the one-dimensional Schrödinger free particle wave packet with form factor sech α( k - k 0) is obtained as a function of time.

Quantum Brownian motion of a heavy particle: An adiabatic expansion

Abstract We consider Brownian motion of a heavy particle (mass M ), under conditions where the particle itself displays quantum effects, either because T is smaller than characteristic frequencies, or because the thermal wave packet size, h √MT , is bigger than characteristic lengths. A direct space method is developed which clearly separates wave packet spread from recoil effects (in the coupling to the heat bath): for a heavy particle, the latter are small, while the former are important in a quantum regime (a plain expansion in 1/ M is thus precluded). Expanding in powers of the recoil corresponds to an adiabatic expansion. The zeroth order yields an effective “potential” which is defined unambiguously for arbitrarily strong particle-bath coupling. The next term describes friction. A highly degenerate bath usually results in a collision integral which may be obtained from suitably renormalized golden rule arguments. An exception is the degenerate Fermi gas, in which one cannot separate scattering processes, there being no distinct time scales. Such an approach clearly emphasizes the various time scales of interest; it also points to the importance of coherence effects in the evolution of the Brownian particle wave packet.

Time-dependent approach to semiclassical dynamics

In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.