A formalism to calculate the time delay experienced by a particle undergoing scattering is presented. The formalism is a natural extension of Wigner’s delay time to wave packets which do not exhibit a well defined angular momentum. The delay depends only on the partial wave phase shifts characterizing the scattering event. The formalism is tested by applying it to the 3D square well. The results are shown to agree with the known properties of resonant scattering as deduced from a study of the poles of the scattering matrix in the complex energy plane.
A formalism to calculate the time delay experienced by a particle undergoing scattering is presented. The formalism is a natural extension of Wigner’s delay time to wave packets which do not exhibit a well defined angular momentum. The delay depends only on the partial wave phase shifts characterizing the scattering event. The formalism is tested by applying it to the 3D square well. The results are shown to agree with the known properties of resonant scattering as deduced from a study of the poles of the scattering matrix in the complex energy plane.
We present stationary expressions for the on-shell time delay operator and for its trace in quantum-mechanical potential scattering. The trace of this operator is shown to represent the time delay for a monoenergetic and well-collimated beam of initial states. For exponentially decaying potentials the total scattering cross section and the trace of the on-shell time delay operator have analytic extensions to complex energies, with poles at most at the usual resonance poles (poles of the analytically continued S -matrix) in the lower half plane and at their mirror images with respect to the real axis. At real energies close to such a pole both the scattering cross section and the time delay are approximately of Breit–Wigner type, the sign of the time delay being related to that of the virial. These results corroborate the idea that a resonance should show up through a peak in the scattering cross section and a large (though positive) time delay.
Two-body scattering is analyzed using a time-dependent formalism and wave packets of arbitrary shape as the initial wavefunctions. In particular, the case of single-channel resonant scattering is discussed in some detail. In addition to the usual amplitudes resulting from the potential term and the resonant term, a third amplitude is found which interferes with the other two. Another interesting result is that in this formalism the final state is not necessarily represented by spherical outgoing waves, but can be interpreted instead as two separate outgoing wave packets whose shape is different from the initial packets. The Wigner time delay is obtained as a natural consequence of the analysis, and the resonant cross sections are expressed as density matrices constructed from the wave packets.
The time delay, as defined via phase shifts, does not apply to atom-atom collisions because of the semiclassical nature of the system. In this limit the contribution of one partial wave to the cross section is negligible. Therefore, we analyze time delay for the scattering amplitude and show that some new phenomena may occur, which cannot be explained by the time delay for a single phase shift. The time delay, which is averaged over all scattering angles, shows structure corresponding to several delay mechanisms. We also show that the lifetime of a resonance state, formed in a collision, may be considerably shorter than expected from the theory of resonance scattering.
We study non-relativistic propagation of Gaussian wave packets in one-dimensional Eckart potential, a barrier, or a well. In the picture used, the transmitted wave packet results from interference between the copies of the freely propagating state with different spatial shifts (delays), [Formula: see text], induced by the scattering potential. The Uncertainty Principle precludes relating the particle's final position to the delay experienced in the potential, except in the classical limit. Beyond this limit, even defining an effective range of the delay is shown to be an impracticable task, owing to the oscillatory nature of the corresponding amplitude distribution. Our examples include the classically allowed case, semiclassical tunnelling, delays induced in the presence of a virtual state, and scattering by a low barrier. The properties of the amplitude distribution of the delays, and its pole representation are studied in detail.
In the standard Breit-Wigner approach to scattering the phase shift is to have a form tanδBW=Γ1/(E1−E) at a real energy resonance. This leads to complex energy poles in the scattering amplitude at EBW=E1−iΓ1, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting tanδBW=Γ1/(E1−E) is not always a good description of the real energy scattering amplitude, that Γ1 can be negative, that EBW is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear PT symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs E∓=E2∓iΓ2 with E−=EBW, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.
In this paper we extend our previous analysis of the scattering of wave packets in one dimension to the case of the square-well potential. The analytic properties of the general scattering solution are emphasized thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. The square-well model is particularly interesting because of its application to the deuteron problem. Resonance scattering, barrier penetration, time delay, and line shape are discussed at the level of the first-year graduate student.