Stark effect in Lax–Phillips scattering theory

Abstract

Abstract:The scattering theory of Lax and Phillips, originally developed to describe resonances associated with classical wave equations, has been recently extended to apply as well to the case of the Schrödinger equation in the case that the wave operators for the corresponding Lax-Phillips theory exist. It is known that the bound state levels of an atom become resonances (spectral enhancements) in the continuum in the presence of an electric field (on all space) in the quantum mechanical Hilbert space. Such resonances appear as states in the extended Lax-Phillips Hilbert space. We show that for a simple version of the Stark effect, these states can be explicitly computed, and exhibit the (necessarily) semigroup property of decay in time. The widths and location of the resonances are those given by the poles of the resolvent of the standard quantum mechanical form. 1 IntroductionThe description commonly used for an unstable system is that of Wigner and Weisskopf 1 . They assumed that the unstable system is represented by a vector, say φ, in the Hilbert space of states, and that the Hamiltonian evolution of this state permits the development of other components which represent the "decayed" system. One may assume a Hamiltonian H 0 for which φ is an eigenstate, and the full Hamiltonian is constructed by adding a pertubation V which induces transitions from this state. The basic assumption of their physical model is that these transitions carry the state of the system from that of some specific system to a state in the continuous spectrum which contains the decay products. For example, one may think of an atom in an excited state as the unstable system; in the absence of electromagnetic interaction, this state would be a stable bound state. A term added to the Hamiltonian corresponding to electromagnetic interaction induces a transition from this state, and the square of the corresponding transition amplitudes, which may be generally computed perturbatively, give the probability for decay. This very standard technique is rigorously correct for reversible quantum transitions, according to the laws of quantum theory. When applied to the decay of an unstable system, however, for which the evolution is irreversible, it consitutes an approximation which may be inadequate. In the following we discuss some of the difficulties of the application of the Wigner-Weisskopf method to the treatment of irreversible phenomena, such as particle decay, and in Section 3 we describe the general structure of the Lax-Phillips theory, which yields an exact semigroup evolution and appears to be a more appropriate description of such phenomena. In Section 2, we apply the Wigner-Weisskopf theory to a simple model for the Stark effect, for comparison, and work out the Lax-Phillips theory for this model in Section 4. Some of the calculations in Section 2 coincide with those needed in Section 4; the approximate pole approximation decay law of the Wigner-Weisskopf theory contains exactly the same pole as the singularity of the Lax-Phillips S m...

Semigroup evolution in the Wigner-Weisskopf pole approximation with Markovian spectral coupling

“…This theory achieves an exact semigroup law by imbedding the usual quantum theory in a larger Hilbert space, consisting of a direct integral of a family of Hilbert spaces of usual type, foliated according to the time parameter. The result was particularly effective and straightforward for treating quantum mechanical systems with Hamiltonians of unbounded spectrum [9,10]. Its generalization for problems with semibounded spectrum [8] made it clear that the imbedding associated with the Lax-Phillips theory effectively introduces many additional degrees of freedom.…”