The problem of extending quantum-mechanical formal scattering theory to a more general class of models that also includes quantum field theories is discussed, with the aim of clarifying certain aspects of the definition of scattering states. As the strong limit is not suitable for the definition of scattering states in quantum field theory, some other limiting procedure is needed. Two possibilities are considered, the abelian limit and adiabatic switching. Formulas for the scattering states based on both methods are discussed, and it is found that generally there are significant differences between the two approaches. As an illustration of the application and the features of these formulas, S-matrix elements and energy corrections in two quantum field theoretical models are calculated using (generalized) old-fashioned perturbation theory. The two methods are found to give equivalent results.
The exact expression for the transition rate for quantum-mechanical scattering problems is derived formally without recourse to adiabatically switching on the interaction or averaging the initial state in time. The equivalence is shown of the stationary definition of the
S matrix and an appropriate definition by the time-dependent method.
There exists the well-known approximate expression describing the large time behavior of matrix elements of the evolution operator in quantum theory: < U(t)>≃ exp (at). This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf–Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: <U(t)>= exp (At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.
There exists the well known approximate expression describing the large time behaviour of matrix elements of the evolution operator in quantum theory: =exp(at)+... This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf-Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: =exp(At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.
This paper is concerned with the problem of calculating amplitudes for nonadiabatic transitions induced by a time-dependent Hamiltonian, in the semiclassical limit h/→0, with emphasis on questions relevant to semiclassical theories of electronically inelastic scattering. For this problem the semiclassical limit is mathematically equivalent to the adiabatic limit, and the adiabatic theorem says that all these transition amplitudes vanish in the limit; the question is, what is the asymptotic form of the nonadiabatic amplitudes, as they go to zero? We consider Hamiltonia that are analytic matrix functions of time. We prove a generalization of the adiabatic theorem to the complex time plane; paradoxically, the adiabatic theorem in the complex plane gives us directly the nonadiabatic amplitudes along the real time axis. We derive Dykhne’s remarkable formula for the two-state case, which says that the limiting form of the transition amplitude depends only on the energy curves of the two states, not on the nonadiabatic coupling which is responsible for transition between them. We discuss the three-state problem at length and show that the obvious generalization of the Dykhne formula is sometimes true, sometimes false. To indicate the scope of methods based on the adiabatic theorem in the complex plane, we give an elementary proof of the semiclassical formula for above-barrier reflection of a one-dimensional particle.
Within the framework of two-Hilbert space scattering theory the existence of the strong Abel limit of a certain operator is proved, leading to the following results. A generalized Lippmann identity is derived that is valid for all channels, rather than only two-body channels. On shell equivalence of the prior, post and AGS transition operators is rigorously proved, thus closing a gap in previous proofs. Results concerning the existence of the scattering operator as a strong, rather than weak, Abel limit are presented, and their implications with respect to the problem of unitarity are discussed. Finally, the possibility of exploiting operator limits of the Obermann–Wollenberg type is studied, with negative results.
Given two Hamiltonians
H0,H1=H0+V corresponding respectively to a free particle Hamiltonian and the free particle Hamiltonian plus its interaction energy with a scattering centre, we can compute the S-matrix, i.e., the matrix that determines the quantum mechanical amplitude of the process involving a particle coming from a free particle state at an infinite distance from the scattering centre at time
t=−∞, entering into an initial scattered state, interacting with the scattering centre, entering into a final scattered state, and then escaping away to infinity at time
t→∞ to a final free particle state. Such amplitudes were first studied by Lippmann and Schwinger (Steven Weinberg, The Quantum Theory of Fields, Vol. 1, W.O. Amrein, Hilbert Space Methods in Quantum Mechanics) and then analyzed in a mathematically precise way by [T. Kato], [W. Amrein], and [K.B. Sinha]. In this paper, we first derive the Lippmann-Schwinger equation for the input and output scattered states in terms of the input and output free particle states using intuitive arguments originally due to Lippmann and Schwinger, and then, using rigorous operator theoretic arguments revolving around the spectral theorem for unbounded self-adjoint operators in a Hilbert space, we obtain an elegant formula for the scattering matrix elements in terms of the interaction potential and the free particle Hamiltonian between two free particle states corresponding to the same measure. We then discuss a computationally more efficient method based on the Dyson series expansion (Steven Weinberg, The Quantum Theory of Fields, Vol. 1), which works even in the case when the interaction Hamiltonian is time varying, in contrast to the Lippmann-Schwinger method, for calculating the S-matrix. We assume that the scattering potential, which can be time varying, can be controlled by incorporating control parameters and explain how the resulting unitary S-matrix can be made to approximate a given unitary gate in infinite-dimensional Hilbert space by optimizing the Hilbert-Schmidt/Frobenius norm distance between the given gate and the S-matrix gate, in which the latter is approximated by a truncated Dyson series. Of course, the S-matrix given by the Dyson series is a Taylor series functional expansion in the interaction potential and hence is a highly nonlinear function of the control parameters, and hence we propose numerical methods for such optimization. We then observe that if the control parameters are taken as random variables with joint moments, then the TPCP map involving statistical averaging of the adjoint action of the S-matrix acting on an initial mixed state w.r.t. the probability distribution of the control parameters will result in a final mixed state that is linear in the statistical moments of the control parameters, and hence optimization w.r.t. these moments can be easily carried out. Our method of TPCP map design is to optimize the sum of Frobenius distance squares between a sequence of desired output mixed states and the S-matrix based formula for the TPCP map acting on the corresponding input mixed states w.r.t. the statistical moments of the control parameters. We then generalize this method to include the case of TPCP maps obtained from the Hudson-Parthasarathy quantum noisy Schrödinger equation. In summary, we simulate the S-matrix gate for independent realizations of the control parameters having a probability distribution defined by their joint moments obtained by the above optimization procedure and apply the adjoint action of these independent S-matrices on a given input state, and take their ensemble average to obtain a good approximation to the action of the desired TPCP map on any given input state. This suggests that TPCP maps of arbitrarily large size can be realized using scattering experiments in the laboratory.