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Progress of Theoretical Physics

Scattering Matrix in the Heisenberg Representation for a System with Bound States

Abraham Klein

Published Dec 1, 1955• 5 citations
Publisher

Abstract

A definition is formulated of the scattering matrix for a closed physical system with bound states which makes use throughout only of the assumed observable properties of the system. A direct product space, X, is defined in which the ingredient factor space comprises the steady states—vacuum, one-particle, and bound states—of the physical system. It is argued that the boundary conditions for a scattering experiment are suitably expressed in terms of vectors in X and that these stand in unitary correspondence, U, to the Heisenberg states. Indeed, one defines two operators U(±) to express outgoing and ingoing wave boundary conditions, and the scattering matrix is constructed from these in the usual way. A suitable Yang-Feldman formalism is then developed in which the operators in the remote past and future also describe the bound states of the system. A representation of the framework thus constructed in terms of field operators for individual fields results in the well-known formulas for S-matrix elements in terms of covariant amplitudes.

DOI
10.1143/ptp.14.580

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A definition is formulated of the scattering matrix for a closed physical system with bound states which makes use throughout only of the assumed observable properties of the system. A direct product space, X, is defined in which the ingredient factor space comprises the steady states—vacuum, one-particle, and bound states—of the physical system. It is argued that the boundary conditions for a scattering experiment are suitably expressed in terms of vectors in X and that these stand in unitary correspondence, U, to the Heisenberg states. Indeed, one defines two operators U(±) to express outgoing and ingoing wave boundary conditions, and the scattering matrix is constructed from these in the usual way. A suitable Yang-Feldman formalism is then developed in which the operators in the remote past and future also describe the bound states of the system. A representation of the framework thus constructed in terms of field operators for individual fields results in the well-known formulas for S-matrix elements in terms of covariant amplitudes.
This chapter focuses on the scattering matrix, or S-matrix, an infinite-dimensional matrix or operator that expresses the state of a scattering system consisting of waves or particles or both in the far future in terms of its state in the remote past. In the case of electromagnetic (or acoustic) waves, the S-matrix connects the intensity, phase, and polarization of the outgoing waves in the far field at various angles to the direction and polarization of the beam pointed toward an obstacle. The chapter first considers the problem of scattering by a square well, located symmetrically with respect to the origin, before discussing bound states and a heuristic derivation of the Breit-Wigner formula. It als describes the Watson transform and Regge poles before concluding with an analysis of the time-independent radial Schrödinger equation and Levinson's theorem.
Abstract When creation and annihilation of particles is forbidden, the general features of a manyparticle collision are illustrated by a scattering system containing two particles and a fixed scattering centre. The term in the wave function, which is called by Moller the ‘outgoing’ part, is shown to contain a term representing a totally outgoing wave, but also terms describing the interference effects between the incident wave of one particle and the outgoing wave of the other. Corresponding to the latter terms, there exist singular eigenstates of the S matrix which are simultaneous eigenstates of the separate kinetic energies of the particles. The remaining eigenstates are called non-singular, and for these only the sum of the kinetic energies can be given a definite value. Analytic continuation of the non-singular eigenstates in the complex plane of total kinetic energy shows that the corresponding eigenvalues of S can be used to determine the energy levels of states with both particles bound to the centre of force. The eigenvalues for the singular eigenstates will lead to the bound energy of a single particle in the scattering field of the other. The formalism is extended to include singular eigenstates which describe the scattering of one particle on a compound centre made up of the other particle bound to the scatterer.
This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.
Quantum mechanics can be formulated either in terms of matrix mechanics or in terms of wave mechanics. This chapter centers on a discussion of the historical origins, mathematical structure, and extension to non-Hermitian quantum theory of matrix mechanics. The development of atomic discrete spectra and the old quantum theory showed that classical mechanics could not account for the discreteness of atomic energy levels and spectral lines, which motivated Heisenberg, Born, Jordan, and others to establish matrix mechanics centered on noncommuting physical quantities. Combining the Bohr atomic model, angular momentum quantization, and radiative transitions, one can see how matrix mechanics describes the energy level structure and quantum transitions of atomic systems, and how the Heisenberg equation of motion gives the fundamental form of quantum dynamics. The example of the harmonic oscillator further shows that the discreteness of the energy spectrum can be obtained directly from the matrix method, thereby establishing a clear connection among transition frequencies, matrix elements, and radiation-induced transitions. The matrix representation corresponds to the operator theory in Hilbert space, where state vectors, inner products, complete bases, and spectral decomposition constitute the fundamental mathematical framework of quantum mechanics. For closed quantum systems, physical observables are generally required to be represented by Hermitian or self-adjoint operators, so as to ensure that their eigenvalues possess the physically acceptable property of being real numbers. The Hermiticity of an operator is also closely related to its domain and boundary conditions, which provides a mathematical possibility for the emergence of non-Hermitian Hamiltonians. Based on these discussions, this chapter further explains how non-Hermitian dynamics is connected with open systems, dissipative processes, and special boundary conditions, thereby demonstrating the natural connection between traditional matrix mechanics and modern non-Hermitian quantum theory.
The scattering matrix is used to relate particle states before and after undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory. When quantum theory got developed, the classical descriptions for scattering were no longer sufficient and other methods began to develop. One of them being the S-matrix. In this thesis we present assumptions made in order to define the S-matrix, as well as its different properties and some one-dimensional examples. The definition of the S-matrix relates to the probability amplitude for different outcomes in a scattering experiment and the elements of the matrix are called scattering amplitudes. The S-matrix is unitary and if the scattering potential V (x) is real, it is also time-reversal invariant. In one dimension it is showcased how particles can transmit through or reflect off of a potential as well as how the symmetry of the potential effects the S-matrix. Lastly we give some examples of the S-matrix in one dimension and we end with a brief outlook of how it is defined and applied in quantum field theory where special relativity is taken into account.
The connection between causality and analyticity in scattering theory is formulated in terms of Hilbertspace concepts. The usual rules of nonrelativistic quantum mechanics are assumed to hold for the "in" and "out" states of the scattering system. We show that there is a (physically verifiable) causality condition which implies that each diagonal SSS-matrix element ${S}_{\ensuremath{\alpha}\ensuremath{\alpha}}(E)$ must be the limit of an analytic function of the energy EEE, regular in ImE>0\mathrm{Im}E>0ImE>0. The implications for partial-wave amplitudes and for the forward scattering amplitude in elastic two-body collisions are discussed.