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Journal of Mathematical Physics

Scattering theory in arrangement channel quantum mechanics

James W. Evans, David K. Hoffman, Donald Jack Kouri

Published Mar 1, 1983• 8 citations
Publisher

Abstract

The time-independent scattering theory associated with the non-self-adjoint matrix Hamiltonians H of arrangement channel quantum mechanics is presented in detail first using the 3-particle case as an example. A key feature is the biorthogonality of a suitably constructed set of scattering eigenvectors and duals. Channel space Möller operators, S- and T-matrices are defined and a variety of properties investigated including the way multichannel unitarity is embedded into the theory. Some remarks on the time-dependent theory are also made. A detailed discussion of channel space density matrix scattering theory (of interest, e.g., in reactive kinetic theory) is presented using the Liouville representation. We describe some special cases including the exclusion of breakup and 2×2 choices of three particle H.

DOI
10.1063/1.525730

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The time-independent scattering theory associated with the non-self-adjoint matrix Hamiltonians H of arrangement channel quantum mechanics is presented in detail first using the 3-particle case as an example. A key feature is the biorthogonality of a suitably constructed set of scattering eigenvectors and duals. Channel space Möller operators, S- and T-matrices are defined and a variety of properties investigated including the way multichannel unitarity is embedded into the theory. Some remarks on the time-dependent theory are also made. A detailed discussion of channel space density matrix scattering theory (of interest, e.g., in reactive kinetic theory) is presented using the Liouville representation. We describe some special cases including the exclusion of breakup and 2×2 choices of three particle H.
A non-Hermitian matrix Hamiltonian H appears in the wavefunction form of a variety of many-body scattering theories. This operator acts on an arrangement channel Banach or Hilbert space 𝒞 = ⊕αℋ where ℋ is the N-particle Hilbert space and α are certain arrangement channels. Various aspects of the spectral and semigroup theory for H are considered. The normalizable and weak (wavelike) eigenvectors of H are naturally characterized as either physical or spurious. Typically H is scalar spectral and ’’equivalent’’ to H on an H-invariant subspace of physical solutions. If the eigenvectors form a basis, by constructing a suitable biorthogonal system, we show that H is scalar spectral on 𝒞. Other concepts including the channel space observables, trace class and trace, density matrix and Möller operators are developed. The sense in which the theory provides a ’’representation’’ of N-particle quantum mechanics and its equivalence to the usual Hilbert space theory is clarified.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.
A rigorous derivation of the reactive quantum Boltzmann equations is presented for systems where breakup and recombination are excluded. The use of an arrangement channel space representation allows an exact decomposition of the N particle density matrix into components for different chemical compositions and an exact definition of reduced species density matrices (as opposed, e.g., to standard projection operator techniques). This necessitates the use of the combinatorially complex arrangement channel BBGKY hierarchy which, however, avoids the need for the usual heuristic specification of collision terms. Another advantage is that scattering equations generated for the reactive and nonreactive many body T matrices appearing in the Boltzmann equations have ‘‘well-behaved’’ kernels (unlike the corresponding Lippmann–Schwinger equations). From the derived equations we readily obtain, e.g., reaction-diffusion equations and nonequilibrium expressions for the chemical reaction rates.
An alternative formulation is presented of the formal theory of multi-channel scattering in nonrelativistic quantum mechanics. We start by defining spaces of state vectors, where two particles either stay together or separate in the limit t →+∞ (or — ∞), when the state vector develops in time by e–i H t (H is the complete Hamiltonian of the n-particle system). A channel is defined as a space of state vectors with the following property: Developing in time by e-i H t they asymptotically describe a state of the n-particle system, where the particles are grouped in fragments. Defining a Hamiltonian Hγ for each channel, in which—compared to H—the interactions acting between particles from different fragments are missing, it is physically plausible that lim eiH e—iHt Ψ exists for vectors Ψ in the channel. Having discussed the limit vectors (asymptotic states), the S-matrix formalism can be introduced as usual. Finally the introduction of the exclusion principle is discussed.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.