Although the standard formalism of quantum mechanics is based on the requirement of the physical operators to be Hermitian, the use of non-Hermitian operators in the study of different types of phenomena is not uncommon. One of the most well-known non-Hermitian potentials is the optical potential where for any given choice of N channels the exact eigenvalue is a solution of a single-channel problem. The optical potential is a non-Hermitian, non-local and energy-dependent operator. In his book on scattering theory Taylor writes: “In practice, the optical potential is far too complicated for exact use in actual calculations”. It is often believed that the complex energies which are obtained by the use of optical potentials result from the approximations in the calculations. However, this is not true. The complex energy obtained by solving the one-channel problem with an optical potential is the exact eigenvalue of the original N -channel problem which is obtained by imposing outgoing boundary conditions on the eigenfunctions of the time-independent Schrödinger equation. In this chapter we wish to show that the study of the resonances in multi-channel problems (so-called Feshbach resonances) can be considered as the point where quantum mechanics branches into the standard (Hermitian) and non-standard (non-Hermitan) formalisms. Feshbach resonances Quite a long time ago Feshbach showed that the exact energy spectrum of the full physical problem can be obtained by solving two different self-energy problems.
Although the standard formalism of quantum mechanics is based on the requirement of the physical operators to be Hermitian, the use of non-Hermitian operators in the study of different types of phenomena is not uncommon. One of the most well-known non-Hermitian potentials is the optical potential where for any given choice of N channels the exact eigenvalue is a solution of a single-channel problem. The optical potential is a non-Hermitian, non-local and energy-dependent operator. In his book on scattering theory Taylor writes: “In practice, the optical potential is far too complicated for exact use in actual calculations”. It is often believed that the complex energies which are obtained by the use of optical potentials result from the approximations in the calculations. However, this is not true. The complex energy obtained by solving the one-channel problem with an optical potential is the exact eigenvalue of the original N -channel problem which is obtained by imposing outgoing boundary conditions on the eigenfunctions of the time-independent Schrödinger equation. In this chapter we wish to show that the study of the resonances in multi-channel problems (so-called Feshbach resonances) can be considered as the point where quantum mechanics branches into the standard (Hermitian) and non-standard (non-Hermitan) formalisms. Feshbach resonances Quite a long time ago Feshbach showed that the exact energy spectrum of the full physical problem can be obtained by solving two different self-energy problems.
As has been shown long time ago by Feshbach, the exact energy spectrum of the full problem can be obtained by solving two different self-energy problems. In spite of the fact that the two effective Hamiltonians are derived in very similar ways in one case, the exact energy spectrum of the full problem can be either real or complex (depending on the boundary conditions), whereas the exact energy spectrum associated with the second effective Hamiltonian has to be complex (excluding bound states in the continuum). The focus of this paper is on the fact that in both cases the complex eigenvalues result from the same requirement of an out-going boundary condition. The branching of quantum mechanics to standard (Hermitian) formalism and non-Hermitian formalism is associated with the decision to express the exact energy spectrum with one of the two possible self-consistent like problems where the use of the Green operator imposes an outgoing boundary condition on the solutions of the time-independent Schrodinger equation. Our analysis is made for the case where an ABC molecule has sufficient energy to dissociate to A + BC but not to A + B + C and not to AB + C or to AC + B.
The Hermitian properties of the Hamiltonian are related not only to the operator itself but also to the functions on which it acts. Hermitian Hamiltonians operate on functions in the L 2 Hilbert space which correspond to boundary conditions which vanish at infinity. In this chapter, in order to move into the non-Hermitian domain, we will impose on the solutions to the time-independent Schrödinger equation (TISE) different boundary conditions which lead to solutions which can be associated with different types of the complex poles of the scattering matrix. These solutions will contain information which was not available within the scope of functions in L 2 . By imposing outgoing boundary conditions on the eigenfunctions of the timeindependent Hamiltonian complex eigenvalues, E res = ε − (i/2)Г, are obtained. These complex energies are associated with decaying resonance states which were discussed in the previous two chapters. The bound states (if they exist) appear as real eigenvalues since they result from exactly such outgoing boundary conditions which appear under the threshold energy. When incoming boundary conditions are imposed two kind of solution are obtained. One type of solution is the complex conjugates of the decay resonance solutions mentioned above. In scattering theory text books (see Taylor for example) the physical resonance solutions are associated with the poles of the scattering matrix which are embedded in the lower half of the complex energy plane. However, in nuclear physics the complex poles embedded in the upper half of the complex energy plane, so-called virtual states , are denoted as capture resonances .
This book is dedicated to the non-Hermitian formalism of quantum mechanics. In this chapter we wish to give the motivation and the rational for developing a non-Hermitian formalism to quantum mechanics. Therefore this chapter will not explain how non-Hermitian calculations are carried out or in what way the non-Hermitian formalism is analogous to the standard (Hermitian) formalism of quantum mechanics. It is important to emphasize that there is no (known) transformation which enables one to map results which were obtained using one formalism to the other one. Yet, the same physical results should be obtained by studying the same phenomenon using the two formalisms. If this is the case, why should one bother to develop an alternative formalism to the standard Hermitian formalism of quantum mechanics? There are several reasons for doing this and here we shall focus on five of those reasons. (1) There are phenomena which can be explained in a straightforward fashion using the non-Hermitian formalism but are very hard and often impossible to explain within the framework of the standard (Hermitian) formalism of quantum mechanics . In particular in Chapter 9 we will describe several physical phenomena which are associated with the self-orthogonality where two or more degenerate resonance states are coalesced. (2) There are physical phenomena which one might not immediately associate with quantum behavior where the quantum language can be used to describe the physics . The studied problem may be, for example, in systems described in terms of classical statistical mechanics, diffusion in biological systems, or propagation of light in waveguides (WG). […]
We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schrödinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.
Although the non-Hermitian formalism of quantum mechanics which is developed in this book is not limited to specific examples and is applicable to problems which are not necessarily quantum mechanical (such as problems which require the solution of the Maxwell equation rather than of the Schrödinger equation) we dedicate an entire chapter to resonance phenomena in nature since they are related to a broad range of subjects and fields in physics, chemistry, molecular biology and technology. In this chapter we will introduce two different types of resonances, so called shape-type and Feshbach-type resonances, as they appear in different fields of science. The resonance phenomenon is associated with metastable states of a system that as time passes breaks into several subsystems. That is, even though the system has sufficient energy to break apart, this does not happen instantly but requires quite a long time with respect to the characteristic time scale of the system. In Table 2.1 we give several examples of resonance phenomena, where we specify the decaying systems, the resulting subsystems, and classification in terms of shape and Feshbach resonances. (These concepts will be explained more formally later.) Each of the listed systems has a typical time scale and in some of these cases the lifetime of the system is less than one nano-second while in other cases it takes more than several thousand years for the system to decay.
As discussed in the previous chapter, the poles of the S-matrix are identified with discrete eigenvalues of the time-independent Schrödinger equation, where the asymptotes of the corresponding eigenfunctions are either purely outgoing waves or purely incoming waves. More specifically, the bound and decay resonance poles are obtained by imposing the outgoing boundary conditions on the solutions of the time-independent Schrödinger equation, while the anti-bound and virtual states (sometimes referred to as capture resonances) are associated with the solutions obtained under the requirement of the incoming boundary conditions. Except for the bound states, all other poles of the S-matrix are associated with exponentially divergent wavefunctions which by definition do not belong to the Hilbert space of conventional Hermitian quantum mechanics. This fact represents a major difficulty for the development of a non-Hermitian quantum mechanical formalism. Consequently, one may wonder, for example, how to properly define an inner product in non-Hermitian quantum mechanics (NHQM) if the wavefunctions diverge asymptotically. We recall in this context that the concept of an inner product constitutes a fundamental building block of standard (Hermitian) quantum mechanics (QM), by means of which one defines the quantum mechanical expectation values of physically observable quantities over the quantum states under consideration. An inner product for NHQM is necessary in order to accommodate the tools of conventional QM. Furthermore, one might anticipate that an appropriate NHQM inner product would also facilitate practical numerical calculation of the S-matrix poles for those cases where the eigenvalues of the Hamiltonian do not possess an analytical closed form expression (unlike the cases studied in the previous chapter).