In the standard (Hermitian) formalism of quantum mechanics, usually when a potential parameter is varied the crossing of energy levels with the same symmetry is avoided. However, within the framework of the non-Hermitian formalism it is possible that two (or even more) complex eigenvalues with the same symmetry will cross. At the crossing point the eigenvalues are degenerate and this is accompanied by the coalescence of the eigenfunctions (or eigenvectors). Therefore, we may term this special situation as a non-Hermitian degeneracy. This special situation is associated with a branch point in the complex energy plane which is commonly termed an “exceptional point” in the spectrum of the non-Hermitian Hamiltonian. With respect to the c-product defined for non-Hermitian operators (matrices) in Chapter 6, the degenerate eigenstate is self-orthogonal. Since a branch point in the spectrum is removed by any infinitesimally small external perturbation, it seems to be inaccessible experimentally and may be considered just as a mathematical object rather than a physical one. However, as we will show here, by varying the potential parameters the existence of a branch point is reflected in the measurement of the geometrical phases also known as Berry phases. It should be stressed here that while in our case the geometrical phase results from a coalescence of eigenfunctions of a non-Hermitian Hamiltonian, the so-called Berry phase phenomenon occurs also within the Hermitian formalism of quantum mechanics when the eigenvalues of the molecular Hamiltonian in the Born–Oppenheimer approximation are degenerate for specific geometry of the poly-atomic molecule.

We begin by considering the following scattering experiment. A projectile, e.g., an atom A in a given electronic state, collides with a target which we will take as a diatomic molecule BC in its ground electronic, vibrational and rotational state. For a short period of time an activated complex [ABC]# is generated. As time passes the activated complex can break into different products. For instance, in our example these products will be A + BC, B + AC, C + AB and A + B + C. Each one of the possible products can be in different electronic, vibrational and rotational quantum states. The total energy which is originally the sum of the electronic and translational energies of the projectile A and the electronic, vibrational, rotational and translational energies of target BC is conserved during the scattering process.

Time-independent scattering theory enables one to calculate the probability of obtaining the specific products in given quantum states and the kinetic energy distribution of the products as a function of the total energy of the system without the need to solve the time-dependent Schrödinger equation. The time-independent formulation of scattering theory is based on the ability to propagate analytically an initial given wavepacket, Φ(0), to infinite times. That is, we need to get a closed form expression for limt→±∞ e−iĤt/ħ|Φ(t = 0)〉. To quote from the introduction of the excellent book on scattering theory written by Taylor: “The most important experimental technique in quantum physics is the scattering experiment. That this is so is clear from even the briefest review of modern physics”.

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).

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.

This is the first book ever written that presents non-Hermitian quantum mechanics (NHQM) as an alternative to the standard (Hermitian) formalism of quantum mechanics. Previous knowledge of the basic principles of quantum mechanics and its standard formalism is required.

The standard formalism is based on the requirement that all observable properties of a dynamic nature are associated with the real eigenvalues of a special class of operators, called Hermitian operators. All textbooks use Hermitian Hamiltonians in order to ensure conservation of the number of particles. See, for example, the monumental book of Dirac on The Principles of Quantum Mechanics.

The motivation for the derivation of the NHQM formalism is twofold.

The first is to be able to address questions that can be answered only within this formalism. For example:

• –in optics, where complex index of refraction are used;

• – in quantum field theory, where the parity–time (PT) symmetry properties of the Hamiltonian are investigated;

• – in cases where the language of quantum mechanics is used, even though the problems being addressed are within classical statistical mechanics or diffusion in biological systems;

• – in cases where complex potentials are introduced far away from the interaction region of the particles. This approach simplifies the numerical calculations and avoids artificial interference effects caused by reflection of the propagated wave packets from the edge of the grid.

The second is the desire to tackle problems that can, in principle, also be solved within the conventional Hermitian framework, but only with extreme difficulty, whereas the NHQM formalism enables a much simpler and more elegant solution.

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.

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 L2 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 L2.

By imposing outgoing boundary conditions on the eigenfunctions of the timeindependent Hamiltonian complex eigenvalues, Eres = 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. (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.

2. 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.

3. (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.

4. 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). […]

This chapter is divided into several sections which together represent one of the fundamental concepts in the non-Hermitian formalism of quantum mechanics (NH QM). First we discuss the need to replace the inner product used in the standard (Hermitian) formalism of quantum mechanics by another construct which was termed the c-product by Moiseyev, Certain and Weinhold in 1978. Unlike the standard situation where the eigenfunctions (eigenvectors) of an Hermitian operator (matrix) form a complete set which can be used to expand a wavepacket which describes the system at a given time, in NH QM it might happen that the eigenfunctions make up an incomplete set since several (usually two) eigenfunctions (eigenvectors) coalesce to generate a self-orthogonal state. We need completeness and closure relations in order to develop, for example, perturbation theory and scattering theories for non-Hermitian Hamiltonians and in order to be able to solve the Schrödinger equation by numerical methods. Therefore, the second section of this chapter is devoted to the completeness of the spectrum in NH QM. Other aspects of the non-Hermitian formalism which stem from this issue deal with the advantages of using a non-Hermitian formalism for a time-dependent description of a decaying system as well as its application to time-periodic systems. Accordingly, the discussion in one of the sections will encompass the propagation of wavepackets in non-Hermitian quantum mechanics whereas another will elaborate on the benefits of the formalism for the description of the interaction of matter with intense laser radiation.

Introduction

Irving John Good once titled a paper, “46,656 Varieties of Bayesians.” With little stretch of the imagination, one can believe that a similar paper on the interpretations of quantum mechanics exists somewhere out there too. And that may be true even though 46,656 is greater than the whole membership of the American Physical Society! It is no wonder that taking a stand on quantum foundations is treacherous business. Nevertheless, in the March 2000 issue of Physics Today, Asher Peres and I flouted the usual rewards of a calm life and did just that. Our article was titled, “Quantum Theory Needs No ‘Interpretation’.”

This chapter is about the evolution of that article and a later “reply to critics.” When I was an undergraduate at the University of Texas, Bryce DeWitt made a great impression on me by saying something I have not forgotten: “We learn mathematics so that we don't have to think when we do physics.” My paper with Asher contained no mathematics. We had to think, and think very hard. The fruits of that labor, for my part, were in a greatly enhanced keenness for seeing through various foundational issues. How I wanted to share that newfound keenness with the reader! But it was difficult, if not impossible, to convey such an aspect of the paper-writing process in the very paper itself.