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.
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 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). […]
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 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 .
A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, and acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena and subjects in quantum regimes, such as quantum resonances, superradiance, the continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, first given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.
For the understanding of irreversibility at the quantum level, the formation and decay of transient (unstable) states play a fundamental role. If the system is treated within Hermitian quantum mechanics, the resulting energy distribution of the resonance state, whose Fourier transform yields the time-dependent probability of decay, P(t), is real. The physical constraint of the lower bound in the energy spectrum introduces ``memory,'' and causes nonexponential decay (NED) to set in after $t\ensuremath{\gg}\ensuremath{\tau},$ where \ensuremath{\tau} is the lifetime defined by exponential decay. The closer to threshold the decaying state is, the earlier NED appears. Apart from the constraint of E>0, the constraint of t>0 must be accounted for at the same time. It results from the discontinuity at t=0 of the solution of the time-dependent Schr\"odinger equation, which breaks the unitarity associated with the S matrix and gives rise to a complex energy distribution, as a manifestation of the non-Hermitian property of the decaying states. For a narrow isolated resonance state, for which the self-energy is essentially energy-independent, analytic results for PNED(t) obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints E>0 and t>0 are taken into account. As an example of the difference in the magnitude of the PNED(t) when using a real and a complex energy distribution, application is made to the decay of the unstable ${\mathrm{He}}_{2}^{2+}{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ ground molecular state.
It is critical to understand the nature of the propagation medium in terms of the available resonances which couple to the electromagnetic field. The detailed specification of the complex index of refraction as a function of frequency is the subject of spectroscopy. Since propagation media cover all phases of matter, the spectroscopy of gases, solids, and liquids is introduced in this chapter. Topics are not exhaustively covered, but rather with sufficient depth to support the practical applications that will come later. The chapter begins with a formal review of topics covered in stationary-state quantum mechanics necessary for the development of spectroscopy. The next section develops models to calculate spectral line positions, which are necessary to characterize a spectral line. The full development of spectral line parameters is not only the goal of this chapter but the next two as well. The remaining sections address the essential elements of the spectroscopy of gases, solids and liquids. Both classical and quantum models are used. Time-independent quantum mechanics allows the description of the quantized energy level structure of matter. This is the central topic of this chapter. Time-dependent quantum mechanics is necessary for the development of a theory covering transitions between stationary-state energy levels and is the topic of Chapter 5. The classical concept of light is that of an oscillating continuous wave field, as presented in the first two chapters. This is consistent with the classical electrodynamics based on Maxwell’s equations. However, the work by Planck on blackbody radiation and Einstein on the photoelectric effect showed that optical fields also have a quantized or particle-like nature. Planck, in 1900, suggested that the energy of light, E, be quantized according to . . . E = h f [J] (3.1) . . . where f is frequency (sec−1) and h is Planck’s constant (h = 6.6260755(40) × 10−34 J-sec). This allowed a theoretical description of blackbody radiation for the first time. However, it was Einstein, who, a few years later, made Eq. 3.1 more credible by applying it to explain the photoelectric effect.