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.
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.
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).
The solutions of the time-independent Schr\"odinger equation for non-Hermitian (NH) Hamiltonians have been extensively studied and calculated in many different fields of physics by using L2 methods that originally have been developed for the calculations of bound states. The existing non-Hermitian formalism breaks down when dealing with wave packets (WPs). An open question is how time-dependent expectation values can be calculated when the Hamiltonian is NH? Using the F-product formalism that was recently proposed by Moiseyev and Lein [J. Phys. Chem. 107, 7181 (2003)] we calculate the time-dependent expectation values of different observable quantities for a simple well-known study test case model Hamiltonian. We carry out a comparison between these results and those obtained from conventional (i.e., Hermitian) quantum mechanics (QM) calculations. The remarkable agreement between these results emphasizes the fact that in NH QM, unlike standard QM, there is no need to split the entire space into two regions, i.e., the interaction region and its surrounding. Our results open a door for a type of WP propagation calculations within the NH QM formalism that until now were impossible. In particular our work is relevant to the many different fields in physics and chemistry where complex absorbing potentials are introduced in order to reduce the propagation calculations to a restricted region in space where the artificial reflections from the edge of the numerical grid or box are avoided.
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). […]
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.
Linear algebra is the natural mathematical language of quantum mechanics. For this reason, the present chapter starts with a review of Hilbert spaces for vectors and operators. We introduce vector and Hilbert spaces along with inner products and metrics. The Dirac notation is developed for the Euclidean vector spaces as a starting point for the concepts of complete orthonormal sets of vectors, closure, dual vector spaces, and adjoint operators. The Dirac delta function in various forms and the principal part is introduced in Appendix 5 as an essential tool. The chapter turns to the main use of Dirac notation for function spaces; the concepts of norm, inner product, and closure are discussed. Fourier, Cosine, and Sine series are discussed as examples of expansions in complete orthonormal sets of functions. Although Hilbert spaces are interesting mathematical objects with important physical applications, the study of linear algebra remains incomplete without a study of linear operators (i.e., linear transformations). In fact, the set of linear transformations itself forms a vector space and therefore has a basis set. The basis set for the operator is linked with the basis sets for the spaces that it operates between. The linear operator can be discussed as an abstract operator or through an isomorphism as a matrix or as a generalized expansion in operator space. A Hermitian (a.k.a., self-adjoint) operator produces a basis set within a Hilbert space. The basis set comes from the eigenvector equation for the particular operator. The fact that a Hermitian operator produces a complete set (of orthornomal vectors) has special importance for quantum mechanics. Observables such as energy or momentum correspond to Hermitian operators. Complete sets make it possible to represent every possible result of a measurement of the observable by an object (vector) in the theory. The Hermitian operators have real eigenvalues which represent the results of the measurement.