This chapter deals with the time-independent Schrödinger equation. The introduction of complex ‘Hamiltonians’ is essential for treating resonances that are inherently different from genuine (physical) bound states. True square-integrable bound-state eigenvectors are associated with discrete negative eigenenergies and Hermitean Hamiltonians. Resonant states are also square-integrable, but they are linked to complex energies with positive real parts that belong to a continuum. The concept of the absorbing boundary condition comes to rescue the situation via the introduction of complex ‘Hamiltonians’ with normalizable scattering wavefunctions as proper physical states. A method which can provide an adequate spectral representation of the total Green operator or resolvent G^(ω) would be one of the key inputs into an invaluable practical quantum-mechanical theory for scattering (E = 0) and spectroscopy (E < 0).
This chapter deals with the time-independent Schrödinger equation. The introduction of complex ‘Hamiltonians’ is essential for treating resonances that are inherently different from genuine (physical) bound states. True square-integrable bound-state eigenvectors are associated with discrete negative eigenenergies and Hermitean Hamiltonians. Resonant states are also square-integrable, but they are linked to complex energies with positive real parts that belong to a continuum. The concept of the absorbing boundary condition comes to rescue the situation via the introduction of complex ‘Hamiltonians’ with normalizable scattering wavefunctions as proper physical states. A method which can provide an adequate spectral representation of the total Green operator or resolvent G^(ω) would be one of the key inputs into an invaluable practical quantum-mechanical theory for scattering (E = 0) and spectroscopy (E < 0).
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 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 .
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.
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.
Many physical systems cannot be solved exactly. In this case, we look for approximate methods to gain insight into their properties. Perturbation theory is one such method. When the Hamiltonian of the system doesn't change over time, one approximate method is the Time Independent Perturbation Theory, which is the subject of Chapter 10 . The infinite square well, the harmonic oscillator, and the hydrogen atom are among the few physical systems for which the Schrodinger equation can be solved exactly. For this reason, they are described in detail in most of the introductory quantum mechanics textbooks. However, most physical systems are not exactly solvable, i.e., they are described by the Schrodinger equation, whose solutions cannot be expressed in terms of commonly used functions. In order to study such systems, we try to decompose the corresponding Hamiltonian into a solvable Hamiltonian plus a small perturbation. For such cases, we say that the Hamiltonian is “close enough” to an exactly solvable Hamiltonian to warrant using perturbative methods to approximate its eigenenergies and eigenfunctions.
In many situations in physics, the Hamiltonian of a given system is so complicated that the solution to the stationary Schrodinger equation is practically impossible or very difficult. Therefore, some approximation methods are required. This chapter presents the time-independent perturbation theory, the procedure of finding corrections to non-degenerate eigenvalues and eigenvectors to a part of the Hamiltonian of a given system. The perturbation theory is appropriate when the Hamiltonian can be split into two parts. The perturbation theory can be applied to analyze the quantum properties of particles trapped in two closely coupled potential wells. By closely coupled wells, it is meant that these two wells are separated by a barrier. This is a typical situation in the studies of quantum dynamics of Bose–Einstein condensates. Using the first-order perturbation theory, one can find the eigenvalues and eigenvectors.