Abstract In this chapter we discuss a number of methods for handling stationary state problems in quaternionic quantum mechanics and then consider more generally methods in which the phase of the wave function plays a primary role. We first describe exact methods for reducing the anti Hermitian quaternionic Hamiltonian H to a complex Hermitian Hamiltonian H for an equivalent complex quantum mechanics problem. We then develop stationary state perturbation theory, and as an application compute the leading approximation to the Hamiltonian phase operator IH. We also give a quaternionic analog of the Rayleigh-Ritz variational principle. We proceed next to an extended discussion of the geometric phase in quaternionic quantum mechanics, in both the adiabatic and the nonadiabatic cases. Finally, we give the quaternionic generalization of the eikonal or WKB approximation.
Abstract In this chapter we discuss a number of methods for handling stationary state problems in quaternionic quantum mechanics and then consider more generally methods in which the phase of the wave function plays a primary role. We first describe exact methods for reducing the anti Hermitian quaternionic Hamiltonian H to a complex Hermitian Hamiltonian H for an equivalent complex quantum mechanics problem. We then develop stationary state perturbation theory, and as an application compute the leading approximation to the Hamiltonian phase operator IH. We also give a quaternionic analog of the Rayleigh-Ritz variational principle. We proceed next to an extended discussion of the geometric phase in quaternionic quantum mechanics, in both the adiabatic and the nonadiabatic cases. Finally, we give the quaternionic generalization of the eikonal or WKB approximation.
Abstract In this chapter we develop methods, both exact and approximate, for timedependent problems (Adler, 1988), which complement the methods for stationary state problems given in Chapter 5 and their application to scattering theory in Chapter 6. We begin by giving the general equations for time-dependent perturbation theory in quaternionic quantum mechanics. By making an appropriate choice of initial conditions, these equations are applied successively to the problems of scattering theory and of decaying state theory. We next discuss the use of the interaction and Heisenberg pictures, particularly in the context of the quaternionic forced harmonic oscillator. We conclude by briefly discussing the use of quaternionic quantum mechanics as a model for time reversal violation in elementary particle physics (Adler, l 986a,b).
In this chapter, we present some fundamental issues about approximation methods that are often used when a quantum-mechanical system is perturbed and about the relationship between classical and quantum mechanics. In Sec. 10.1 we introduce the stationary perturbation theory, while Sec. 10.2 is devoted to time-dependent perturbations. In Sec. 10.3 we briefly examine the adiabatic theorem. In Sec. 10.4 we introduce the variation method, an approximation method that is not based on perturbation theory. In Sec. 10.5 we discuss the classical limit of the quantum-mechanical equations, whereas in Sec. 10.6 we deal with the semiclassical approximation, in particular the WKB method. On the basis of the previous approximation methods in Sec. 10.7 we present scattering theory. Finally, in Sec. 10.8 we treat a method that has a wide range of applications: the path-integral method. Stationary perturbation theory Perturbation theory is a rather general approximation method that may be applied when a small additional force (the perturbation) acts on a system (the unperturbed system), whose quantum dynamics is fully known. If the disturbance is small, it modifies both the energy levels and the stationary states. This allows us to make an expansion in power series of a perturbation parameter, which is assumed to be small. Perturbation theory may be applied both to the case where the additional force is time-independent (in which case a stationary treatment suffices – the subject of the present section) as well as to the case where it explicitly depends on time.
The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method , using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics. Stationary perturbation theory for a system with non-degenerate states This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.
Perturbation theory for non-degenerate and degenerate systems, variational methods, and WKB approximation are covered in this chapter. First, second, and kth order corrections to energy and wave function ket of non-degenerate states, energy eigenvalues, and wave functions kets of anharmonic oscillator, first-order perturbation theory for degenerate states and linear stark effect are written under perturbation theory. Second part of the chapter includes the variational methods for computation of the energy for ground state of helium atom, Rayleigh-Ritz variational method and its application to hydrogen molecule ion, calculation of energy of excited states, and its application to 1D harmonic oscillator. The topics which are covered under WKB approximation are the classical and non-classical regions of particle motion, alternative derivation of WKB formulae, connecting formulae, and quantum condition for bound state. The kth order corrections to energy and wave function kets for non-degenerate perturbation theory, treatment of WKB approximation from two different approaches and the detailed derivation of connecting formulae, application of variational method to compute energy for excited states are topics which are not generally found in existing text books on quantum mechanics. A good number of solved examples illustrating the importance of approximation methods in contemporary fields are added at the end of chapter.
By using the recent mathematical tools developed in quaternionic differential operator theory, we solve the Schrödinger equation in the presence of a quaternionic step potential. The analytic solution for the stationary states allows one to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart. A brief discussion on reflected and transmitted times, performed by using the stationary phase method, and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented. The analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem (up to now solved by numerical method).
Solving the Schrödinger equation in quaternionic quantum mechanics (QQM) has been a nut to crack. In complex quantum mechanics (CQM), approximation methods have been used to find the solutions of the Schrödinger equation when exact solutions are difficult to obtain. This study challenges the earlier Eikonal (or WKB) approximation formalism [S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995)] developed for an imaginary quaternion potential. In this paper, the correct form of coupled equations, obeying the noncommutative structure and approximated solution for quaternionic time-independent Schrödinger equation (QTISE), has been obtained.