Methods for Time Development

Stephen L. Adler

Published Apr 27, 1995• 0 citations

Abstract

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).

DOI: 10.1093/oso/9780195066432.003.0007

Related papers

Stationary State Methods and Phase Methods

Stephen L. Adler

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.

Apr 27, 19950 citations

Scattering Theory and Bound States

Stephen L. Adler

Abstract We proceed in this chapter to discuss scattering theory and related aspects of the bound-state problem in quaternionic quantum mechanics. We begin with a detailed discussion of a simple one-dimensional example, the delta function potential model. From this model will emerge the result (Adler, 1988) that will be the principal theme of this and the succeeding two chapters: with the standard ray representative choice of Sec. 4.2, for short-range potentials the asymptotic scattering states and the S-matrix in quaternionic quantum mechanics are always complex ℂ(1, i).

Apr 27, 19950 citations

Perturbations and approximation methods

Gennaro Auletta, Mauro Fortunato, Giorgio Parisi A

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.

Cambridge University Press eBooksApr 16, 20090 citations

Approximation Methods in Quantum Mechanics

Kevin F. Brennan

In the previous chapters, we have solved a series of problems that all have an exact solution. However, there are only a select few problems in quantum mechanics that provide exact analytical solutions. Those problems that can be solved exactly are the free particle, the one-dimensional barrier potential, the finite and the infinite square wells, the infinite triangular well, the harmonic oscillator, and the hydrogen atom, including both its nonrelativistic and relativistic solutions. However, there are many problems different from the above-mentioned few as well as problems that cannot be approximated by these solutions. What happens then if we want to solve a different problem, such as the helium atom or the hydrogen atom in an applied electric field? To solve these problems it is necessary to use approximation techniques. In this chapter, we consider some of these techniques, the most important being the time-independent and the time-dependent perturbation theories. We begin with a discussion of time-independent disturbances in a nondegenerate system. Time-independent Perturbation Theory: The Nondegenerate Case We seek to determine the solution to a problem in which a small disturbance is applied to an otherwise determinable system. Perturbation theory is used most often in cases for which the solution to the undisturbed (unperturbed) system is known and the perturbation is small. For example, perturbation theory is useful in solving for the energy levels in a hydrogen atom in the presence of a small external electric field.

Cambridge University Press eBooksFeb 13, 19998 citations

Quantum dynamics using the time-dependent Schrödinger equation

Abraham Nitzan

Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.

Jun 14, 20241 citation

Time-Dependent Systems

Walter Wilcox

Synopsis: The time development of quantum systems with time-dependent Hamiltonians or potentials will be discussed in this chapter. First, we set up the formalism of time development in a very general way and discover that there are two basic Hilbert space descriptions of these processes. As discussed earlier, exact solutions for physical systems are rare, and we therefore find approximate ways of describing fast and slow time developments. Concentrating on two-state systems, we investigate the important role complex phases play in the slow evolution of quantum systems. In this context, we learn how to apply perturbation theory to our formalism, resulting in wonderfully practical formulas that can readily be applied in the lab, including elastic scattering and spontaneous emission rates. Finally, the important topic of time decay of systems and associated uncertainties in the energies of such states will be discussed.

CRC Press eBooksAug 23, 20190 citations