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Quantal trajectories and geometric phase

Henrik Carlsen

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

This thesis concerns the following topics: geometric phase in the context of Galilean invariance and quantum measurements, Rydberg states of hydrogen atoms, vibronic coupling in the E A e Jahn-Teller system and realism in quantum computations. In the analyses the de Broglie-Bohm pilot-wave formulation of quantummechanics is mainly used.It is shown that geometric phase is not Galilean invariant. Experimental implications are discussed and it is found that the experiments performed to date are frame independent. An experiment which is in principle able to detect the noninvariance is sketched. By adopting the measurement theory of the pilot-wave formulation it is shown how the measurement induced geometric phase continuously emerges. The Samuel-Bhandari geometric phase is identified as the nonrandom part of the total geometric phase induced in the measurement.Ensembles of particles for a circular Rydberg wave packet are studied. The trajectories of pilot-wave particles are shown to accurately imitate the behaviour of the wave packet in the high quantum number limit. The nonclassical features of the wave packet are intuitively explained by the nonvanishing quantum potential.Vibronic coupling in the Longuet-Higgins model of the E A e Jahn-Teller system is investigated by means of quantal trajectories. The pilot-wave picture provides an intuitive tool for discussing time-scales. An argument based on ergodicity leads to an understanding of the averaging procedure over the electronic motion which provides the approximate nuclear motion.The existence of efficient quantum algorithms triggers questions on Natures ability of storing and processing information during quantum computations. The role of elements of reality in quantum computations is addressed using quantal trajectories. It is found that there is a many-to-one relationship between quantal trajectories and performed computations when quantum parallelism is utilized.

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The dynamics of circular Rydberg wave packets for the hydrogen atom with particular emphasis on fractional and full revivals previously studied by Gaeta and Stroud [Phys. Rev. A 42, 6308 (1990)], is developed within the de Broglie-Bohm pilot-wave theory. The classical limit at high quantum numbers as described within the correspondence principle is compared with the classical limit of pilot-wave theory. By studying ensembles of particles it is found that pilot-wave theory is able to deliver a consistent picture of the particle dynamics even in the high-quantum-number limit, where classical and quantum features coexist.

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This thesis concerns the following topics: geometric phase in the context of Galilean invariance and quantum measurements, Rydberg states of hydrogen atoms, vibronic coupling in the E A e Jahn-Teller system and realism in quantum computations. In the analyses the de Broglie-Bohm pilot-wave formulation of quantummechanics is mainly used.It is shown that geometric phase is not Galilean invariant. Experimental implications are discussed and it is found that the experiments performed to date are frame independent. An experiment which is in principle able to detect the noninvariance is sketched. By adopting the measurement theory of the pilot-wave formulation it is shown how the measurement induced geometric phase continuously emerges. The Samuel-Bhandari geometric phase is identified as the nonrandom part of the total geometric phase induced in the measurement.Ensembles of particles for a circular Rydberg wave packet are studied. The trajectories of pilot-wave particles are shown to accurately imitate the behaviour of the wave packet in the high quantum number limit. The nonclassical features of the wave packet are intuitively explained by the nonvanishing quantum potential.Vibronic coupling in the Longuet-Higgins model of the E A e Jahn-Teller system is investigated by means of quantal trajectories. The pilot-wave picture provides an intuitive tool for discussing time-scales. An argument based on ergodicity leads to an understanding of the averaging procedure over the electronic motion which provides the approximate nuclear motion.The existence of efficient quantum algorithms triggers questions on Natures ability of storing and processing information during quantum computations. The role of elements of reality in quantum computations is addressed using quantal trajectories. It is found that there is a many-to-one relationship between quantal trajectories and performed computations when quantum parallelism is utilized.
The dynamics of circular Rydberg wave packets for the hydrogen atom with particular emphasis on fractional and full revivals previously studied by Gaeta and Stroud [Phys. Rev. A 42, 6308 (1990)], is developed within the de Broglie-Bohm pilot-wave theory. The classical limit at high quantum numbers as described within the correspondence principle is compared with the classical limit of pilot-wave theory. By studying ensembles of particles it is found that pilot-wave theory is able to deliver a consistent picture of the particle dynamics even in the high-quantum-number limit, where classical and quantum features coexist.
In this thesis, properties of particle trajectories associated with the de Broglie-Bohm causal interpretation of quantum mechanics are studied. These trajectories are shown to exhibit deterministic chaos and adiabatic invariance under certain conditions. The very basic elements of the causal interpretation are presented in the first chapter. These include the equations of motion for the particle and the quantum potential. A brief discussion of the philosophically agreeable features of the theory is also included. In Chapter 2, properties of chaotic systems are studied. We define deterministic chaos for a flow and present methods for calculating the maximum Lyapunov exponent. The properties of the different types of systems and the conditions that lead to chaos in these systems are analyzed. We study in detail the specific example of the two-dimensional harmonic oscillator in Chapter 3. We find that different types of trajectories include those which are periodic and chaotic. The necessary conditions for obtaining chaos are determined for a superposition of stationary states. Systems which are qualitatively similar to the harmonic oscillator are covered in Chapter 4. These include the two-dimensional infinite well, an infinite well bisected by a finite barrier, and a Rydberg atom in an external electromagnetic field. In Chapter 5, the effect of a spin 1/2 wavefunction is considered. The causal equations of motion for a spin 1/2 particle are introduced. We find that chaotic trajectories are easily obtained. The causal analogue of the geometric phase is defined in Chapter 6. This phase is shown to be an adiabatic invariant for periodic trajectories. We define the geometric frequency for both periodic and aperiodic trajectories. Finally, in Chapter 7 we examine trajectories associated with stationary states. We define necessary conditions for chaos to arise in the trajectories. The properties of entangled boson and fermion systems are analyzed.
Kinematical and dynamical aspects of the noncyclic geometric phase for coherent states are analyzed. It is shown for Glauber and SU(2) coherent states that the geometric phase obeys a geodesic closure rule associated with the coset space of the respective Lie group. In the former case the geodesic is a straight line in phase space, while in the latter the geodesic is a great circle on the Bloch sphere (S2).({\mathcal{S}}^{2}).(S2). An alternative expression for the geometric phase is derived from the time-dependent variational principle by making a specific choice of gauge. The noncyclic geometric phase is numerically calculated in a time-dependent variational treatment of the $E\ensuremath{\bigotimes}\ensuremath{\varepsilon}$ Jahn-Teller model first introduced by Longuet-Higgins and co-workers. The electronic and nuclear degrees of freedom are parametrized by SU(2) and Glauber coherent states respectively. In particular, the adiabatic limit is studied and shown to yield the anticipated Berry geometric phase.
The de Broglie-Bohm “pilot-wave” theory replaces the paradoxical wave-particle duality of ordinary quantum theory with a more mundane and literal kind of duality: each individual photon or electron comprises a quantum wave (evolving in accordance with the usual quantum mechanical wave equation) and a particle that, under the influence of the wave, traces out a definite trajectory. The definite particle trajectory allows the theory to account for the results of experiments without the usual recourse to additional dynamical axioms about measurements. Instead, one need simply assume that particle detectors click when particles arrive at them. This alternative understanding of quantum phenomena is illustrated here for two elementary textbook examples of one-dimensional scattering and tunneling. We introduce a novel approach to reconcile standard textbook calculations (made using unphysical plane-wave states) with the need to treat such phenomena in terms of normalizable wave packets. This approach allows for a simple but illuminating analysis of the pilot-wave theory's particle trajectories and an explicit demonstration of the equivalence of the pilot-wave theory predictions with those of ordinary quantum theory.
An approach to the measurement induced geometric phase based on the de Broglie--Bohm hidden-variable model is developed. The analysis involves an evolving geometric phase connecting the initial and final states of an individual experimental run. As an illustration the geometric phase produced by a cyclic sequence of spin-12\frac{1}{2}21​ filtering experiments is considered. It is argued, in this case, that the evolving geometric phase can be made close to the Samuel-Bhandari phase.
This paper presents a geometric and wave interpretation of energy quantization in the hydrogen atom, based on the de Broglie closure condition of the electron wave in a circular orbit. In this concept, energy quantization is a secondary phenomenon resulting from the fact that the electron wave in each orbit consists of exactly n full periods, and the transition to level n+1 corresponds to the addition of one full period. Combining the wave condition with the classical equilibrium of the Coulomb and centripetal forces leads to values of the orbital radii and a discrete energy spectrum consistent with solutions of the Schrödinger equation for the hydrogen atom. It is shown that energy quantization can be interpreted as a consequence of the resonant nature of the electron's wave nature and the condition of phase uniqueness after a complete orbit around the nucleus.1 The model under consideration is semiclassical in nature and serves as an intuitive representation of known results from quantum mechanics. It is assumed that the allowable states correspond to configurations in which the de Broglie wave forms a standing wave containing an integer number of full periods around the orbital circumference. This condition leads directly to the quantization of angular momentum according to the Bohr model. The waveband model provides a one-dimensional analogy of the full quantum description and can serve as a teaching tool to facilitate understanding the geometric aspects of energy quantization in the hydrogen atom. It demonstrates that energy quantization is a natural consequence of the standing wave geometry and the addition of successive full periods along the orbit as the system transitions to the next energy eigenstate after activation. The electron is a stable eigenstate of a quantum field whose behavior in bound systems can be geometrically interpreted as the self-resonance of a de Broglie wave satisfying the condition of single-valent phase, rather than as a local particle with a classical trajectory. Self-resonance of a wave means that the condition of single-valent phase of the wave function after a complete orbit around the nucleus is satisfied, i.e., the requirement that the phase change be an integer multiple of 2π; this is equivalent to the Bohr–de Broglie condition for the closure of the de Broglie wave.