Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field

The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck’s constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution

Studies relating to ground anchorage systems

This thesis comprises 28 papers which illustrate the nature and direction of development work and associated research undertaken between 1965 and 1993 on soil and rock anchorage systems. The research was performed in order to obtain a basic understanding of the behaviour of newly developed anchorage systems in a variety of ground types and conditions, in order to improve anchorage designs, construction methods and testing procedures, and thereby encourage the safe and economic application of ground anchorages worldwide. Field development of anchorage construction methods in gravels, sand, clays, marls and chalk using cement grout injection techniques is described together with equations evolved to estimate the ultimate resistance to withdrawal for each ground type, based on systematic testing of full scale anchorages. A new design method for single and multi tied stiff retaining walls installed in any soil is detailed and validated by large scale tests and closely monitored case histories. The interactions between wall, anchorage and soil are illustrated, coupled with the refinement of overall stability analyses in cohesionless soils using wedge and log spiral based mechanisms of failure. For the rapid installation of anchorages in granular soils, vibratory driving is investigated in the laboratory and two distinct types of motion are found to exist. Theoretical equations of motion are developed to define the penetration processes and facilitate the design of vibrodrivers and vibrohammers. World practice in relation to the design, construction, testing and behaviour of rock anchorages is appraised, and field studies permit an improved understanding of uplift capacity by general shear failure, load transfer mechanisms, bond at rock/grout and grout/tendon interfaces, debonding, service performance and post-failure behaviour. Acceptance criteria related to service behaviour are created for load relaxation and creep displacement with time, which are independent of ground type and potentially of short duration. The extent and nature of steel tendon corrosion is described based on an international study of the corrosion performance of post-tensioned anchorages. Guidance is provided on class of protection, design principles and acceptable protective systems. For rock tunnelling by drill and blast methods of excavation, a fundamental understanding of rock bolt behaviour under static and dynamic loading is provided. Field, laboratory and finite element studies are combined to investigate the character of blast induced wave forms within a rock mass and the effect of these signatures on the rock bolt system. Attenuation relationships for peak particle velocity and peak dynamic bolt load are presented together with effect of bolt prestress, bolt length, and both single and two speed resin systems. Observations confirm that resin bonded rock bolts have a remarkable resilience to close proximity blasting, and the data provide a new understanding of stress transfer in tensioned bolts under static and dynamic conditions. A simple device to control rock bolt tensioning is developed and applied as a result of observed variations in prestress during production bolting. Ground anchorage technology is reviewed to highlight areas where further investigation and study would enhance understanding of anchorage behaviour and improve standards of practice.PUBLICATIONS INCLUDED IN THESIS: #2. HSU, T.C., LITTLEJOHN, G.S. & MARCHBANK, B.M. (1965) "Elongation in the Tension Test as a Measure of Ductility" Proc. Amer. Soc. Test. Mat., 65, 874-98. (Hsu wrote the paper and supervised the research of Littlejohn and Marchbank who provided the data.) #3. LITTLEJOHN, G.S. (1968) "Recent Developments in Ground Anchor Construction" Ground Engineering, 1(3), 32-36 & 46. #4. HANNA, T.H. & (1969) "Retaining Wall Tie-Backs" LITTLEJOHN, G.S. Consulting Engineer, May, 50-53, June, 49-52. (Hanna and Littlejohn contributed equally to this paper.) #6. LITTLEJOHN, G.S. #8. LITTLEJOHN, G.S., JACK, B.J. & SLIWINSKI, Z.J. (1971) "Anchored Diaphragm Walls - Some Design and Construction Considerations" Journal of the Institution of Highway Engineers, 18 (4), 15-29. (Littlejohn co-ordinated this paper and contributed half the data.) #13. LITTLEJOHN, G.S. & (1974) "A Case History Study of Multi-Tied Diaphragm MACFARLANE, I.M. Walls" Proc. Conf. on Diaphragm Walls and Anchorages, 113-121, Institution of Civil Engineers, London. (Littlejohn wrote the paper. Macfarlane provided technical data.) #14. LITTLEJOHN, G.S. & (1974) "Ground Anchors at Devonport Nuclear Complex" TRUMAN-DAVIES, C. Ground Engineering, 7 (6), 19-24. (Littlejohn wrote the paper. Truman-Davies supervised the field work.) #16. LITTLEJOHN, G.S. & (1975) "Rock Anchors - State-of-the-Art Part 1: Design" BRUCE, D.A. Ground Engineering, 8 (3), 25-32, 8 (4), 41-48. (Littlejohn wrote the paper and supervised Bruce's Ph.D. programme.) #18. LITTLEJOHN, G.S. & (1975) "Rock Anchors - State-of-the-Art Part 2; BRUCE, D.A. Construction" Ground Engineering, 8 (5), 34-45, 8 (6), 36-45. (Littlejohn wrote the paper and supervised Bruce's Ph.D. programme.) #19. LITTLEJOHN, G.S. & (1976) "Rock Anchors - State-of-the-Art Part 3: Stressing BRUCE, D.A. and Testing" Ground Engineering, 9 (2), 20-29, 9 (3), 55-60, 9 (4), 33-44. (Littlejohn wrote the paper and supervised Bruce's Ph.D. programme.) #20. LITTLEJOHN, G.S., (1977) "A Study of Rock Slope Reinforcement at Westfield NORTON, P.J. & Open Pit and the Effect of Blasting on Prestressed TURNER, M.J. Anchors" Proc. Conference on Rock Engineering, University of Newcastle-upon-Tyne, England. (Littlejohn wrote the paper. Norton and Turner provided technical data.) #22. LITTLEJOHN, G.S., (1977) "Anchor Field Tests in Carboniferous Strata" BRUCE, D.A. & Revue Francaise de Geotechnique No. 3, DEPPNER, W. January 1978, 82-86. (Littlejohn wrote the paper. Bruce and Deppner provided field data.) #28. LITTLEJOHN, G.S. & (1979) "Long Term Performance of High Capacity Rock BRUCE, D.A. Anchors at Devonport" Ground Engineering, 12 (7), 25-33. (Littlejohn and Bruce jointly wrote the paper. Bruce provided the field data under the supervision of Littlejohn.) #29. LITTLEJOHN, G.S. #30. LITTLEJOHN, G.S. #32. RODGER, A.A. & (1980) "A Study of Vibratory Driving in Granular Soils" LITTLEJOHN, G.S. Geotechnique, 30 (3), 269-93. (Rodger and Littlejohn jointly wrote the paper. Littlejohn supervised Rodger's Ph.D. programme) #34. LITTLEJOHN, G.S. (1981) "Acceptance Criteria for the Service Behaviour of Ground Anchorages" Ground Engineering, 14 (3), 26-29 & 36. #36. LITTLEJOHN, G.S. #50. LITTLEJOHN, G.S. (1987) "Ground Anchorages : Corrosion Performance" Proc. Instn. Civ. Engrs., Part 1, 82, 645-662. #51. LITTLEJOHN, G.S., (1987) "Monitoring the Influence of Blasting on the RODGER, A.A., Performance of Rock Bolts at Penmaenbach MOTHERSILLE, D.K.V. Tunnel" & HOLLAND, D.C. Proc. Int. Conf. on Foundations and Tunnels, University of London, 1,99-106. (Littlejohn and Rodger jointly wrote the paper and supervised Mothersille's and Holland's research work.) #52. LITTLEJOHN, G.S. (1988) "Sprayed Concrete for Underground Support" Proc. 3rd Int. Conf. on Underground Space and Earth Sheltered Buildings, Shanghai, China. #53. LITTLEJOHN, G.S. (1988) "Rock Anchorages for Underground Support" Proc. 3rd Int. Conf. on Underground Space and Earth Sheltered Buildings, Shanghai, China. "Thermal Behaviour of Grouted Supports for Pipelines" Department of Energy - Offshore Technology Report 'Grouts and Grouting for Construction and Repair of Offshore Structures', OTH 88289, 111-120. (Littlejohn and Hughes jointly wrote the paper. Hughes provided the laboratory data under Littlejohn's supervision.) #56. RODGER, A.A., (1989) "Instrumentation Used to Monitor the LITTLEJOHN, G.S., Influence of Blasting on the Performance of HOLLAND, D.C. & Rock Bolts at Penmaenbach Tunnel" MOTHERSILLE, D.K.V. Proc. Conf. on Instrumentation in Geotechnical Engineering, University of Nottingham, 267-279. (Rodger and Littlejohn jointly wrote the paper and supervised Mothersille's and Holland's research work.) #58. LITTLEJOHN, G.S., (1989) "Dynamic Response of Rock Bolt Systems" RODGER, A.A., Proc. Int. Conf. on Foundations and Tunnels, MOTHERSILLE, D.K.V. University of London (Sept.), 2, 57-64. & HOLLAND, D.C. (Littlejohn and Rodger wrote the paper and supervised Mothersille's and Holland's research work.) #61. LITTLEJOHN, G.S. #62. LITTLEJOHN, G.S. #64. LITTLEJOHN, G.S. #74. RODGER, A.A., (1993) "Dynamic Response of Rock Bolts at Pen y Clip LITTLEJOHN, G.S., Tunnel in North Wales" HOLLAND, D.C. & Proc. Int. Cong, on Options for Tunnelling, Int. XU, H. Tunnelling Assoc., Amsterdam. (Rodger and Littlejohn jointly wrote the paper and supervised Holland's and Xu's research.) #75. LITTLEJOHN, G.S. (1993) "A Simple Device to Control Rock Bolt Tensioning" & CONWAY, J. Tunnels & Tunnelling, (in press). (Littlejohn and Conway jointly wrote the paper. Littlejohn planned and supervised the experimental work.

Quantum Charged Spinning Particles in a Strong Magnetic Field (a Quantal Guiding Center Theory)

A quantal guiding center theory allowing to systematically study the separation of the different time scale behaviours of a quantum charged spinning particle moving in an external inhomogeneous magnetic filed is presented. A suitable set of operators adapting to the canonical structure of the problem and generalizing the kinematical momenta and guiding center operators of a particle coupled to a homogenous magnetic filed is constructed. The Pauli Hamiltonian rewrites in this way as a power series in the magnetic length $l_B= \sqrt{\hbar c/eB}$ making the problem amenable to a perturbative analysis. The first two terms of the series are explicitly constructed. The effective adiabatic dynamics turns to be in coupling with a gauge filed and a scalar potential. The mechanism producing such magnetic-induced geometric-magnetism is investigated in some detail.Comment: LaTeX (epsfig macros), 27 pages, 2 figures include

Diagonalization of multicomponent wave equations with a Born-Oppenheimer example

A general method to decouple multicomponent linear wave equations is presented. First, the Weyl calculus is used to transform operator relations into relations between c-number valued matrices. Then it is shown that the symbol representing the wave operator can be diagonalized systematically up to arbitrary order in an appropriate expansion parameter. After transforming the symbols back to operators, the original problem is reduced to solving a set of linear uncoupled scalar wave equations. The procedure is exemplified for a particle with a Born-Oppenheimer-type Hamiltonian valid through second order in h. The resulting effective scalar Hamiltonians are seen to contain an additional velocity-dependent potential. This contribution has not been reported in recent studies investigating the adiabatic motion of a neutral particle moving in an inhomogeneous magnetic field. Finally, the relation of the general method to standard quantum-mechanical perturbation theory is discussed

Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles

We investigate the Dirac equation in the semiclassical limit \hbar --> 0. A semiclassical propagator and a trace formula are derived and are shown to be determined by the classical orbits of a relativistic point particle. In addition, two phase factors enter, one of which can be calculated from the Thomas precession of a classical spin transported along the particle orbits. For the second factor we provide an interpretation in terms of dynamical and geometric phases.Comment: 8 pages, no figure

Product rule for gauge invariant Weyl symbols and its application to the semiclassical description of guiding center motion

We derive a product rule for gauge invariant Weyl symbols which provides a generalization of the well-known Moyal formula to the case of non-vanishing electromagnetic fields. Applying our result to the guiding center problem we expand the guiding center Hamiltonian into an asymptotic power series with respect to both Planck's constant $\hbar$ and an adiabaticity parameter already present in the classical theory. This expansion is used to determine the influence of quantum mechanical effects on guiding center motion.Comment: 24 pages, RevTeX, no figures; shortened version will be published in J.Phys.

Boundary Conditions on Internal Three-Body Wave Functions

For a three-body system, a quantum wave function $\Psi^\ell_m$ with definite $\ell$ and $m$ quantum numbers may be expressed in terms of an internal wave function $\chi^\ell_k$ which is a function of three internal coordinates. This article provides necessary and sufficient constraints on $\chi^\ell_k$ to ensure that the external wave function $\Psi^\ell_m$ is analytic. These constraints effectively amount to boundary conditions on $\chi^\ell_k$ and its derivatives at the boundary of the internal space. Such conditions find similarities in the (planar) two-body problem where the wave function (to lowest order) has the form $r^{|m|}$ at the origin. We expect the boundary conditions to prove useful for constructing singularity free three-body basis sets for the case of nonvanishing angular momentum.Comment: 41 pages, submitted to Phys. Rev.

Poincar\'e Husimi representation of eigenstates in quantum billiards

For the representation of eigenstates on a Poincar\'e section at the boundary of a billiard different variants have been proposed. We compare these Poincar\'e Husimi functions, discuss their properties and based on this select one particularly suited definition. For the mean behaviour of these Poincar\'e Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincar\'e Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincar\'e Husimi functions in the case of ergodic systems is shown.Comment: 17 pages, 5 figures. Figs. 1,2,5 are included in low resolution only. For a version with better resolution see http://www.physik.tu-dresden.de/~baecker

Semiclassical approximations for Hamiltonians with operator-valued symbols

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $\varepsilon\ll 1$ controls the separation of time scales and the limit $\varepsilon\to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\varepsilon\to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\varepsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an \epsi-dependent classical Hamilton function and an $\varepsilon$-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order $\varepsilon^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.Comment: Final version to appear in Commun. Math. Phys. Results have been strengthened with only minor changes to the proofs. A section on the Hofstadter model as an application of the general theory was added and the previous section on other applications was remove

Quantum dynamics and breakdown of classical realism in nonlinear oscillators

The dynamics of a quantum nonlinear oscillator is studied in terms of its quasi-flow, a dynamical mapping of the classical phase plane that represents the time-evolution of the quantum observables. Explicit expressions are derived for the deformation of the classical flow by the quantum nonlinearity in the semiclassical limit. The breakdown of the classical trajectories under the quantum nonlinear dynamics is quantified by the mismatch of the quasi-flow carried by different observables. It is shown that the failure of classical realism can give rise to a dynamical violation of Bell's inequalities.Comment: RevTeX 4 pages, no figure