DOCSLIB.ORG

Complete Closed-Form Solutions of the Stark Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Complete Closed-Form Solutions of the Stark Problem Celest Mech Dyn Astr (2011) 109:333–366 DOI 10.1007/s10569-010-9331-1 ORIGINAL ARTICLE Complete closed-form solutions of the Stark problem Gregory Lantoine · Ryan P. Russell Received: 16 April 2010 / Revised: 25 August 2010 / Accepted: 16 December 2010 / Published online: 5 February 2011 © Springer Science+Business Media B.V. 2011 Abstract Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direc- tion. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions. Keywords Stark problem · Kepler problem · Perturbation · Integrability · Analytical solution · Low-thrust 1 Introduction 1.1 Motivations The purpose of this paper is to describe the so-called Stark problem, the perturbed two-body problem where the perturbation is a constant force in magnitude and direction. This problem is interesting because it is one of the few exactly solvable problems (i.e. systems of governing ordinary differential equations) in Celestial Mechanics. G. Lantoine (B) · R. P. Russell School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] R. P. Russell e-mail: [email protected] 123 334 G. Lantoine, R. P. Russell This problem is named after the German physicist Johannes Stark who discovered in 1913 the Stark effect (Stark 1914), i.e. the shifting and splitting of spectral lines of atoms and molecules in the presence of an external static electric field. Understanding the Stark effect is important for the analysis of atomic and molecular rotational spectra (Chattopadhyay and Boxer 1995; Pierce and Boxer 1995). Thus, apart from the intrinsic interest of its own, the Stark problem can play the role of a model problem, from which one can obtain information about the properties of atoms when strong electric fields are present. Exploiting this effect can also greatly enhance the utilities of some molecules (Ishigami et al. 2005). By analogy between atoms and planets, systems described by the Stark problem are also common in orbital mechanics. We can mention the case of low-thrust trajectories where a spacecraft engine can generate a continuous acceleration of constant magnitude and direc- tion. The Stark problem can therefore model a low-thrust trajectory discretized by segments of constant thrust (Lantoine and Russell 2009). Another important potential application of this problem is the study of the influence of the solar pressure on the orbit of a satellite, since the corresponding perturbation force can be approximated as constant over a short time inter- val. This is a particularly valid approximation when the satellite does not enter often into the shadow of the Earth, for instance in the cases of sun-synchronous orbits above the terminator. However, in the present work we stop short of these considerations; our aim is confined to the derivation of explicit representations for the solutions of the Stark problem. Next, we give a thorough review of previous work regarding the Stark model over the past three centuries. 1.2 Historical survey The main integrable problems of Celestial Mechanics are easily counted, namely, the Kepler problem, the Euler problem (two center Newtonian gravitational motion), and the Stark problem. This small number of integrable problems along with the interest from the physics community explains why the Stark problem has received special attention over almost two and one-half centuries, with occasionally periods of intense studies. The Stark problem was shown to be integrable first by Lagrange who reduced it to quadr- atures at the end of the eighteenth century (Lagrange 1788). Although elliptic functions were not known at his time, Lagrange’s analysis and reduction is very elegant, and he demon- strates the intuition that the solution can be expressed with some transcendental functions (“rectification of conic sections”). Lagrange also points out that the Stark problem differs significantly from the Euler problem and that a dedicated analysis is necessary. In the middle of the nineteenth century, two mathematicians, Jacobi and Liouville, give crucial contributions towards a more a rigorous treatment of the Stark problem. Their work represents the mathematical foundation that all later studies build upon, including this paper. Following the work of Hamilton on his ‘General Method in Dynamics’ (Hamilton 1835), Jacobi derives a general procedure for the study of dynamical systems through the Hamilton-Jacobi equation. Jacobi also found that the Stark system admits the separation of its variables in the parabolic coordinates and formulates the Hamilton-Jacobi equation for the problem in these coordinates. Complementing the ideas of Jacobi, Liouville derives sufficient conditions for separability of dynamical systems (at the origin of the notion of Liouville integrability), and noted that most of the known integrable problems, including the Stark problem, met his conditions for separability (Liouville 1849). At the beginning of the twentieth century, the Stark problem received other attention due to the first observations of the Stark effect. Within a decade of the appearance of Bohr’s quantum theory, this effect was first explored to explain some characteristics of the 123 Complete closed-form solutions of the Stark problem 335 hydrogen atom in a state excited by a homogeneous electric field (Banks and Leopold 1978a,b; Murray-Krezan 2008; Hezel et al. 1992; Byrd and Mitchell 2004). Most authors relied on the same parabolic coordinates and separation of variables to characterize the motion of the electron in a hydrogen atom, whether from a classical or quantum mechanics perspective. On one hand, Born stays in a classical mechanics context to show that previous theories of the Stark problem are particularly powerful to find transition frequencies of excited systems, but he only finds approximate solutions using Fourier series (Born 1960). Slightly later Epstein treated the same expressions by successive expansions and obtained results up to second order in the electric field strength (Epstein 1916). On the other hand, Froman provides a comprehensive review of all major studies of the Stark effect from a quantum mechanics point of view (Froman 2008). In connection with the development of wave mechanics, it is shown that the Schrodinger equation for a one-electron system in a homogeneous electric field is also separable in parabolic coordinates. This analogy between the treatment of the Stark problem in quantum and classical mechanics is remarkable. More recently, the Stark problem is extended by considering a charged particle moving in the field of one localized dyon inside a homogeneous electric field (referred to as the MICZ-Kepler-Stark system) (Nersessian and Ohanyan 2008). As in the nominal case, this also is an integrable system, which allows separation of variables in parabolic coordinates. Furthermore, the advent of the space age in the 1950s following the launch of Sputnik led to an increase of interest in the Stark problem. There were investigations in two areas: a theo- retical examination of the solutions of the Stark problem; and an investigation of the potential use of solutions of the Stark problem as a basis of approximation for specific solutions in orbital dynamics. In the former, we mention especially the work of Isayev who derived an analytical theory of motion for a balloon-satellite perturbed by a constant solar pressure (Isayev and Kunitsyn 1972); and in the latter we note the interesting work of Beletskii about the planar Stark problem (Beletsky 2001). Following his discussion of the accessible/non- accessible regions, the solution forms are presented in terms of Jacobian elliptic functions based on the analysis of the integrals of motion. The author also characterizes (incompletely) the different types of orbits that can be encountered in the two-dimensional Stark problem and provides representative planar trajectories. However, one negative feature of the form of the Beletskii’s solutions is that some of the parameters in the solutions tend to infinity in the Kepler limit (as the perturbation approaches zero). At about the same time, Vinti investigated the effect of a constant force on a Keplerian orbit with the introduction of Delaunay variables (a common set of variables for perturbed Kepler problems) (Vinti 1964). Note that his result is approximate as he eliminates short periodic terms to focus on secular terms only. Another important milestone in the history of the Stark problem is the work of Kirchgraber in the seventies. He is concerned (like us) with handling perturbations of the Kepler prob- lem and contrary to previous authors who mainly use parabolic coordinates, he uses the so-called KS-variables to show that the Stark problem is separable and to describe it ana- lytically (Kirchgraber 1971). However, the required change of variables is a complicated nonlinear transformation which is likely to be computationally inefficient and presents less physical insight than previous transformations. Also, the final closed-form expressions of the solutions are unfortunately not given. Having arrived at the quadrature, he states that this last integral can be solved by invoking the Jacobi elliptical functions. But there is no indication of how this last step is accomplished. Using Kirchgraber’s method, Rufer applies the Stark formulation to low-thrust trajectory optimization (Rufer 1976).
Load more

Recommended publications
  • Electric Dipoles in Atoms and Molecules and the Stark Effect Publicação IF – 1663/2011
    Electric Dipoles in Atoms and Molecules and the Stark Effect Instituto de Física, Universidade de São Paulo, CP 66.318 05315-970, São Paulo, SP, Brasil Publicação IF – 1663/2011 21/06/2011 Electric Dipoles of Atoms and Molecules and the Stark Effect M. Cattani Instituto de Fisica, Universidade de S. Paulo, C.P. 66318, CEP 05315−970 S. Paulo, S.P. Brazil . E−mail: [email protected] Abstract. This article was written for undergraduate and postgraduate students of physics. We analyze the electric dipole moments (EDM) of atoms and molecules when they are isolated and when are placed in a uniform static electric field. This was done this because usually in text books and articles this separation is not clearly displayed. Key words: Electric dipole moments of atoms and molecules; Stark effect . (I) Introduction . Our goal is to write an article for undergraduate and postgraduate students of physics to study the electric dipole moments (EDM) of atoms and molecules when they are isolated and when they are placed in a uniform static electric field . We have done this because many times in text books and articles this separation is not clearly displayed. The dipole of an isolated system will be named natural or permanent EDM and that one which is generated by an external field will be named induced EDM. So, we begin recalling the definition of EDM adopted in basic physics courses 1−4 for an isolated aggregate of charges. If in a given system positive charges + q and negative −q are concentrated at different points we say that it has an EDM.
    [Show full text]
  • 22.51 Course Notes, Chapter 11: Perturbation Theory
    11. Perturbation Theory 11.1 Time-independent perturbation theory 11.1.1 Non-degenerate case 11.1.2 Degenerate case 11.1.3 The Stark effect 11.2 Time-dependent perturbation theory 11.2.1 Review of interaction picture 11.2.2 Dyson series 11.2.3 Fermi’s Golden Rule 11.1 Time-independent perturbation theory Because of the complexity of many physical problems, very few can be solved exactly (unless they involve only small Hilbert spaces). In particular, to analyze the interaction of radiation with matter we will need to develop approximation methods36 . 11.1.1 Non-degenerate case We have an Hamiltonian = + ǫV H H0 where we know the eigenvalue of the unperturbed Hamiltonian and we want to solve for the perturbed case H0 = 0 + ǫV , in terms of an expansion in ǫ (with ǫ varying between 0 and 1). The solution for ǫ 1 is the desired Hsolution. H → We assume that we know exactly the energy eigenkets and eigenvalues of : H0 k = E(0) k H0 | ) k | ) As is hermitian, its eigenkets form a complete basis k k = 11. We assume at first that the energy spectrum H0 k | )( | is not degenerate (that is, all the E(0) are different, in the next section we will study the degenerate case). The k L eigensystem for the total hamiltonian is then ( + ǫV ) ϕ = E (ǫ) ϕ H0 | k)ǫ k | k)ǫ where ǫ = 1 is the case we are interested in, but we will solve for a general ǫ as a perturbation in this parameter: (0) (1) 2 (2) (0) (1) 2 (2) ϕ = ϕ + ǫ ϕ + ǫ ϕ + ..., E = E + ǫE + ǫ E + ..
    [Show full text]
  • Qualification Exam: Quantum Mechanics
    Qualification Exam: Quantum Mechanics Name: , QEID#43228029: July, 2019 Qualification Exam QEID#43228029 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1=2 particles interacting with one another and with an external uniform magnetic field B~ directed along the z-axis. The Hamiltonian is given by ~ ~ ~ ~ ~ H = −AS1 · S2 − µB(g1S1 + g2S2) · B where µB is the Bohr magneton, g1 and g2 are the g-factors, and A is a constant. 1. In the large field limit, what are the eigenvectors and eigenvalues of H in the "spin-space" { i.e. in the basis of eigenstates of S1z and S2z? 2. In the limit when jB~ j ! 0, what are the eigenvectors and eigenvalues of H in the same basis? 3. In the Intermediate regime, what are the eigenvectors and eigenvalues of H in the spin space? Show that you obtain the results of the previous two parts in the appropriate limits. Problem 2. 1983-Fall-QM-U-2 ID:QM-U-20 1. Show that, for an arbitrary normalized function j i, h jHj i > E0, where E0 is the lowest eigenvalue of H. 2. A particle of mass m moves in a potential 1 kx2; x ≤ 0 V (x) = 2 (1) +1; x < 0 Find the trial state of the lowest energy among those parameterized by σ 2 − x (x) = Axe 2σ2 : What does the first part tell you about E0? (Give your answers in terms of k, m, and ! = pk=m). Problem 3. 1983-Fall-QM-U-3 ID:QM-U-44 Consider two identical particles of spin zero, each having a mass m, that are con- strained to rotate in a plane with separation r.
    [Show full text]
  • Chapter 8 Perturbation Theory, Zeeman Effect, Stark Effect
    Chapter 8 Perturbation Theory, Zeeman Effect, Stark Effect Unfortunately, apart from a few simple examples, the Schr¨odingerequation is generally not exactly solvable and we therefore have to rely upon approximative methods to deal with more realistic situations. Such methods include perturbation theory, the variational method and the WKB1-approximation. In our Scriptum we, however, just cope with perturbation theory in its simplest version. It is a systematic procedure for obtaining approximate solutions to the unperturbed problem which is assumed to be known exactly. 8.1 Time{Independent Perturbation Theory The method of perturbation theory is that we deform slightly { perturb { a known Hamil- tonian H0 by a new Hamiltonian HI, some potential responsible for an interaction of the system, and try to solve the Schr¨odingerequation approximately since, in general, we will be unable to solve it exactly. Thus we start with a Hamiltonian of the following form H = H0 + λ HI where λ 2 [0; 1] ; (8.1) where the Hamiltonian H0 is perturbed by a smaller term, the interaction HI with λ small. The unperturbed Hamiltonian is assumed to be solved and has well-known eigenfunctions and eigenvalues, i.e. 0 (0) 0 H0 n = En n ; (8.2) where the eigenfunctions are chosen to be normalized 0 0 m n = δmn : (8.3) The superscript " 0 " denotes the eigenfunctions and eigenvalues of the unperturbed system H0 , and we furthermore require the unperturbed eigenvalues to be non-degenerate (0) (0) Em 6= En ; (8.4) 1, named after Wentzel, Kramers and Brillouin. 151 152CHAPTER 8. PERTURBATION THEORY, ZEEMAN EFFECT, STARK EFFECT otherwise we would use a different method leading to the so-called degenerate perturbation theory.
    [Show full text]
  • Stark Effect
    Stark Effect The Stark effect is the shift in atomic energy levels caused by an external electric field. There are various regimes to consider. The one treated here is the so-called strong field case, where the shift in energy levels due to the external electric field is large compared to fine structure (although still small compared to the spacings between the unperturbed atomic levels.) In the strong field limit, the Stark effect is independent of electron spin. We start with the ordinary hydrogen Hamiltonian, p2 e2 H = − 0 2m r and add a term arising from a uniform electric field along the z axis. H0 = eEz: Note the + sign on this term. It is easily checked by remembering that the force on the electron due to this term would be obtained by taking −@z; which gives a force along the −z axis, as it should for an electron. To understand the matrix elements that are non-zero, it is useful to temporarily give the external electric field an arbitrary direction, H0 = eE·~ ~x The selection rules on the matrix elements of ~x are < n0; l0; m0j~xjn; l; m >=6 0; l0 = l ± 1: These follow from angular momentum conservation (~x has angular momentum 1), and parity (~x is odd under parity). Returning to the case of the electric field along the z axis, we have an additional selection rule on m; < n0; l0; m0jzjn; l; m >=6 0; l0 = l ± 1; m0 = m From these selection rules we see that non-zero matrix elements require different values of l.
    [Show full text]
  • Higher Orders of Perturbation Theory for the Stark Effect on an Atomic Multiplet I
    Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1006–1018. Translated from Zhurnal Éksperimental’noÏ i TeoreticheskoÏ Fiziki, Vol. 123, No. 6, 2003, pp. 1145–1159. Original Russian Text Copyright © 2003 by Bolgova, Ovsyannikov, Pal’chikov, Magunov, von Oppen. ATOMS, SPECTRA, RADIATION Higher Orders of Perturbation Theory for the Stark Effect on an Atomic Multiplet I. L. Bolgovaa, V. D. Ovsyannikova, V. G. Pal’chikovb,*, A. I. Magunovc, and G. von Oppend aPhysics Department, Voronezh State University, Voronezh, 394006 Russia bNational Research Institute for Physical–Technical and Radiotechnical Measurements, Mendeleevo, Moscow oblast, 141570 Russia *e-mail: [email protected] cInstitute of General Physics, Russian Academy of Sciences, Moscow, 119991 Russia dTechnical University, Berlin, D-10623, Germany Received November 28, 2002 Abstract—The contribution of higher order corrections to the Stark energy is calculated in the anticrossing region of atomic multiplet sublevels. Perturbation theory for close-lying levels is presented that is based on the Schrödinger integral equation with a completely reduced Green’s function. Analytic formulas are obtained for the splitting of two interacting fine-structure sublevels as a function of the field strength. These formulas take into account fourth-order resonance and nonresonance corrections to both the diagonal and the off-diagonal matrix elements of the dipole moment operator. By the method of the Fues model potential, a numerical anal- ysis of radial matrix elements of the second, third, and fourth orders is carried out that determine a variation in 3 3 the transition energy between n P0 and n P2 sublevels of a helium atom for n = 2, 3, 4, 5 in a uniform electric field.
    [Show full text]
  • Atomic Physics
    Atomic Physics High-precision quantum systems and the interaction of light and matter Dr Andrew Steane April 10, 2002 Contents 1 Themes in Atomic Physics 6 1.1 Some mathematical notations . 7 1.2 Atomic physics|some preliminaries . 8 1.2.1 The role of classical and quantum mechanics . 9 1.3 Introducing the atom . 9 2 Hydrogen 10 2.1 SchrÄodingerequation solution: Main features . 10 2.2 Comment . 13 2.3 Orbital angular momentum notation . 13 2.4 Some classical estimates . 14 2.5 How to remember hydrogen . 15 2.6 Hydrogen-like systems . 15 2.7 Main points . 16 2.8 Appendix on series solution of hydrogen equation, o® syllabus . 16 3 Grating spectroscopy and the emission and absorption spectrum of hydrogen 18 3.1 Main points for use of grating spectrograph . 18 1 3.2 Resolution . 19 3.3 Usefulness of both emission and absorption methods . 20 3.4 The spectrum for hydrogen . 20 3.5 What is going on when atoms emit and absorb light . 20 3.6 Main points . 21 4 How quantum theory deals with further degrees of freedom 22 4.1 Multi-particle wavefunctions . 22 4.2 Spin . 23 4.3 Main points . 24 5 Angular momentum in quantum systems 25 5.1 Main points . 28 6 Helium 29 6.1 Main features of the structure . 29 6.2 Splitting of singlets from triplets . 30 6.3 Exchange symmetry and the Pauli exclusion principle . 31 6.4 Exchange symmetry in helium . 34 6.5 Main points . 35 6.5.1 Appendix: detailed derivation of states and splitting .
    [Show full text]
  • A Stark-Effect Modulator for CO2 Laser Free-Space Communications
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 5-2005 A Stark-Effect Modulator for CO2 Laser Free-Space Communications Ryan Lane Holloman University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Physics Commons Recommended Citation Holloman, Ryan Lane, "A Stark-Effect Modulator for CO2 Laser Free-Space Communications. " Master's Thesis, University of Tennessee, 2005. https://trace.tennessee.edu/utk_gradthes/2004 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Ryan Lane Holloman entitled "A Stark-Effect Modulator for CO2 Laser Free-Space Communications." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Physics. Stuart Elston, Major Professor We have read this thesis and recommend its acceptance: Donald Hutchinson, Robert Compton Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Ryan Lane Holloman entitled “A Stark- Effect Modulator for CO2 Laser Free-Space Communications.” I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Physics.
    [Show full text]
  • Quantum Mechanics Department of Physics and Astronomy University of New Mexico
    Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions: • The exam consists two parts: 5 short answers (6 points each) and your pick of 2 out 3 long answer problems (35 points each). • Where possible, show all work, partial credit will be given. • Personal notes on two sides of a 8X11 page are allowed. • Total time: 3 hours Good luck! Short Answers: S1. Consider a free particle moving in 1D. Shown are two different wave packets in position space whose wave functions are real. Which corresponds to a higher average energy? Explain your answer. (a) (b) ψ(x) ψ(x) ψ(x) ψ(x) ψ = 0 ψ(x) S2. Consider an atom consisting of a muon (heavy electron with mass m ≈ 200m ) µ e bound to a proton. Ignore the spin of these particles. What are the bound state energy eigenvalues? What are the quantum numbers that are necessary to completely specify an energy eigenstate? Write out the values of these quantum numbers for the first excited state. sr sr H asr sr S3. Two particles of spins 1 and 2 interact via a potential ′ = 1 ⋅ 2 . a) Which of the following quantities are conserved: sr sr sr 2 sr 2 sr sr sr sr 2 1 , 2 , 1 , 2 , = 1 + 2 , ? b) If s1 = 5 and s2 = 1, what are the possible values of the total angular momentum quantum number s? S4. The energy levels En of a symmetric potential well V(x) are denoted below. V(x) E4 E3 E2 E1 E0 x=-b x=-a x=a x=b (a) How many bound states are there? (b) Sketch the wave functions for the first three levels (n=0,1,2).
    [Show full text]
  • Summer Lecture Notes Hydrogen-Like Atoms, Transition Theory
    Summer Lecture Notes Hydrogen-like Atoms, Transition Theory Andrew Forrester September 7, 2006 Contents 1 Lecture 7 3 2 Preliminaries and Notation 3 3 Hydrogen-Like Atoms to Zeroth Order 4 3.1 The Hamiltonian . 4 3.2 Bound State Energy Eigenfunctions and Spectra . 5 3.3 Special Functions . 5 3.4 Useful Integrals and Expectation Values . 6 4 Approximation Methods for Bound States 8 4.1 Bound State Perturbation Theory . 8 4.1.1 Time-Independent, Non-Degenerate . 8 4.1.2 Time-Independent, Degenerate . 8 4.1.3 Time-Dependent (Non-Degenerate/Degenerate)? . 8 4.2 Variational Method . 8 4.3 Born-Oppenheimer Approximation (is this used inside other methods?) . 8 4.4 WKB (Wentzel-Kramers-Brillouin) Method . 8 5 Hydrogen-like Atom Effects and Corrections 9 5.1 General or External Effects . 9 5.1.1 (Electric Coupling?): Stark Effect . 9 5.1.2 (Magnetic Coupling): Zeeman Effect, Anomalous Zeeman Effect . 9 5.1.3 Angular Momentum Coupling . 10 5.1.4 van der Waals Effect . 10 5.1.5 Minimal Coupling(?) . 10 5.2 Internal Effects, or Corrections . 10 5.2.1 Reduced Mass Correction . 10 5.2.2 Relativistic Kinetic Energy Correction . 10 5.2.3 (Magnetic-(Electric/Magnetic) Coupling?): Spin-Orbit (LS) Coupling, Fine Structure . 10 5.2.4 (Magnetic Coupling?): Spin-Spin (SS) Coupling, Hyperfine Structure (A Per- manent Zeeman Effect) . 10 5.2.5 Finite Size of the Nucleus Correction . 10 5.2.6 Lamb Shift . 10 6 Examples 11 6.1 Helium Ground State . 11 6.2 Molecules: Born-Oppenheimer Approximation, Hydrogen Molecular Ion .
    [Show full text]
  • Theory of the Ac Stark Effect on the Atomic Hyperfine Structure And
    University of Nevada, Reno Theory of the ac Stark Effect on the Atomic Hyperfine Structure and Applications to Microwave Atomic Clocks A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics by Kyle Beloy Dr. Andrei Derevianko/Dissertation Advisor August, 2009 THE GRADUATE SCHOOL We recommend that the dissertation prepared under our supervision by KYLE BELOY entitled Theory of the ac Stark Effect on the Hyperfine Structure and Applications to Microwave Atomic Clocks be accepted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Andrei Derevianko, Ph. D., Advisor Jonathan Weinstein, Ph. D., Committee Member Peter Winkler, Ph. D., Committee Member Sean Casey, Ph. D., Committee Member Stephen Wheatcraft, Ph. D., Graduate School Representative Marsha H. Read, Ph. D., Associate Dean, Graduate School August, 2009 i Abstract Microwave atomic clocks are based on the intrinsic hyperfine energy interval in the ground state of an atom. In the presence of an oscillating electric field, the atomic system|namely, the hyperfine interval|becomes perturbed (the ac Stark effect). For the atomic sample in a clock, such a perturbation leads to an undesired shift in the clock frequency and, ultimately, to an inaccuracy in the measurement of time. Here a consistent perturbation formalism is presented for the theory of the ac Stark effect on the atomic hyperfine structure. By further implementing relativistic atomic many-body theory, this formalism is then utilized for two specific microwave atomic clock applications: a high-accuracy calculation of the blackbody radiation shift in the 133Cs primary frequency standard and a proposal for microwave clocks based on atoms in an engineered optical lattice.
    [Show full text]
  • Arxiv:1308.5205V2 [Physics.Atom-Ph] 11 Nov 2013
    Double Bragg diffraction: A tool for atom optics E. Giese,1 A. Roura,1 G. Tackmann,2 E. M. Rasel,2 and W. P. Schleich1 1Institut f¨urQuantenphysik and Center for Integrated Quantum Science and Technology IQST , Universit¨atUlm, Albert-Einstein-Allee 11, D-89081, Germany. 2Institut f¨urQuantenoptik, Leibniz Universit¨atHannover, Welfengarten 1, D-30167 Hannover, Germany. The use of retro-reflection in light-pulse atom interferometry under microgravity conditions nat- urally leads to a double-diffraction scheme. The two pairs of counterpropagating beams induce simultaneously transitions with opposite momentum transfer that, when acting on atoms initially at rest, give rise to symmetric interferometer configurations where the total momentum transfer is automatically doubled and where a number of noise sources and systematic effects cancel out. Here we extend earlier implementations for Raman transitions to the case of Bragg diffraction. In contrast with the single-diffraction case, the existence of additional off-resonant transitions between resonantly connected states precludes the use of the adiabatic elimination technique. Nevertheless, we have been able to obtain analytic results even beyond the deep Bragg regime by employing the so-called \method of averaging," which can be applied to more general situations of this kind. Our results have been validated by comparison to numerical solutions of the basic equations describing the double-diffraction process. I. INTRODUCTION ability to manipulate light. It should be noted that all cases mentioned so far involve static potentials (or effec- There has recently been a growing interest in the tive potentials) for the dynamics of the matter waves, possibilities that (ultra)cold atoms in microgravity offer which implies that the kinetic energy (the modulus of for long-time interferometry and its application to high- the wave-vector) before and after scattering remains the precision measurements [1{4].
    [Show full text]