DOCSLIB.ORG

The Stark Effect in Rydberg States of a Highly Polar Diatomic Molecule: Caf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Stark Effect in Rydberg States of a Highly Polar Diatomic Molecule: Caf The Stark effect in Rydberg states of a highly polar diatomic molecule: CaF The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Petrovic#, Vladimir S. et al. “The Stark Effect in Rydberg States of a Highly Polar Diatomic Molecule: CaF.” The Journal of Chemical Physics 131.6 (2009) : 064301. © 2009 American Institute of Physics As Published http://dx.doi.org/10.1063/1.3179942 Publisher American Institute of Physics Version Final published version Citable link http://hdl.handle.net/1721.1/65894 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. THE JOURNAL OF CHEMICAL PHYSICS 131, 064301 ͑2009͒ The Stark effect in Rydberg states of a highly polar diatomic molecule: CaF ͒ Vladimir S. Petrović, Jeffrey J. Kay, Stephen L. Coy, and Robert W. Fielda Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA ͑Received 10 October 2008; accepted 25 June 2009; published online 10 August 2009͒ The Stark effect in molecular Rydberg states is qualitatively different from the Stark effect in atomic Rydberg states because of the anisotropy of the ion core and the existence of rotational and vibrational degrees of freedom. These uniquely molecular features cause the electric-field-induced decoupling of the Rydberg electron from the body frame to proceed in several stages in a molecule. Because the transition dipole moment among the same-n* Rydberg states is much larger than the permanent dipole moment of the ion core, the decoupling of the Rydberg electron from the ion core proceeds gradually. In the first stage, analyzed in detail in this paper, ᐉ and N are mixed by the external electric field, while N+ is conserved. In the further stages, as the external electric field increases, N+, n*, and v+ are expected to undergo mixing. We have characterized these stages in n*=13, v+ =1 states of CaF. The large permanent dipole moment of CaF+ makes CaF qualitatively different from the other molecules in which the Stark effect in Rydberg states has been described ͑ ͒ H2,Na2,Li2, NO, and H3 and makes it an ideal testbed for documenting the competition between the external and CaF+ dipole electric fields. We use the weak-field Stark effect to gain access to the lowest-N rotational levels of f, g, and h states and to assign their actual or nominal N+ quantum number. Lowest-N rotational levels provide information needed to disentangle the short-range and long-range interactions between the Rydberg electron and the ion core. We diagonalize an effective Hamiltonian matrix to determine the ᐉ-characters of the 3ഛᐉഛ5 core-nonpenetrating 2⌺+ states and to characterize their mixing with the core-penetrating states. We conclude that the mixing of the ᐉ=4, N−N+ =−4͑g͑−4͒͒ state with lower-ᐉ 2⌺+ states is stronger than documented in our previous multichannel quantum defect theory and long-range fits to zero-field spectra. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3179942͔ I. INTRODUCTION foundly from the mixing caused by the quadrupole moment and polarizabilities. Since its discovery in 1913, the Stark effect has been ᐉ-mixing in molecular Rydberg states results from both widely used to probe and exert control over atoms and mol- the incomplete shielding of the ion core ͑also present in at- ecules. The Stark effect in Rydberg states has received espe- oms͒, as well as the departure from spherical symmetry due cially close attention due to the exceptionally large transition to the presence of more than one nucleus ͑with ᐉ being the dipole moments among Rydberg states, especially those be- orbital angular momentum for the Rydberg electron͒. For longing to the same n* ͑which scales approximately ϰn*2, lower-ᐉ states, core penetration of the Rydberg electron is the where n* is the effective principal quantum number͒. Large dominant cause of ᐉ-mixing. For core-nonpenetrating states, transition dipole moments make it possible to exert profound ᐉ-mixing caused by the ion core can be understood in terms external influence over atoms or molecules even with rela- of the static long-range properties of the ion core, such as tively weak electric fields. Both the strength of the interac- multipole moments and polarizabilities. The interaction se- tion with a static electric field ͑discussed in this paper͒ and lection rules are such that the dipole field of the ion core the intensities of radiative transitions among Rydberg states causes interaction among states with ᐉ differing by 1, while are determined by the magnitude of the same transition di- the quadrupole field of the ion core mixes states of ᐉ differ- pole moment. Nevertheless, due to the complex patterns of ingby2or0͑see Table I͒. In CaF, due to the large CaF+ multistate interactions among Rydberg states, high-resolution dipole and quadrupole moments, ᐉ-mixing is profound. It is spectroscopic studies of the Stark effect in molecular Ryd- not limited only to s-d mixing, as in homonuclear molecules berg states have been limited to only a handful of cases. All but affects all ᐉഛ3 states.1 of the molecules in which the Stark effect in Rydberg states In a nonrotating molecule the information about the dif- has been studied have a molecular ion with either a small or ferent mechanisms of the interaction of the Rydberg electron zero permanent dipole moment. However, even in the ab- with the ion core is not contaminated by rotation-electronic sence of an external electric field the mechanism by which interactions. However, to be able to identify assignment- the dipole moment mixes the electron orbital angular mo- aiding patterns, spectra usually must be recorded from sev- mentum quantum number in Rydberg states differs pro- eral rotational levels of an intermediate electronic state. Ro- tational assignments are then made by identifying pairs of ͒ a Electronic mail: rwfi[email protected]. spectral features ͑combination differences͒ with the same N 0021-9606/2009/131͑6͒/064301/14/$25.00131, 064301-1 © 2009 American Institute of Physics Downloaded 20 Jul 2011 to 18.74.1.24. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 064301-2 Petrović et al. J. Chem. Phys. 131, 064301 ͑2009͒ TABLE I. Selection rules for interactions discussed in the text. Matrix elements for these types of interactions are nonzero when the conditions from the table are satisfied. Interaction ⌬ᐉ ⌬N+ ⌬Na Ion-core dipole field operating on Rydberg electron Ϯ1b Ϯ10 Ion-core quadrupole field operating on Rydberg electron 0, Ϯ2b 0, Ϯ20 External homogeneous electric field operating on Rydberg electron Ϯ100,Ϯ1 External homogeneous electric field operating on the ion core 0 Ϯ10,Ϯ1 aElectric field polarization can further affect these selection rules. bThese matrix elements are zero when ⌬n*=0. but different NЈ ͑N=J−S; primed values pertain to the inter- sis of the Stark effect in the spectrum of the CaF molecule, mediate state, unprimed to a Rydberg state͒.AsN increases, which has an ion core, CaF+, with one of the largest perma- for a given N+ value, the rotation-electronic interaction nent dipole moments known in a diatomic molecular ion, ͑ᐉ-uncoupling͒ results in the evolution from Hund’s coupling Ϸ9D͑Ref. 4͒͑origin at center of mass of CaF+͒. case ͑b͒ to Hund’s case ͑d͒, and the intensity patterns ex- We describe several stages of the decoupling of the mo- pected for Hund’s case ͑b͒ are distorted. As a result, combi- lecular Rydberg electron from the body frame caused by the nation differences in N become systematically unobservable. external electric field. The weak-field Stark effect is em- The distortion of the relative intensity patterns from those ployed to gain access to the f, g, and h states and to aid the expected for Hund’s case ͑b͒ to Hund’s case ͑d͒ transitions assignment of the actual or nominal value of N+, the rota- can be understood in terms of the interference effects2 ͑see tional quantum number of the ion core ͑N+ =N−ഞ͒, for the Fig. 1͒. When the Rydberg state into which a transition is lowest-N rotational levels. From the analysis of the lowest-N observed is not in the Hund’s case ͑b͒ limit, we can decom- rotational levels, we learn about the interactions of the core- pose that state into Hund’s case ͑b͒ basis states with different nonpenetrating f, g, and h states with the lower-ᐉ core- values of the ⌳ quantum number. The transformation be- penetrating states. tween the Hund’s case ͑b͉͒␩,ᐉ,⌳,N͘ and Hund’s case ͑d͒ Two models, based on the experimental data that cover ͉␩,ᐉ,N,N+͘ basis states is given by the following expres- the region surrounding n*=13, have been developed previ- sion: ously to describe the Rydberg states of CaF. A global multi- + channel quantum defect theory ͑MQDT͒ model was used to + N ᐉ N ͉␩,ᐉ,⌳,N͘ = ͚ ͑−1͒N −ᐉ+⌳͑2N+ +1͒1/2ͩ ͪ fit the states from n*Ϸ12−18, N=0−14, v+ =1, n*Ϸ9−10, ⌳ − ⌳ 0 + + N+ N=0−14, v =2, and n*Ϸ7, N=3−10, v =3 regions, and generated 0ഛᐉഛ3 quantum defect matrix and its derivatives ϫ͉␩,ᐉ,N,N+͘. ͑1͒ with respect to the internuclear distance.5 A complementary The low-n* ͑and low-NЈ͒ intermediate state through which extended long-range model, which included the 3ഛᐉഛ5, the Rydberg state is accessed is usually in the Hund’s case v+ =1 states from the region of 13ഛn*ഛ17, yielded the val- ͑a͒ or ͑b͒ limit and has a definite value of the ⌳Ј quantum ues for dipole and quadrupole moments and the isotropic number.
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]