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Volumn 77, Issue 4, 2008, Pages

Semiclassical theory of the structure of the hydrogen spectrum in near-perpendicular electric and magnetic fields: Derivations and formulas for Einstein-Brillouin-Keller-Maslov quantization and description of monodromy

Author keywords

[No Author keywords available]

Indexed keywords

ELECTRIC FIELDS; HAMILTONIANS; MAGNETIC FIELDS; MOLECULAR STRUCTURE; PERTURBATION TECHNIQUES; PHASE SPACE METHODS; TOPOLOGY;

EID: 43049147131     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.77.043422     Document Type: Article
Times cited : (12)

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    • Nearly perpendicular weak fields means |χ| ωf N3 1.
    • Nearly perpendicular weak fields means |χ| ωf N3 1.
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  • 63
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    • This 2-form is undefined at points where | μi | =±N/2, but all necessary integrals are well defined.
    • This 2-form is undefined at points where | μi | =±N/2, but all necessary integrals are well defined.
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    • These circles have coordinates ψ1 or ψ2, whichever survives according to the discussion in Sec. 3.
    • These circles have coordinates ψ1 or ψ2, whichever survives according to the discussion in Sec. 3.
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    • The space of Pauli orbits at fixed Q [the (W, δW) surface] is homeomorphic to a sphere, but we do not establish that it is diffeomorphic to a sphere.
    • The space of Pauli orbits at fixed Q [the (W, δW) surface] is homeomorphic to a sphere, but we do not establish that it is diffeomorphic to a sphere.
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    • The classical spectrum is also known as the image of the energy momentum map.
    • The classical spectrum is also known as the image of the energy momentum map.
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    • For some parameters, a dashed red curve appears on classically doubly degenerate points of the classical spectrum. Specific examples are the upper boundary of the inner triangular region for the Stark region, and the lower boundary of the inner triangular region for the Zeeman region. In these cases, only one of the two disjoint components of the level set is not a 2-torus. Such cases will always be evident when inspecting the contours of h2 on the (W, δW) surface.
    • For some parameters, a dashed red curve appears on classically doubly degenerate points of the classical spectrum. Specific examples are the upper boundary of the inner triangular region for the Stark region, and the lower boundary of the inner triangular region for the Zeeman region. In these cases, only one of the two disjoint components of the level set is not a 2-torus. Such cases will always be evident when inspecting the contours of h2 on the (W, δW) surface.
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    • Equations 40 cannot be linearized at the poles of the (W, δW) surface, and extrema at the poles are not quadratic.
    • Equations 40 cannot be linearized at the poles of the (W, δW) surface, and extrema at the poles are not quadratic.
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    • The reference point of the action may be taken to be at Q=0. All other choices of reference point would yield an action that differs from Eq. 43 by an additive constant.
    • The reference point of the action may be taken to be at Q=0. All other choices of reference point would yield an action that differs from Eq. 43 by an additive constant.
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    • For 0< |Q|
    • For 0< |Q|
  • 72
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    • By construction, the action is defined only for areas bounded by contours of h2 (W, δW) which do not contain fixed points. See Chap. 10 of Ref..
    • By construction, the action is defined only for areas bounded by contours of h2 (W, δW) which do not contain fixed points. See Chap. 10 of Ref..
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    • The quantum basis must include several n -manifolds or important effects, second order in the electric field, will be neglected.
    • The quantum basis must include several n -manifolds or important effects, second order in the electric field, will be neglected.
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    • In the definitions of the Q and W operators, the field vectors F and B can be replaced with arbitrary vectors V1 and V2 such that the operators H 0, Q, and W will form a complete set of commuting observables on a basis of bound states of the unperturbed hydrogen atom.
    • In the definitions of the Q and W operators, the field vectors F and B can be replaced with arbitrary vectors V1 and V2 such that the operators H 0, Q, and W will form a complete set of commuting observables on a basis of bound states of the unperturbed hydrogen atom.
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    • A lattice can be constructed from a list of energy levels alone. See Ref..
    • A lattice can be constructed from a list of energy levels alone. See Ref..
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    • In the Stark limit, the equator W=0 has exceptional structure. It is composed entirely of relative fixed points. Similar exceptional structure is found on ΓN,Q=0 for F and B that lie on the (black and blue online) curves dividing regions of the map.
    • In the Stark limit, the equator W=0 has exceptional structure. It is composed entirely of relative fixed points. Similar exceptional structure is found on ΓN,Q=0 for F and B that lie on the (black and blue online) curves dividing regions of the map.
  • 78
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    • Quantum calculations also show that states belonging to different q - and n -manifolds also exhibit anticrossings with variations in ωf.
    • Quantum calculations also show that states belonging to different q - and n -manifolds also exhibit anticrossings with variations in ωf.
  • 79
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    • www.chrisschleif.com, E-PLRAAN-77-087804 for a detailed derivation. For more information on EPAPS, see
    • See EPAPS Document No. E-PLRAAN-77-087804 for a detailed derivation. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html. This material is also available for download from www.chrisschleif.com
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    • See Chap. 22 of Ref. or Chap. 3.8 of Ref..
    • See Chap. 22 of Ref. or Chap. 3.8 of Ref..
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    • See Fig. 13 of Ref..
    • See Fig. 13 of Ref..
  • 82
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    • See Chap. 4.4 of Ref..
    • See Chap. 4.4 of Ref..
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    • In the six-dimensional phase space 2n, either K is diffeomorphic to a circle (L 0) or its compactification is. Each point x K 2n is labeled by a value of the Delaunay coordinate N.
    • In the six-dimensional phase space 2n, either K is diffeomorphic to a circle (L 0) or its compactification is. Each point x K 2n is labeled by a value of the Delaunay coordinate N.


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