Quantization in generalized coordinates III-Lagrangian formulation

It is shown that if one incorporates the generalized coordinate quantum velocitiesQ ¹ as given byQ ¹=l[H,Q ¹](h=1) into the generalized classical Lagrangian for a free particle (the total energy),L=1/2Q ¹ g _tk Q ^k one does not obtain (no matter what ordering of the operatorsq ^l ,q ^k andg _lkwe choose the correct quantum Lagrangian operator which is a transformation from -1/2V² to generalized coordinates (Gruber, 1971, 1972).q ^l as given byq ^l=i[H,q ^l] turns out to be the Hermitian part of a more generaiized operator which we call the total generalized velocity operator similar to the notation in ear previous articles (Gruber, 1971, 1972). This total velocity operator really determines the fundamental structure governing our system in the Lagrangian formulation. We show that ft is through the total velocity operator that we make the transition from classical to quantum mechanics and through our procedure we arrive at the correct quantum Lagrangian operator.     

Core polarization effects and the random phase approximation solution

A metricg _ik=η _ik+Hξ _iξ_k+2Jξ _(iPk) is investigated. WhenJ=0 this reduces to the well-known Kerr metric. Conditions on the vectorp _i are obtained under which a geodetic, shear-free null congruenceξ _i in the Minkowskian space-time (with metricη _ik) will continue to remain geodetic and shear-free in the Riemannian space-time ofg _ik. A general solution of Einstein’s equationR _ik=σξ _iξ_k is obtained whenp _iξⁱ=0 andξ _i is twist-free.     

Mößbauereffekt in157Gd und155Gd

Using the Mössbauer technique electric hyperfine interactions of the first excited non-rotational states at 64 keV in¹⁵⁷Gd and at 86.5 keV in¹⁵⁵Gd have been determined in GdF₃ and GdCl₃· 6H₂O. For the ratio of quadrupolmoments the ratiosQ ₆₄ ¹⁵⁷ /Q _g ¹⁵⁷ =1.74±0.04,Q _86.5 ¹⁵⁵ /Q _g ¹⁵⁷ =?0.07 ± 0.21 andQ _g ¹⁵⁵ /Q _g ¹⁵⁷ =0.78 ± 0.06 were found. In addition isomer shifts were observed from which a ratio δ〈r²〉 ₆₄ ¹⁵⁷ /δ〈r²〉 _86.5 ¹⁵⁵ =?2.6±0.15 can be inferred.     

The construction of the eigenstates for nonlinear Schrödinger model with supermatrices and attractive coupling

A quantum nonlinear Schrödinger model with supermatrices and attractive coupling is studied by using the quantum inverse scattering method. The eigenstates of the Hamiltonian and the infinite number of the conserved quantities of the system are constructed. In particular, theN-particle bound states with the mixture of bosons and fermions are found. The energy of theN-particle eigenstate are Σ _i=1 ^N andNp ² ?N(N ²?1)c ²/12 for the scattering state and the bound state respectively.     

Investigation of the two quasiparticle rotational band at 1172 keV in172Yb and determination of theg R -factor of the 3+ 3 band head

Investigations of the rotational band based on the 3⁺ 3 two quasiparticle state in¹⁷²Yb have shown that some of its properties can be well described by the collective model of Bohr and Mottelson. As a result of these investigations we have derived the ratio (g _k – g _R )/Q _o =? (0.0136 ± 0.0007). The magneticg-factor of the band head has also been determined by an integral angular correlation measurement perturbed by an external magnetic field. As a result we foundg=0.201 ± 0.030. Using this value and our result of (g _k – g _R )/Q _o we have calculated theg _R -factor of the two quasiparticle state asg _R =0.283 ± 0.018 which is out of the errors smaller than theg _R -factor of the ground state rotational band. An analysis of our angular correlation experiments gave for the mixing parameter δ of theK-forbidden 1094 keV intraband transition: δ=? (3.63 _?0.06 ^+0.14 ) in agreement with the results of other authors.     

On the signatures of the canonical and the metrical energy densities of vector fields

The energy-density components Θ ₀ ⁰ andT ₀ ⁰ of the canonical and of the metrical energy-momentum tensors Θ _k ⁱ andT _k ⁱ for a statical field of vector mesons have opposite signatures: Θ ₀ ⁰ =H=?T ₀ ⁰ =?L. From this property some relativistic and field-theoretical theorems can be deduced in an elementary way.     

Optical spectroscopy of Dysprosium Aluminum Garnet (DAG)

The absorption spectrum of antiferromagnetic dysprosium aluminium garnet (DAG) (T _N =2.50 °K) has been investigated at low temperatures. The groundstate splitting due to all interactions in the antiferromagnetic state is (5.27±0.10) cm^?1 extrapolated to 0 °K. The temperature dependence of the lineshift of the absorption lines is measured. Zeeman effect studies give theg-tensor of the groundstate asg _z =18.4±0.5,g _x =g _y =0.5±0.2. The studies also allow the determination of the critical fields asH _c ^[100] =(5.0±0.1) koe,H _c ^[111] =(3.9±0.2) koe andH _c ^[110] =(4.9±0.6) koe. In addition an investigation of a number of satellite lines is performed. Some of them can be interpreted as spin wave sidebands (Stokes and anti-Stokes); others obviously come from dysprosium ions which have impurity ions on regular lattice sites as neighbours.     

Quantum deformation of compactified Minkowski space

The quantum GrassmanianG(2|0; ? _q ^4|0 ) of “quantum 2-planes ? _q ^2|0 in the quantum 4-plane ? _q ^4|0 ”, which provides aq-deformation of compactified complexified Minkowski space, is proposed. A quantum analogue of classical Plücker embedding of the usual GrassmanianG(2; ?²) into a non-degenerate quadric in ??⁵ is found.     

Many-atom interactions in the theory of higher order elastic moduli: A general theory

The total potential energy of a crystal U({r _ik }) as a function of the vectors r _ik connecting centers of equilibrium positions of the ith and kth atoms is assumed to be represented as a sum of irreducible interaction energies in clusters containing pairs, triples, and quadruples of atoms located in sites of the crystal lattice A2: U({r _ik }) ≡ Σ _N=1 ⁴ E _N ({r _ik }). The curly brackets denote the “entire set.” A complete set of invariants {I _j ({r _ik })} _N , which determine the energy of each individual cluster as a function of the vectors connecting centers of equilibrium positions of atoms in the cluster E _N ({r _ik }) ≡ E _N ({I _j ({r _ik })} _N ), is obtained from symmetry considerations. The vectors r _ik are represented in the form of an expansion in the basis of the Bravais lattice. This makes it possible to represent the invariants {I _j ({r _ik })} _N in the form of polynomials of integers multiplied by τ ₂ ^m . Here, τ₂ is one-half of the edge of the unit cell in the A2 structure and m is a constant determined by the model of interaction energy in pairs, triples, and quadruples of atoms. The model interaction potential between atoms in the form of a sum of the Lennard-Jones interaction potential and similarly constructed interaction potentials of triples and quadruples of atoms is considered as an example. Within this model, analytical expressions for second-order and third-order elastic moduli of crystals with the A2 structure are obtained.     

Hyperfeinstruktur des 82P3/2-Terms von Cäsium 134

The electric quadrupole interaction constantB of the 8p²P_3/2 state of Cs¹³⁴ has been determined by an optical double resonance measurement of the hyperfine structure transition v_F=11/2?F=9/2=47.84(12) Mc/s. The results are: B _8P ¹³⁴ =8.06(20) Mc/s and Q _8P ¹³⁴ =+ 0.427(8) · 10^?24cm². Comparison is made between the measurements in the 7p and 8p electronic states: Q _8P ¹³⁴ /Q _7P ¹³⁴ =0.977(20). The ratio of the corresponding Sternheimer correction factors yields the value C_7p/C_8P=0.982.     

Hyperfeinstrukturaufspaltung und Lebensdauer des 72 P 3/2-Terms von Rb85 und Rb87

The hyperfine structure splitting of the 7² P _3/2 state of Rb⁸⁵ and Rb⁸⁷ has been measured with optical double resonance. The following hfs interaction constants have been obtained: Rb⁸⁵:A _7p ⁸⁵ =3.71(1) MHz;B _7p ⁸⁵ =3.68 (8)MHz. Rb⁸⁷:A _7p ⁸⁷ =12.57(1) MHz;B _7p ⁸⁷ =1.71 (3)MHz. These values yield quadrupole moments ofQ _7p,hfs ⁸⁵ =+0.316(7) barn andQ _7p,hfs ⁸⁷ =+0.147(2)barn which are reduced by application of the Sternheimer correction toQ _7p ⁸⁵ =+0.267(6) barn andQ _7p ⁸⁷ =+0.124(2) barn. The averaged ratio of the uncorrected values of Q_hfs in the 5p and 7p ² P _3/2 stake isQ _5p,hfs/Q _7p,hfs=1.07(3). This has to be compared with the correction factors for polarization of the core of electrons by the nuclear quadrupole moment (1?R)_5p/(1?R)_7p=1.07. The agreement between the measured and calculated ratio indicates a net antishielding of the nuclear quadrupole moments of rubidium by the core of electrons as has been predicted bySternheimer. The lifetime of the 7² P _3/2 state of the RbI-spectrum is:τ(7² P _3/2, Rb)=2.4(2)·10^?7 sec.     

Quantum theory of gravitation and locality postulate

A quantized space-time metricg _ik (x) is investigated within a suitably modified axiomatic approach. Coordinate distancesdx are called absolutely space-like ifg _ik (x)dx ⁱ dx ^k is negative definite. For such distances, fields are assumed to commute or anticommute, respectively (generalized locality). The quantum fluctuations of the light coneg _ik (x)dx ⁱ dx ^k =0 are shown to extend to distancesdx which are space-like with respect to the Minkowski metric. Generalized locality is therefore weaker than the usual locality postulate.Supported in part by the Deutsche Forschungsgemeinschaft.     

The determination by an independent method ofP k electron capture probabilities in133Ba and139Ce decays gamma decay of133Ba

The electron capture decays of¹³³Ba (10.4 y) and¹³⁹Ce (140 d) were investigated with high resolution Ge(Li) detectors. By x-ray — γ-ray coincidences values of the quantityP _k ω _k ? _k for various transitions in the decay of¹³³Ba and¹³⁹Ce are obtained. The factor ω _k ? _k was measured independently by means ofK-conversion electron-K x-ray coincidences and was then used to determineP _k , theK-capture probability. The independent measurement of the product ω _k ? _k , together with recent accurate values of ω _k , provides a new method for the accurate calibration of semiconductor x-ray detectors. In the decay of¹³⁹Ce, a value ofP _k =0.78±0.03 is found (where the error represents twice the standard deviation) from whichQ _ec =326 _?30 ⁺⁷⁰ keV to the¹³⁹La ground-state is found by use of theory. In¹³³Ba decay, values are found ofP _k 1=0.72±0.04 for theEC transition to the 437 keV level in¹³³Cs,P _k 2=0.80±0.07 to the 384 keV level, and from an independent measurement, the ratioP _k 1/P _k 2=0.87±0.07. From these results the ground-state value ofQ _ec =522 _?10 ⁺²⁰ keV is derived from theory for¹³³Ba decay. The gamma spectrum of¹³³Ba also was remeasured. From the present gamma intensities and previous conversion electron intensities, new values forK-shell conversion coefficients are obtained. Previously reported γ-rays at 35 and 391 keV are not confirmed.     

Multilevel systems and generalizations of the superalgebra

A new approach to constructing the various generalizations of the one-dimensional supersymmetric quantum mechanics is proposed, including the parasupersymmetric quantum mechanics constructed by Rubakov and Spiridonov as the special case. In particular, we derive the generalized superalgebra, which possesses the features both of the familiar superalgebra and of the parasuperalgebra. Namely, the generalized supercharges Q_i ^± and the Hamiltonian H forms the generalized superalgebra, where Q_i ^±2=0 (as for ordinary superalgebra), but the triple products of generalized supercharges obey the relations Q₁ ⁺Q_j ^–Q_j ⁺=Q_i ⁺H (i, j=1, 2) and Q_i ⁺Q_i ^–Q_j ⁺=(1/4)kQ_i ⁺, Q_i ⁺Q_i ^–Q_j ⁺=(1/4)kQ_i ⁺(i, j=1, 2; ij) (analogous to the parasuperalgebra). Furthermore, the generalized supercharges are conserved, i.e. [H, Q_i ^±]=0.Presented at the International Workshop on Squeezed and Correlated States, Moscow, December 3–7, 1990.     

K-capture probabilities in the electron capture decay of159Dy

TheK-electron capture probabilities (P _k) for the 58 keV state and the ground state of¹⁵⁹Tb in the decay of¹⁵⁹Dy have been determined. The measured values,P _k ⁵⁸ =0.81±0.05 andP _k ⁰ =0.79±0.06 are found to be in excellent agreement with theoretical results.     

The spectral class of the quantum-mechanical harmonic oscillator

The purpose of this paper is to study the so-calledspectral class Q of anharmonic oscillatorsQ=?D ²+q having the same spectrum λ _n =2n (n≧0) as the harmonic oscillatorQ ⁰=?D ²+x ²?1. Thenorming constants \(t_n = \mathop {\lim }\limits_{x \uparrow \infty } \ell g[( - 1)^n {{e_n (x)} \mathord{\left/ {\vphantom {{e_n (x)} {e_n }}} \right. \kern-0em} {e_n }}( - x)]\) of the eigenfunctions ofQ form a complete set of coordinates inQ in terms of which the potential may be expressed asq=x ²?1?2D ² ?g? with $$\theta = \det \left[ {\delta _{ij} + (e^{ti} - 1)\int\limits_x^\infty {e_i^0 e_j^0 :0 \leqq i,j,< \infty } } \right],$$ e _n ⁰ being then ^th eigenfunctionQ ⁰. The spectrum and norming constants are canonically conjugate relative to the bracket [F, G]=∫ΔFDΔGdx,to wit: [λ _i , λj=0, [t _i, 2λ _j ]=1 or 0 according to whetheri=j or not, and [t _i,t _j]=0. This prompts an investigation of the symplectic geometry ofQ. The function ? is related to the theta function of a singular algebraic curve. Numerical results are also presented.     

Polarized leptoproduction and dynamically generated spin-dependent parton distributions

The first moment of the polarized parton distributions of the proton is evaluated dynamically using the assumption that at some low resolution scale the proton consists of valence quarks only. The spin carried by gluons, ΔG, at the EMC resolution scaleQ ₀ ² =10 GeV² ispredicted to be ΔG/s _p ?5 withs _p=1/2 denoting the spin of the proton. This result furthermoreexplains the dynamical origin of the recent EMC measurement of the first moment of the polarized structure functiong ₁ ^P (x, Q ²).