Lagrangian formulation of the Schrödinger equation

We recast the Schrödinger equation in a new Lagrangian formulation. The equation is —i?dψ (x,t)/dt = Lψ (x,t), whereL is the Lagrangian operator. Expressions forL and ford/dt — ⊥ are derived in terms of coordinate and momentum operators.     

On the temperature behavior of second sound and Poiseuille flow

The linearized Peierls equation for the phonon densityN (k λ,r t) is solved by replacing the collision operator in the subspace orthogonal to the collision invariants byk-dependent relaxation rates. For the normal process relaxation time the behaviorτ _N (k λ)∝|k|^?p for smallk is assumed. Taking into account thisk-dependence ofτ _N explicitly and avoiding an expansion with respect toΩτ _N (kλ) before performing the necessary integration overk yields new, non-analytic, terms in the hydrodynamic equations describing second sound and Poiseuille flow. It is shown that this may lead to a temperature dependence of second sound damping and thermal conductivity in the Poiseuille flow region differing from the usual theoretical predictions and in better agreement with experiments.     

On the number of negative eigenvalues for a Schrödinger operator with magnetic field

We consider the Schrödinger operator with magnetic field $$H = \sum\limits_{j = 1}^n {\left( {\frac{1}{i}\frac{\partial }{{\partial x_j }} - a_j } \right)^2 + Vin\mathbb{R}^n .} $$ Under certain conditions on the magnetic fieldB=curla, we generalize the Fefferman—Phong estimates (Bull. A. M. S.9, 129–206 (1983)) on the number of negative eigenvalues for ?Δ+V to the operatorH. Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions onB in particular are satisfied if the magnetic potentialsa _j (x) are polynomials.     

Das Fluktuationsspektrum von Elektronen in einem stationären Nichtgleichgewichtszustand

The Boltzmann equation is used to calculate the time correlation function and the fluctuation spectrum for electrons obeying classical statistics. The stationary joint distribution for one electron to be initially ink ₀=k(0) and finally ink=k(t) is given by the product of the conditional probability and the stationary distribution. These quantities can be found from the Boltzmann equation if there exists, for any initial distribution, a unique solution which satisfies the Markov equation and tends to a stationary solution for large times under stationary conditions. It is proved that these conditions hold for linear collision operators and in the relaxation approximation. General operator expressions for the fluctuation spectrum and the differential conductivity in a stationary electric field are given, which can be evaluated within the usual approximation schemes known for the stationary, nonequilibrium solutions of the Boltzmann equation. In equilibrium they reproduce the classical fluctuation dissipation theorem. In a nonequilibrium state they define a noise temperature depending on the field. In the relaxation approximation and for polynomial band structure the exact solution can be found. For parabolic and biparabolic spherical bands the result is discussed explicitly.     

Bis-Benzimidazolyl Diamide Based Fluorescent Probe for Copper(II): Synthesis, Structural and Fluorescence Studies

A new fluorescent probe based on a bis-benzimidazole diamide N ²,N ^2′-bis[(1-ethyl-benzimidazol-2-yl)methyl]biphenyl-2,2′-dicarboxamide ligand L ₁ with a biphenyl spacer group and a Copper(II) trinuclear metallacycle has been synthesized and characterized by X-ray single crystallography, elemental and spectral (FT-IR, ¹H & ¹³C NMR, UV-Visible) analysis. The fluorescence spectra of L ₁ in MeOH show an emission band centered at 300 nm. This band arises due to benzimidazolyl moiety in the ligating system. The diamide L ₁ in the presence of Cu²⁺ show the simultaneous ‘quenching’ of (300 nm) and ‘enhancement’ of (375 nm) emission band. Similar fluorescence behavior was found in water–methanol mixture (9:1). The new emission band at 375 nm is attributed to intra ligand π–π* transition of the biphenyl moiety. L ₁ exhibited high selectivity and sensitivity towards Cu²⁺ in both the medium over other common metal ions like Ni²⁺, Co²⁺, Mn²⁺, Mg²⁺, Zn²⁺, Pb²⁺ and Hg²⁺. The binding constant with Cu²⁺ was calculated by the Benesi-Hildebrand equation. Selective “off-on-off” behavior of L ₁ in methanol has also been studied. The fluorescent intensity of 375 nm bands in L ₁ enhances (turns-on) upon addition of Cu²⁺ and quenches (turn-off) upon addition of Na₂-EDTA.     

Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity

The wave and scattering operators for the equation $$\left( {\square + m^2 } \right)\varphi + \lambda \varphi ^2 = 0$$ withm>0 and λ>0 on four-dimensional Minkowski space are analytic on the space of finite-energy Cauchy data, i.e.L ₂ ¹ (R ³)⊕L ₂(R ³).     

A Clifford Algebra Approach to the Classical Problem of a Charge in a Magnetic Monopole Field

The motion of an electric charge in the field of a magnetic monopole is described by means of a Lagrangian model written in terms of the Clifford algebra of the physical space. The equations of motion are written in terms of a radial equation (involving r=|r|, where r(t) is the charge trajectory) and a rotor equation (written in terms of an unitary operator spinor R). The solution corresponding to the charge trajectory in the field of a magnetic monopole is given in parametric form. The model can be generalized in order to describe the motion of a charge in the field of a magnetic monopole and other additional central forces, and as an example, we discuss the classical ones involving linear and inverse square interactions.     

Photoionisation of atoms and Möller operators

The motion of an hydrogenoïd atom in a laser field is usually given by the time-dependent hamiltonian H(t)=[p?A(t)]²/2+V(r) where V(r) is the atomic potential whileA(t) is to be connected with the laser field. The existence and unicity for the Cauchy problem of the solutions of the corresponding Schrödinger equation are established under mild conditions onA(t) and V(r). The existence of Möller operators is investigated in two cases, namely, when the laser field is a function of time only and when it vanishes asymptotically in time. Special attention is paid for the Coulomb case for which a “distorted” Möller operator is derived. Finally, when the laser field vanishes ast→∞, the photoionisation probability is properly defined by means of the Möller operator $$\Omega (H_{At} ,H) = s - \mathop {\lim }\limits_{t \to \infty } U_{At} (t)^{ - 1} U(t)$$ , whereU(t) is the evolution operator for the system whileU _Att (t) is the evolution operator for the atom.     

Integrable structure of conformal field theory,quantum KdV theory and Thermodynamic Bethe Ansatz

We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as “T-operators,” act in highest weight Virasoro modules. TheT-operators depend on the spectral parameter λ and their expansion around λ=∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. TheT-operators can be viewed as the continuous field theory versions of the commuting transfermatrices of integrable lattice theory. In particular, we show that for the values $c = 1 - 3\frac{{3(2n + 1)^2 }}{{2n + 3}}$ ,n=1,2,3 .... of the Virasoro central charge the eigenvalues of theT-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of the massless Thermodynamic Bethe Ansatz for the minimal conformal field theoryM _2,2n+3; in general they provide a way to generalize the technique of the Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator Φ_1,3. The relation of theseT-operators to the boundary states is also briefly described.     

Single-vacancy induced motion of a tracer particle in a two-dimensional lattice gas

We study the random motion of a tracer particle in a two-dimensional dense lattice gas. Repeated encounters of asingle vacancy displace the tracer particle from its initial position by a vector y of which we calculate the time-dependent distributionP _t(y). On an infinite lattice and for large times $$P_t (y) \simeq \frac{{2(\pi - 1)}}{{\ln t}}K_0 \left( {\left( {\frac{{4\pi (\pi - 1)}}{{\ln t}}} \right)^{1/2} y} \right)$$ whereK ₀ is a modified Bessel function. The same problem is studied on a finiteL×L lattice with periodic boundary conditions; thereP _t(y) is shown to be a Gaussian on a time scaleL ² InL. On an ∞×L strip and for large times,P _t(y) is an explicitly given (but nonelementary) function of the scaling variable ξy ₁/t ^1/4, identical to the function occurring in the problem of a random walker on a random one-dimensional path.     

Evaluating the validity of parametrized relativistic wave equations

We wish to determine the correct partial differential equation(s) for describing a relativistic particle. A physical foundation is presented for using a parametrized wave equation with the general form $$i\frac{{\partial \psi }}{{\partial s}} = K\psi$$ where s is the invariant evolution parameter. Several forms have been proposed for the generator K of evolution parameter translations. Of the proposed generators, only the generator K ₂ which is proportional to the inner product P _μ P ^μ of fourmomentum operators can be derived from first principles, notably a probabilistic basis. Although experimental tests must be made to establish the validity of K ₂,we conclude that K ₂ is the leading theoretical candidate for the form of a generator of evolution parameter translations.     

Spatial representation of groups of automorphisms of von Neumann algebras with properly infinite commutant

Theorem. Let a topological groupG be represented (a→φ _a ) by *-automorphisms of a von Neumann algebraR acting on a separable Hilbert spaceH. Suppose that

1. G is locally compact and separable,
2. R′ is properly infinite,
3. for anyT∈R,x,y∈H the function

$$a \to \left\langle {\phi _a (T)x,y} \right\rangle _H $$ is measurable onG. Then there exists a strongly continuous unitary representation ofG onH,a→U _a , such that forT∈R,a∈G, $$\phi _\alpha (T) = U_a TU_a *.$$ .     

Rotation of the plane of polarization and linear birefringence of sound in hematite below the morin point

The linear antiferromagnetic birefringence of sound in hematite (α-Fe₂O₃) residing in the collinear easy-axis phase (L ‖ C ₃) below the Morin point is experimentally studied. The plane of polarization of a linearly polarized transverse acoustic wave propagating along the trigonal axis C ₃ of a hematite crystal placed in a magnetic field H applied in the basal plane (H ⊥ C ₃, 3.5 ≤ H ≤ 15 kOe) is found to rotate after a temperature-driven orientational phase transition to the easy-axis state. The angle of rotation exhibits a 180° angular dependence on the direction of the magnetic field in the basal plane and varies from zero to ～π/2. Numerical estimates suggest that the conditions necessary for rotation of the plane of polarization through appreciable angles (～π/2) can be satisfied in the easy-axis phase at orientational phase transition temperatures close to the Morin point, which actually takes place in the fields employed. The results obtained are described sufficiently well by the theory of linear antiferromagnetic birefringence of sound (E.A. Turov) and confirm its main conclusions.     

Properties describing the dynamics of crystal electrons for the case of a non-local potential

Thek-p formalism ist generalized to allow for non-local one-electron Hamiltonains, too. The one-electron energiesE _nk are found to be continuous functions ofk and to be analytic provided different bands do not touch. In addition, af-sum rule is derived and it is shown that the expectation value of the velocity operator coincides with the usual definition of group velocity. Explicit expressions for the non-diagonal elements of the velocity operator and of the effective mass tensor are given.     

A study on various expansions of the angular momentum projector in terms of powers ofJ +,J − andJ 0

This note contains a discussion of various expansions of the angular momentum projection operator in terms of the operatorsJ ₊,J _? andJ ₀. This gives us opportunity to rederive the famous Löwdin's shift operator formula. As a starting point the integral representation of the projector is used, which is done in a somewhat different way than that proposed by Raynal.     

Resonant dislocation motion in NaCl crystals in the EPR scheme in the Earth’s magnetic field with pulsed pumping

Resonant dislocation motions in NaCl(Ca) crystals under the simultaneous action of the Earth’s magnetic field B _Earth (～66 μT) and a pulsed pump field $\tilde B$ of sufficient amplitude $\tilde B_m $ and certain duration τ have been detected and studied. The measured dislocation path peaks l(τ) have a maximum at τ = τ _r ≈ 0.53 μs. The resonance criterion has been found to be the ordinary EPR condition in which the g-factor is close to 2 and the optimum inverse pulse duration τ _r ^?1 is used instead of the harmonic pump field frequency ν _r . The largest peak l(τ) height is reached at mutually orthogonal dislocation (L) and magnetic field (B _Earth and $\tilde B$ ) orientations. Pulsed field rotation to the position $\tilde B$ ‖ B _Earth significantly decreases but does not “kill” the effect. For dislocations parallel to the Earth’s field (L ‖ B _Earth), the resonance almost disappears even at $\tilde B$ ⊥ B _Earth. In the optimum geometry of experiments, as the pump field amplitude $\tilde B_m $ decreases from 17.6 to 10 μT, the path peak height l _r = l(τ _r ) decreases only by 7.5%, remaining at the level of l _r ～ 10² μm, and at a $\tilde B_m $ further fall-off to 4 μT, it rapidly decreases to background values. In this case, the relative density of mobile dislocations similarly decreases from ～90 to 40%. Possible physical mechanisms of the observed effect have been discussed.     

Synthesis and Spectroscopy of Novel Branched Fluorescent Dyes Containing Benzophenone Parts and the Possibility as Fluorescence Probes

This paper describes a new fluorescent family of branched dyes containing benzophenone unit including 4-N, N-diphenylamino-4??-phenacyl-stilbene (C1), 4,4??-di(4-benzoylphenylethylene)yl-triphenylamine (C2) and 4,4??,4??-tri(4-benzoylphenylethylene)yl-triphenylamine (C3). Benzophenone part is coupled with core through C?CC double bond. The chemical structures of the derivatives are characterized with ¹H and ¹³C nuclear magnetic resonance and elemental analysis. Strong ?ШC?? stacking interactions are discovered with the analysis of the X-ray crystallographic data of C1. The absorption maxima and emission maxima of the derivatives exhibit gradual bathochromic shift from C1 to C3. The optical density of C1, C2 and C3 are shown to be related to the number of branches. The changes of dipole moments between the excited and ground states for C1, C2 and C3 were estimated to be 4.356, 8.091 and 8.479 Derby, respectively by Lippert equation, confirming that the internal charge transfer (ICT) dominates the process of excited singlet state. The possibility as fluorescence probes of the derivatives on the estimation of what region of micelles interacting with samples was evaluated.     

Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

Exact solution of the Boltzmann equation in the relaxation approximation,existence of negative differential conductivity for certain types of energy bands and its implication for the fluctuation spectrum and the noise temperature

The Boltzmann equation for the distributionf _k of a system of charged particles obeying classical statistics in a uniform fieldF, $$\frac{{\partial f_k }}{{\partial t}} + F\frac{{\partial f_k }}{{\partial k}} = \smallint d^3 k'(W_{kk'} f_{k'} - W_{k'k} f_k ),$$ will be solved analytically for a special class of transition ratesW _kk′=const·h _k ·ν _k ·ν _k′ for any initial distribution.h _k is the Maxwell distribution andν _k >0 can be interpreted as ak-dependent relaxation frequency. The constant relaxation approximation (ν _k =ν) will be used to discuss the drift velocitiesu for all the fields and temperaturesT for certain types of band structuresE(k). Bands with lineark-dependence for largek give rise to drift velocities saturating for large fields. For bands with the periodicity of the reciprocal lattice, the zero drift-theorem has been proved. It states that $$\mathop {\lim }\limits_{F \to \infty } u (F,T) = \mathop {\lim }\limits_{T \to \infty } u (F,T) = 0$$ for all the periodic band structures. This theorem is even correct for a generalW _kk′ if certain restrictions are made. Finally, making use of the Markov character of the conditional probability (Green's function) solution of the Boltzmann equation, the velocity fluctuation spectrumS is calculated forE(k)=A(1?cosa k). It will be shown thatS(F, T, 0) remains positive for the critical field and all temperatures, and therefore the noise temperature diverges on approaching the critical field.