A Particle in a Magnetic Field of an Infinite Rectilinear Current

We consider the Schrödinger operator H=(i+A)² in the space L ₂(R ³) with a magnetic potential A created by an infinite rectilinear current. We show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(–i H t)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.     

Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2m)∑ p _i ² with hermitian form of Cartesian momentum p _i (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2m)∑ (1/f _i )p _i f _i p _i where the functions f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. The existence of non-trivial f _i shows the universality of this constraint induced operator ordering in quantum kinetic energy operator for the constraint systems.     

Quantum Motion on 2D Surfaces of Spherical Topology

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2μ) ∑ p _i ² with hermitian form of Cartesian momentum p _i (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2μ) ∑ (1/f _i )p _i f _i p _i where the functions f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this article, the explicit forms of the dummy functions f _i for quantum motion on some 2D surfaces of revolution of spherical topology are given. PACS numbers: 03.65.-w Quantum mechanics, 04.60.Ds Canonical quantization.     

Spectral Properties of the Periodic Magnetic Schrödinger Operator in the High-Energy Region. Two-Dimensional Case

The goal is to investigate spectral properties of the operator H=(–i +a(x))²+a₀(x) in the two-dimensional situation, a(x), a₀(x)) being periodic. We construct asymptotic formulae for Bloch eigenvalues and eigenfunctions in the high-energy region, describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the Bethe-Sommerfeld conjecture.Research partially supported by USNSF Grant DMS-0201383.Acknowledgements The author is thankful to Konstantin Makarov for very useful discussions and to Young-Ran Lee for her great help with pictures.     

Quantum mechanical version of classical Liouville theorem

In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of coherent state from |z〉to |sz-rz^*〉angle corresponds to the motion from a point z(q,p) to another point sz-rz^* with |s|²-|r|²=1. The evolution is governed by the so-called Fresnel operator U(s,r) recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation and obeys the group product rules. In another word, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.     

A bound on the number of bound states in the Schrödinger equation

A boundS _l is given for the number of bound statesn _i in thelth partial wave corresponding to a spherically symmetric potential in non-relativistic quantum mechanics. This bound is given by whereV _a(l, r) is the attractive part of the effective potentialV(r)+l(l+1)/r ². Extensive comparative study ofS _i and the Bargmann inequality is made.     

Entangled Husimi Distribution and Complex Wavelet Transformation

Similar in spirit to the preceding work (Int. J. Theor. Phys. 48:1539, 2009) where the relationship between wavelet transformation and Husimi distribution function is revealed, we study this kind of relationship to the entangled case. We find that the optical complex wavelet transformation can be used to study the entangled Husimi distribution function in phase space theory of quantum optics. We prove that, up to a Gaussian function, the entangled Husimi distribution function of a two-mode quantum state |ψ〉 is just the modulus square of the complex wavelet transform of e^-|h|2/2e^{-\vert \eta \vert ^{2}/2} with ψ(η) being the mother wavelet.     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

Asymptotic Properties of Solutions to 3-Particle Schrödinger Equations

We construct a generalized Fourier transformation ℱ(λ) associated with the 3-body Schr?dinger operator H=−Δ+Σ ^a V ^a (x ^a ) and characterize all solutions of (H−λ)u= 0 in the Agmon–H?rmander space ℬ^* as the image of ℱ(λ)^*. These stationary solutions admit asymptotic expansions in ℬ^* in terms of spherical waves associated with scattering channels. Received: 20 September 2000 / Accepted: 20 May 2001     

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on with Hamiltonian H(x)=∑ _i,j J _i,j (x(i)−x(j))²+h∑ _i (x(i)−d(i))², where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ ^d . The configuration d is given and considered as observations. The ‘couplings’ J _i,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.     

Fock-Space Projector Studied in Weyl Ordering Approach

We find that the Fock space projector |n〉〈n| is a Weyl ordered Laguerre polynomial 2 ^:_:(-)ⁿL_n ( 4a^fa ) e^-2afa :_:2{\,}^{:}_{:}(-)^{n}L_{n} ( 4a^{\dagger}a ) e^{-2a^{\dagger}a}{\,}^{:}_{:}, where a ^† a is the number operator,^:_: ^:_:,{}^{:}_{:}\ {}^{:}_{:} denotes the Weyl ordering symbol. This brings convenience to derive the Wigner functions of many other quantum states.     

Theory and computer simulation of tweed texture

Abstract

An analytical theory of the ordering interaction J(R _ij ) in structural phase transitions mediated by elastic relaxation in the material is outlined. The ordering process in cell i sets up a local stress field due to the sizes, shapes or displacements of atoms or atomic groups, which is propagated elastically to a distant cell j. The atomistic theory for ferro- and antiferro-elastic transitions takes into account two types of singularity, one due to elastic anisotropy and the other to the Zener interaction J _z of infinite range in ferroelastic transitions. The form of J _k in Fourier space is highly anisotropic with a few “soft” directions coinciding with the orientation of twin boundaries. The asymptoptic J(R) at large R is shown to be very anisotropic as well and decays as R ^?3 in ferroelastic and R ^?5 in antiferroelastic systems.

Computer simulations for a three-dimensional model of about 29,000 particles show a strong tendency to form tweed texture, as observed experimentally. Well above the structural phase transition temperature, the strain fluctuations show well-developed embryos of the tweed texture. On quenching to below the transition temperature, a pronounced micro-twinning appears which follows almost exactly the shape of the embryos and then develops towards a stripe texture. After a certain time needle-shaped domains are formed and a peculiar step-wise process of generating new stripes is observed.     

General quantum mechanical canonical point transformations

Problems related to the operator form of the generalized canonical momenta in quantum mechanics are resolved by use of the general quantum mechanical canonical point transformation method. This method can be applied to any general canonical point transformation irrespective of the relationship between the domains of the original and transformed variables. The differential representation of the original canonical momenta p_i in the original coordinate space is ?i \(\begin{array}{*{20}c} / \\ h \\ \end{array}\) ?/?x _i and of the transformed canonical momentap _i ′ in the transformed coordinate space is ?i \(\begin{array}{*{20}c} / \\ h \\ \end{array}\) ?/?x _i ′. Relationships are derived between the eigenvalues of the original and transformed momenta in either space. The ordering problem for general point transformations is discussed and is shown to be solved. As an example of the generality of the method, it is demonstrated on the point transformation in three dimensions from Cartesian rectilinear to spherical rectilinear coordinates.     

The Geometry of¶(Super) Conformal Quantum Mechanics

N-particle quantum mechanics described by a sigma model with an N-dimensional target space with torsion is considered. It is shown that an SL(2,ℝ) conformal symmetry exists if and only if the geometry admits a homothetic Killing vector D ^a δ _a whose associated one-form D _a dX ^a is closed. Further, the SL(2,ℝ) can always be extended to Osp(1|2) superconformal symmetry, with a suitable choice of torsion, by the addition of N real fermions. Extension to SU(1,1|1) requires a complex structure I and a holomorphic U(1) isometry D ^a I _a ^b δ _b . Conditions for extension to the superconformal group D(2,1;α), which involve a triplet of complex structures and SU(2)×SU(2) isometries, are derived. Examples are given. Received: 3 September 1999 / Accepted: 30 January 2000     

Conservation laws for classical and relativistic collisions. I

On the basis of simple kinematic arguments it is shown that any quantity, depending only on the nature and velocity of a particle, that is conserved in a collision must, in classical mechanics, be of the form λ+Σ_iμ_iυ_i+1/2vυ ² or in relativistic mechanics of the form λ+Σ_iμ_iυ_i[1−(υ ²/c ²)]^−1/2+νc [1−(υ ²/c ²)]^−1/2 where λ,μ _i, andν are particle parameters.     

Zerfallende Zustände als physikalisch nichtisolierbare Teilsysteme

Presently the investigations of decaying quantum mechanical systems lack a well-founded concept, which is reflected by several formal difficulties of the corresponding mathematical treatment. In order to clarify in some respect the situation, we investigate, within the framework of nonrelativistic quantum mechanics, the resonant scattering of an initially well localized partial wave packet ϕ_l(r, t). If the potential decreases sufficiently fast for r → ∞, ϕ_l(r, t) can be expressed at sufficiently long time after the scattering has taken place, as ϕ_l(r, t) = I(r, t) + ∑ N_iϕl(K_i, r) exp {–iK_i² t/2M} × Θ(k_i – γ_i – Mr/t), ϕ_l(K_i, r) being the resonant solution with complex “momentum” K_i = k_i – iγ_i. From this heuristic relation one can deduce not only the probability for the creation of unstable particles but also obtain some hints to a connection between decaying states and physically nonisolable partial systems. On the other hand, this connection can perhaps display the inadequacy of attempts which suggest to solve the problem of decaying states within the usual Hilbert space methods.     

Reentrant transitions in the van Hemmen model of a spin glass

The van Hemmen model of a spin glass, which is an Ising model with random couplings J_ij between sites i and j equal to J₀ + J(ξ_iη_j + ξ_jη_i), where (ξ_i, η_i) are independent, identically distributed random variables, is studied in the pair approximation of the cluster variation method. For the family of probability distributions (1 − p)δ(ξ_i − a) + pδ(ξ_i) + (1 − p)δ(ξ_i + a), where p is varied, phase diagrams are constructed. They are qualitatively different from the mean-field phase diagrams and display a competition between tendencies towards spin-glass and towards ferromagnetic ordering, which results in reentrant transitions. It is argued that the observed effects are not accidental but are borne by the competition of bonds of the underlying lattice system.     

Asymptotics of Determinants of Bessel Operators

 For aL ^∞ (ℝ₊)∩L ¹ (ℝ₊) the truncated Bessel operator B _τ (a) is the integral operator acting on L ² [0,τ] with the kernel