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Top 20 Most Cited Articles
The 20 most cited articles over time based on CrossRef data.
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Time‐Dependent Statistics of the Ising Model J. Math. Phys. 4, 294 (1963); http://dx.doi.org/10.1063/1.1703954 (14 pages) | Cited 1467 times Online Publication Date: 22 December 2004
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The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N‐member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time‐varying magnetic field upon the model is discussed, and the frequency‐dependent magnetic susceptibility is found in the weak‐field limit. Some fluctuation‐dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time‐dependent correlation function of the magnetization at equilibrium. |
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J. Math. Phys. 6, 167 (1965); http://dx.doi.org/10.1063/1.1704269 (15 pages) | Cited 1015 times Online Publication Date: 22 December 2004
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Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points. The number of distinct points visited after n steps on a k‐dimensional lattice (with k ≥ 3) when n is large is a1n + a2n☒ + a3 + a4n−☒ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face‐centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited. The probability F(c) that a walker on a one‐dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c∕(1 − c)] log c. Most of the results in this paper have been derived by the method of Green's functions. |
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An Approach to Gravitational Radiation by a Method of Spin Coefficients J. Math. Phys. 3, 566 (1962); http://dx.doi.org/10.1063/1.1724257 (13 pages) | Cited 964 times Online Publication Date: 22 December 2004
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A new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of Ricci rotation coefficients which give, in effect, the spinor affine connection. It is applied to two problems in radiation theory; a concise proof of a theorem of Goldberg and Sachs and a description of the asymptotic behavior of the Riemann tensor and metric tensor, for outgoing gravitational radiation. |
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Brownian Motion of a Quantum Oscillator J. Math. Phys. 2, 407 (1961); http://dx.doi.org/10.1063/1.1703727 (26 pages) | Cited 897 times Online Publication Date: 22 December 2004
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An action principle technique for the direct computation of expectation values is described and illustrated in detail by a special physical example, the effect on an oscillator of another physical system. This simple problem has the advantage of combining immediate physical applicability (e.g., resistive damping or maser amplification of a single electromagnetic cavity mode) with a significant idealization of the complex problems encountered in many‐particle and relativistic field theory. Successive sections contain discussions of the oscillator subjected to external forces, the oscillator loosely coupled to the external system, an improved treatment of this problem and, finally, there is a brief account of a general formulation. |
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The Painlevé property for partial differential equations J. Math. Phys. 24, 522 (1983); http://dx.doi.org/10.1063/1.525721 (5 pages) | Cited 800 times
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In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems. |
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Periodic Orbits and Classical Quantization Conditions J. Math. Phys. 12, 343 (1971); http://dx.doi.org/10.1063/1.1665596 (16 pages) | Cited 773 times Online Publication Date: 28 October 2003
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The relation between the solutions of the time‐independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium. |
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The Zeno’s paradox in quantum theory J. Math. Phys. 18, 756 (1977); http://dx.doi.org/10.1063/1.523304 (8 pages) | Cited 670 times Online Publication Date: 26 August 2008
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We seek a quantum‐theoretic expression for the probability that an unstable particle prepared initially in a well defined state ρ will be found to decay sometime during a given interval. It is argued that probabilities like this which pertain to continuous monitoring possess operational meaning. A simple natural approach to this problem leads to the conclusion that an unstable particle which is continuously observed to see whether it decays will never be found to decay!. Since recording the track of an unstable particle (which can be distinguished from its decay products) approximately realizes such continuous observations, the above conclusion seems to pose a paradox which we call Zeno’s paradox in quantum theory. The relation of this result to that of some previous works and its implications and possible resolutions are briefly discussed. The mathematical transcription of the above‐mentioned conclusion is a structure theorem concerning semigroups. Although special cases of this theorem are known, the general formulation and the proof given here are believed to be new. We also note that the known ’’no‐go’’ theorem concerning the semigroup law for the reduced evolution of any physical system (including decaying systems) is subsumed under our theorem as a direct corollary. |
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J. Math. Phys. 10, 1458 (1969); http://dx.doi.org/10.1063/1.1664991 (16 pages) | Cited 670 times Online Publication Date: 4 November 2003
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The theory of explicitly time‐dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well‐posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time‐dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time‐dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time‐dependent uniform charge distribution. A class of explicitly time‐dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas. |
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Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension J. Math. Phys. 1, 516 (1960); http://dx.doi.org/10.1063/1.1703687 (8 pages) | Cited 667 times Online Publication Date: 22 December 2004
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A rigorous one‐one correspondence is established between one‐dimensional systems of bosons and of spinless fermions. This correspondence holds irrespective of the nature of the interparticle interactions, subject only to the restriction that the interaction have an impenetrable core. It is shown that the Bose and Fermi eigenfunctions are related by ψB=ψFA, where A(x1 … xn) is +1 or −1 according as the order pq … r, when the particle coordinates xj are arranged in the order xp<xq< … <xr, is an even or an odd permutation of 1 … n. The energy spectra of the two systems are identical, as are all configurational probability distributions, but the momentum distributions are quite different. The general theory is illustrated by application to the special case of impenetrable point particles; the one‐one correspondence between bosons with this particular interaction and completely noninteracting fermions leads to a rigorous solution of this many‐boson problem. |
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J. Math. Phys. 36, 6377 (1995); http://dx.doi.org/10.1063/1.531249 (20 pages) | Cited 642 times
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According to ’t Hooft the combination of quantum mechanics and gravity requires the three‐dimensional world to be an image of data that can be stored on a two‐dimensional projection much like a holographic image. The two‐dimensional description only requires one discrete degree of freedom per Planck area and yet it is rich enough to describe all three‐dimensional phenomena. After outlining ’t Hooft’s proposal we give a preliminary informal description of how it may be implemented. One finds a basic requirement that particles must grow in size as their momenta are increased far above the Planck scale. The consequences for high‐energy particle collisions are described. The phenomenon of particle growth with momentum was previously discussed in the context of string theory and was related to information spreading near black hole horizons. The considerations of this paper indicate that the effect is much more rapid at all but the earliest times. In fact the rate of spreading is found to saturate the bound from causality. Finally we consider string theory as a possible realization of ’t Hooft’s idea. The light front lattice string model of Klebanov and Susskind is reviewed and its similarities with the holographic theory are demonstrated. The agreement between the two requires unproven but plausible assumptions about the nonperturbative behavior of string theory. Very similar ideas to those in this paper have long been held by Charles Thorn. © 1995 American Institute of Physics. |
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Statistical Theory of the Energy Levels of Complex Systems. I J. Math. Phys. 3, 140 (1962); http://dx.doi.org/10.1063/1.1703773 (17 pages) | Cited 641 times Online Publication Date: 22 December 2004
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New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time‐reversal invariance is violated, and the symplectic ensemble applies only to odd‐spin systems without rotational symmetry. The probability‐distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one‐dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group‐theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper. |
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An Exactly Soluble Model of a Many‐Fermion System J. Math. Phys. 4, 1154 (1963); http://dx.doi.org/10.1063/1.1704046 (9 pages) | Cited 629 times Online Publication Date: 22 December 2004
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An exactly soluble model of a one‐dimensional many‐fermion system is discussed. The model has a fairly realistic interaction between pairs of fermions. An exact calculation of the momentum distribution in the ground state is given. It is shown that there is no discontinuity in the momentum distribution in this model at the Fermi surface, but that the momentum distribution has infinite slope there. Comparison with the results of perturbation theory for the same model is also presented, and it is shown that, for this case at least, the perturbation and exact answers behave qualitatively alike. Finally, the response of the system to external fields is also discussed. |
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Exponential Operators and Parameter Differentiation in Quantum Physics J. Math. Phys. 8, 962 (1967); http://dx.doi.org/10.1063/1.1705306 (21 pages) | Cited 546 times Online Publication Date: 21 December 2004
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Elementary parameter‐differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker‐Campbell‐Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)∕dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated‐commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed‐form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed‐form single‐exponential solutions to the differential equation dY(t)∕dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple‐exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic‐oscillator canonical‐ensemble averages of general operators defined by the Weyl prescription. Solutions of the density‐matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian. |
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Completely positive dynamical semigroups of N‐level systems J. Math. Phys. 17, 821 (1976); http://dx.doi.org/10.1063/1.522979 (5 pages) | Cited 544 times Online Publication Date: 28 August 2008
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We establish the general form of the generator of a completely positive dynamical semigroup of an N‐level quantum system, and we apply the result to derive explicit inequalities among the physical parameters characterizing the Markovian evolution of a 2‐level system. |
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Lorentz Invariance and the Gravitational Field J. Math. Phys. 2, 212 (1961); http://dx.doi.org/10.1063/1.1703702 (10 pages) | Cited 535 times Online Publication Date: 22 December 2004
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An argument leading from the Lorentz invariance of the Lagrangian to the introduction of the gravitational field is presented. Utiyama's discussion is extended by considering the 10‐parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables. It is then unnecessary to introduce a priori curvilinear coordinates or a Riemannian metric, and the new field variables introduced as a consequence of the argument include the vierbein components hkμ as well as the ``local affine connection'' Aijμ. The extended transformations for which the 10 parameters become arbitrary functions of position may be interpreted as general coordinate transformations and rotations of the vierbein system. The free Lagrangian for the new fields is shown to be a function of two covariant quantities analogous to Fμν for the electromagnetic field, and the simplest possible form is just the usual curvature scalar density expressed in terms of hkμ and Aijμ. This Lagrangian is of first order in the derivatives, and is the analog for the vierbein formalism of Palatini's Lagrangian. In the absence of matter, it yields the familiar equations Rμν=0 for empty space, but when matter is present there is a difference from the usual theory (first pointed out by Weyl) which arises from the fact that Aijμ appears in the matter field Lagrangian, so that the equation of motion relating Aijμ to hkμ is changed. In particular, this means that, although the covariant derivative of the metric vanishes, the affine connection Γλμν is nonsymmetric. The theory may be reexpressed in terms of the Christoffel connection, and in that case additional terms quadratic in the ``spin density'' Skij appear in the Lagrangian. These terms are almost certainly too small to make any experimentally detectable difference to the predictions of the usual metric theory. |
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A method for determining a stochastic transition J. Math. Phys. 20, 1183 (1979); http://dx.doi.org/10.1063/1.524170 (19 pages) | Cited 490 times Online Publication Date: 29 July 2008
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A number of problems in physics can be reduced to the study of a measure‐preserving mapping of a plane onto itself. One example is a Hamiltonian system with two degrees of freedom, i.e., two coupled nonlinear oscillators. These are among the simplest deterministic systems that can have chaotic solutions. According to a theorem of Kolmogorov, Arnol’d, and Moser, these systems may also have more ordered orbits lying on curves that divide the plane. The existence of each of these orbit types depends sensitively on both the parameters of the problem and on the initial conditions. The problem addressed in this paper is that of finding when given KAM orbits exist. The guiding hypothesis is that the disappearance of a KAM surface is associated with a sudden change from stability to instability of nearby periodic orbits. The relation between KAM surfaces and periodic orbits has been explored extensively here by the numerical computation of a particular mapping. An important part of this procedure is the introduction of two quantities, the residue and the mean residue, that permit the stability of many orbits to be estimated from the extrapolation of results obtained for a few orbits. The results are distilled into a series of assertions. These are consistent with all that is previously known, strongly supported by numerical results, and lead to a method for deciding the existence of any given KAM surface computationally. |
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Solution of a Three‐Body Problem in One Dimension J. Math. Phys. 10, 2191 (1969); http://dx.doi.org/10.1063/1.1664820 (6 pages) | Cited 484 times Online Publication Date: 4 November 2003
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The problem of three equal particles interacting pairwise by inversecube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is solved in one dimension. |
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Thermodynamics of a One‐Dimensional System of Bosons with Repulsive Delta‐Function Interaction J. Math. Phys. 10, 1115 (1969); http://dx.doi.org/10.1063/1.1664947 (8 pages) | Cited 471 times Online Publication Date: 4 November 2003
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The equilibrium thermodynamics of a one‐dimensional system of bosons with repulsive delta‐function interaction is shown to be derivable from the solution of a simple integral equation. The excitation spectrum at any temperature T is also found. |
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The Einstein Tensor and Its Generalizations J. Math. Phys. 12, 498 (1971); http://dx.doi.org/10.1063/1.1665613 (4 pages) | Cited 468 times Online Publication Date: 28 October 2003
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The Einstein tensor Gij is symmetric, divergence free, and a concomitant of the metric tensor gab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors. |
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J. Math. Phys. 12, 419 (1971); http://dx.doi.org/10.1063/1.1665604 (18 pages) | Cited 465 times Online Publication Date: 28 October 2003
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The quantum‐mechanical problems of N 1‐dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and∕or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2∕(4m), the complete energy spectrum (in the center‐of‐mass frame) is given by the formula  |
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