Showing papers on "Quantum state published in 1974"

Generalized quantum mechanics

Abstract: A convex scheme of quantum theory is outlined where the states are not necessarily the density matrices in a Hilbert space. The physical interpretation of the scheme is given in terms of generalized “impossibility principles”. The geometry of the convex set of all pure and mixed states (called a statistical figure) is conditioned by the dynamics of the system. This provides a method of constructing the statistical figures for non-linear variants of quantum mechanics where the superposition principle is no longer valid. Examples of that construction are given and its possible significance for the interrelation between quantum theory and general relativity is discussed.

Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory

Abstract: Roberts' proposal of a rigged Hilbert space Φ⊂G⊂Φ× for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces which might be of interest for the application of this formalism to special quantum systems. It is shown that on the basis of this proposal one also obtains a satisfactory solution for a rigged Hilbert space for composite systems. Another part is concerned with topological properties of the so‐called eigenoperators γ(t) belonging to an A‐eigen‐integral decomposition of Φ with respect to a self‐adjoint operator A on G. We derive a representation of γ(t) in terms of the generalized eigenvectors of A and in the same context give a rough topological characterization of these eigenvectors.

Quantum mechanics of systems periodic in time

Abstract: Some expressions for the time evolution of quantum-mechanical systems with Hamiltonians periodic in time, derivable from the work of Shirley and applied by Young, Deal, and Kestner and Haeberlen and Waugh---all for finite-basis-set systems---are derived for a general system (possibly infinite Hilbert space). These results suggest a new type of approximation to the time-evolution operator, one which is exact at multiples of the period of the Hamiltonian. Comparison is made to an exactly soluble problem, namely, a nonrelativistic hydrogen atom in a circularly polarized monochromatic field.

Hilbert space approach to the quantum mechanical three-body problem

Abstract: 2014 We study the quantum mechanical three-body problem in n-dimensional space (n > 3) with pair potentials that decrease at infinity as x ,-(2+£). We work in configuration space and use only Hilbert space methods, in particular Kato’s theory of smooth operators and Agmon’s a priori estimates in weighted Hilbert spaces. We recover most of Faddeev’s results. We prove in particular that the negative spectrum of H consists, besides the expected absolutely continuous part, of isolated eigenvalues of finite multiplicities which can accumulate at most at zero and at the two-body thresholds from below. The positive singular spectrum is contained in a closed set of measure zero, and the wave operators are asymptotically complete. INTRODUCTION The scattering and spectral theory of the quantum mechanical twobody problem has recently been brought into very satisfactory condition in the case of short range potentials, namely potentials v(x) that decrease at infinity as x ~ ~ 1 + £~, 8 > 0. It has been proved long ago that the wave operators (*) Laboratoire associe au Centre National de la Recherche Scientifique. Postal~address: Laboratoire de Physique Theorique et Hautes Energies, Batiment 211, Université de Paris-Sud, 91405 Orsay (France). Annales de l’Institut Henri PoMcarc-Section A Vol. XXI, n° 2 1974. 8 98 J. GINIBRE AND M. MOULIN where ho is the free hamiltonian and h = ho + v, exist in this case (see for instance [7] and references therein contained). Furthermore, the ranges 9l(W:f:) of the are contained in the subspace of absolute continuity of h : The problem of proving that equality in fact holds in (2), commonly referred to as that of asymptotic completeness, has been studied by several authors and was finally solved for general short range potentials by Kato ([2], p. 206). Under general assumptions on v, it is easy to see that the essential spectrum of h is 6e(h) _ [0, oo). The next question is to determine whether this part of the spectrum is absolutely continuous, as physical intuition suggests. The absence of positive eigenvalues has been proved for short range potentials satisfying mild additional regularity conditions [3] [4]. The absence of singular continuous spectrum (together with the absence of positive eigenvalues and with asymptotic completeness) had been proved earlier in the fundamental paper of Ikebe [5], for potentials decreasing at infinity as x ~ -t2 +E~. The absence of singular continuous spectrum for potentials has been proved recently by Agmon [6], see also [7] and [9]. These results have been extended to some classes of long range potentials under more special assumptions, in particular to repulsive and other potentials by commutator methods [8] [9], and to dilation analytic potentials [10] [77]. The corresponding problems for three-body and more generally N-body systems interacting via two-body forces are still at a much less advanced stage. The existence of the wave operators has been proved by the same methods and for the same potentials as in the two-body case (see for instance [1]). The essential spectrum of the hamiltonian H has been proved to be what physical intuition suggests, namely 6e(H) = [Eo, oo) where Eo is the lowest two-body threshold [12]. For the other (more difficult) problems, the situation is much less satisfactory. In the three-body case, asymptotic completeness and the absence of singular continuous spectrum have been proved in the fundamental work of Faddeev [13] for potentials that decrease at infinity as x 1(3 + £) in three dimensions. These results have been partially extended to the N-body system by Hepp and coworkers [l4] [7~] ] under additional technical assumptions. The results obtained for repulsive or dilation analytic potentials have been extended to the N-body case [8] [76] [17] [77]. In the present paper, we shall take up the three-body problem in the same spirit as Faddeev, but depart from his methods in the following respects : ( 1 ) Faddeev works in momentum space, and uses Banach spaces of Holder-continuous functions of the momenta. These spaces are rather Annales de l’Institut Henri Poincaré Section A 99 HILBERT SPACE APPROACH TO THE QUANTUM MECHANICAL THREE-BODY PROBLEM difficult to handle, and have no direct physical meaning. We shall instead work in configuration space, and use exclusively Hilbert spaces. A number of well established techniques is then available, in particular Kato’s theory of smooth operators [7~] and Agmon’s method using weighted Hilbert spaces [6]. We shall use both. (2) In the two-body problem, it is convenient to use a symmetrized form of the resolvent equation (see equation (3 .1 ) of this paper). This is even more true in the three-body problem, where this idea has already been introduced by Newton for similar purposes [19]. We shall also make use of it. (3) With the methods described above, it turns out that one can handle potentials that decrease at infinity as x ~ ~2 + E~ in n-dimensional space, for n >_ 3. With this assumption (more precise formulations of which are contained in section 1. B), we are able to recover most of Faddeev’s results, with the exception that we are not able to prove the absence of positive singular continuous spectrum for H. We prove in particular that the negative spectrum of H consists, besides the expected absolutely continuous part, of isolated eigenvalues of finite multiplicities which can accumulate only at zero and at the two-body thresholds (i. e. at the bound state energies of the two-body subsystems) from below (proposition (7. 2)). The positive singular spectrum is contained in a closed set of measure zero (proposition (6. 4)), and the wave operators are asymptotically complete (proposition (8.4)). It appears that for negative energy, the analytical difficulties of the three-body problem are essentially the same as those of the two-body problem. In particular, we can also prove that for potentials that decrease at infinity as x ~ ~ 1 + ~}, the negative singular continuous spectrum of H is empty and that the negative point spectrum consists of eigenvalues of finite multiplicities which can accumulate at most at the two-body thresholds and at zero (proposition (7.3)). For such potentials, however, we obtain no information on the positive spectrum.. The paper is organized as follows. Section 1 contains some preliminary definitions and properties, namely kinematics (section 1. A), the conditions on the potentials (section 1. B) and the definition of the Hamiltonian (section 1. C). In section 2, we collect the basic estimates from Agmon’s method in the form that is most useful for our purpose, and some extensions which are needed for the three-body case. In section 3, we study the twobody problem, using exclusively the methods that carry over to the threebody problem. This section therefore contains no new result, except perhaps for the fact that for M ~ 3 the number of bound states, including those with positive energies, is finite for potentials decreasing at infinity as x ~ ~ 2 + ~~. (See [20] for estimates on the number of negative energy bound states under similar assumptions on the potential). In section 4, we begin the study of the three-body problem itself by deriving a modified form of the Vol. XXI, n° 2 1974. 100 J. GINIBRE AND M. MOULIN Faddeev equations and setting up an algebraic formalism to construct the resolvent operator. Special attention is paid to the bound states of the two-body subsystems. In section 5, we derive the basic properties of the kernels of the modified Faddeev equations, namely uniform boundedness, Holder-continuity and compactness in the closed cut plane, and analyticity in the open cut plane. We then apply the analytic Fredholm theorem to these equations. In section 6 we consider the associated homogeneous equations. We prove that outside of the essential spectrum, their solutions are in one to one correspondence with the bound states of H, while on the essential spectrum, they vanish on the energy shell. We also construct the resolvent operator. In section 7, we study the negative part of the spectrum of H and prove the results mentioned above. In section 8, we express the wave operators, or rather their adjoints, and the spectral projectors of H on absolutely continuous subsets of the spectrum, in terms of the resolvent operator. We then prove asymptotic completeness. Technical estimates are collected in Appendices A and B. 1. PRELIMINARIES In this section, we collect some definitions and results which will be used throughout the paper. Section 1. A is devoted to kinematics, section l.B _ contains the conditions fulfilled by the interactions, and section 1. C is devoted to the definition of the Hamiltonian. A Kinematics. We consider a system of three non relativistic particles in n-dimensional space (n ;;::: 3). Particles will be labelled by latin indices i, j, etc. running from 1 to 3. Pairs of particles will be labelled by greek indices oc, /?, etc. running over ( 12), (23), (31 ). We denote by mi the mass of particle i, by M the total mass of the system (M = m1 + m2 + m3), by ma the reduced mass of the pair a (~ 1 = + m 1 if x = (i, j)), and by na the reduced mass of the pair x and of the third particle (na 1 = (m~ + mJ)-’ + 1 We denote by xi the position of particle i, by xa the relative position of the particles in the pair a if a = (i, j)) and by ya the relative position of the third particle with respect to the center of mass of the pair ex. We denote by X the set of internal coordinates of the system : X = (xa, ya) for any oc. The (xa, y03B1) for different values of 03B1 are connected by well-known formulas. Typically : Annales de l’Institut Henri Poincare Section A 101 HILBERT SPACE APPROACH TO THE QUANTUM MECHANICAL THREE-BODY PROBLEM We shall use the volume element dX = dx«dy« (any a) or equivalently dX = dx03B1dx03B2 (any 03B1 ~ 03B2) in coordinate space. One easily checks that the Jacobians are equal to one. After separat

Interfering electron quantum states in ultrapure magnesium

Abstract: A detailed analysis is presented of the large-amplitude temperature-insensitive oscillations in the transverse magnetoresistivity of ultrapure Mg single crystals resulting from the direct interference of electron quantum states. A calculation of the relative harmonic content of these interference signals based on the transmission characteristics of a magnetic breakdown-generated interferometer is used to quantitatively study certain aspects of the electron states in Mg as well as details of magnetic breakdown. In particular, values of magnetic breakdown parametersH 0are determined without invoking the complexities of transport theory. An absolute lower limit for the electron quantum state lifetime of τ ≳ 0.5 nsec is obtained (forT=1.5° K), although a best fit to the data gives a value an order of magnitude larger, τ∼5 nsec, which corresponds to quantum phase coherence extending over a distance of 3 mm in these crystals. In addition, this work provides direct experimental verification of the π/2 phase difference between transmitted and reflected electron states at a magnetic breakdown junction. Comparison of the results of this experiment with previous work via an existing semiempirical band structure calculation demonstrates the complete consistency of these measurements with previous Fermi surface data.

Functional properties of quantum logics

Abstract: A quantum logic is defined as a setL of functions from the set of all statesS into [0, 1] satisfying the orthogonality postulate: for any sequencea1,a2, ... of members ofL satisfyingai+aj≤1 fori≠j there isb∈L such thatb+a1+a2+...=1. Every logicL is in a natural way an orthomodular σ-orthocomplemented partially ordered set (L, ≤, ′) with members ofS inducing a full set of measures onL. It is shown that a logicL is quite full if and only if (L,≤,′) is isomorphic to an orthocomplemented set lattice of subsets ofS. Sufficient conditions are given in order that a quite full logic be representable in the set of projection quadratic formsf(u)=(Pu, u) on a complex Hilbert space, or in the set of trace functionsf(A)=Trace (AP) generated by projectionsP, where the domain off is the set of non-negative self-adjoint trace operators of trace 1 in a complex Hilbert space.

The quantum probability calculus

Abstract: Quantum mechanics has opened a vast sector of physics to probability calculus. In fact most of the physical interpretation of the formalism of quantum mechanics is expressed in terms of probability statements.1

Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. II. Transformation theory in nonrelativistic quantum mechanics

Abstract: Results of a previous paper are used to obtain a rigorous mathematical formulation of the transformation theory of nonrelativistic quantum mechanics within the framework of rigged Hilbert spaces.

Correlations without joint distributions in quantum mechanics

Abstract: The use of joint distribution functions for noncommuting observables in quantum thermodynamics is investigated in the light of L. Cohen's proof that such distributions are not determined by the quantum state. Cohen's proof is irrelevant to uses of the functions that do not depend on interpreting them as distributions. An example of this, from quantum Onsager theory, is discussed. Other uses presuppose that correlations betweenp andq values depend at least on the state. But correlations may be fixed by the state even though the distribution varies from one ensemble to another represented by that state. Taking covariance as a measure of correlation, it is shown that the different commonly used joint distributions yield the same correlations for a given state. A general characterization is given for a family of distributions with this same covariance.

Van Fraassen's Modal Model of Quantum Mechanics

Abstract: Bas van Fraassen in [4] has recently tried to use modal logic to solve the measurement problem of quantum mechanics. His model is based on a method of expressing quantum states developed by Hugh Everett [1] called the "relative state formulation." Unfortunately, Everett's mathematics cannot be generalized as van Fraassen requires. The difficulty itself is elementary enough. But a revision of van Fraassen's postulates (such as the one he adopted on p. 341 after he had been told of the problem) can save the mathematics only on pain of making the whole study irrelevant to the physics. I shall explain the dilemma in this note. In a later paper [5], van Fraassen avoids this dilemma by developing an abstract modal interpretation which does not rely on any expression for quantum states. In van Fraassen's view, the problem of measurement is to reconcile the contradiction between the predictions of the Schrodinger equation for the final state of the combined system, object plus measuring apparatus, with the predictions provided by the "naive" account of measurement described by von Neumann, [6]. For an object initially in the state I Cr,r, the naive account tells us that if the pointer states are {i}, and the eigenstates of the measured observable are {}J, then the final state of the composite will be the mixture

The quantum mechanical Hilbert space formalism and the quantum mechanical probability space of the outcomes of measurements

Abstract: The relation between the quantum mechanical descriptive basis in its Hilbert space formulation and the quantum mechanical probability space is analysed The EVERETT WHEELER GRAHAM DE WITT interpretation of the quantum mechanical formalism, GLEASON's theorem on the measure of the closed subspaces of a Hilbert space, and the theory of probabilities are used as a combined system of reference The conclusion is reached that the quantum mechanical probability law is not obtainable in a purely deductive way from the basic descriptive system of quantum mechanics : the link between the quantum mechanical basis and the quantum mechanical probability law is insured by an independant postulate, of which the complex semantic contents and syntactic structure are established in detail The performed analysis brings forth new perspectives and an efficient methodologic fact

Quantum mechanics presented as harmonic analysis

Abstract: This paper grew out of a desire on the part of its author to be able to explain, for philosophy, the significance of Quantum Mechanics. The traditional formulation, based on Hilbert spaces and operators on them, leaves much to be desired if, for example, one is interested in the theory of physical measurement. (It may fairly be asked what the operation on a state function of partial differentiation really has to do with the actual business of measuring the momentum associated with that state function.)

The relevance of the quantum-mechanical space-time description

Abstract: I t has been pointed out that under the conditions on which the energy spectrum is limited only to positive values, Pauli 's objection becomes inopcrant so that the necessity and possibility of defining a quantum-mechanical time operator may be well stated (1). In this context we shall extend the viewpoint of (1) proving that quantum mechanics permits not only the existence of the time operator, but implicitly offers the possibility of defining an objective imprecision bound on the time measurements. This lat ter idea refers in fact to the problem of the definition of th~ b inary time operators, whose meaning has been previously analysed (2). Besides the possibility of defining the time operator by vir tue of the correspondence principle (3), wc shall actually investigate some other formulations, supporting the programme of the quantummechanical space-time description. For simplicity we shall neglect the spin. The quantum-mechanical momentum and energy operators may be defined as the generators of the infinitesimal space and time translations, if we ohange the topics it is quite natura l to define the space and time operators as the generators of the infinitesimal translations performcd in the p-momentum and po-energy planes, respectively. If we restrict ourselves to positive momentum and energy values, a well-dcfined value of the angular momentum is implied. Thus, the translat ion

Symmetry in Quantum Systems

Abstract: Homological algebra is used to derive some results in the theory of Lie groups.