Showing papers on "Ladder operator published in 1977"

Operator orderings and functional formulations of quantum and stochastic dynamics

Abstract: We study the connection between operator ordering schemes and thec-number formulations of quantum mechanics, which are based on generating functionals and functional integrals. We show by explicit construction that different operator ordering schemes are related to different functional and functional integral formulations of quantum mechanics. The results of these considerations are applied to classical non-linear stochastic dynamics by using the formal analogy between the Fokker-Planck equation and the Schrodinger equation.

Quantum mechanics in discrete space and angular momentum

Abstract: Recently we have studied quantum mechanics of bounded operators with a discrete spectrum. In particular, we derived an expression for the commutator[Q, P] of two bounded operators whose spectrum is discrete, and we showed that in the limit of a continuous spectrum the commutator becomes the standard one of Heisenberg. In this paper we show that the angular momentum operator and the phase operator satisfy the new commutation relation. We also briefly discuss the problem of the canonical phase operator conjugate to the number operator.

The quantum mechanical representations of the anisotropic harmonic oscillator group

Abstract: A group G associated with the n‐dimensional anisotropic harmonic oscillator is constructed : G is essentially a group generated by the position and momentum observables, the identity operator, and the Hamiltonian of the system. All the quantum mechanical irreducible representations of G are evaluated, using Mackey’s theory of induced representations.

Some aspects of the N‐representability problem in finite dimensions. II. Operator endomorphisms which induce sufficient conditions

Abstract: Given the set SN of all operator endomorphisms T with the property that whenever D is any 2‐density operator, then T (D) is N‐representable is considered. We specify a pair {Q2,PN} of extreme points of SN which are related to each other via the particle‐hole duality (Theorem 1.2). The rest of the paper is devoted to a complete characterization of the set ŜN of all operator endomorphisms T with the property that whenever D is any 2‐density operator then T (D) is quasi‐N‐representable, i.e., satisfies the necessary conditions Id, QN, BN, and CN.

Equations-of-motion formulation of many-body perturbation theory

Abstract: A formulation of many-body perturbation theory starting from the operator equation (H,Q+)= omega Q+ is presented. The method of solution is based on an operator scalar product, (X/Y)=Tr(X+Y), which allows the use of resolvent and partitioning techniques to establish Rayleigh-Schrodinger or Brillouin-Wigner perturbation theory for the excitation energy and excitation operator, Q+. The excitation operator contains all information about the two states involved in the transition. Specific results are given for removal or addition of an electron and for excitations of particle-hole type and comparison with the Green's function methods is made.

Operator density current and relativistic localization problem

Abstract: The well‐known difficulties with the relativistic localization in quantum theory of one particle are revisited. Among them the noncausal propagation is shown to be connected with the positivity of the time translations generator. A proposal for solving this problem is presented, namely searching for an operator probability density current being the 4‐vector quantity. The operator of multiplication by the argument of the Dirac wavefunction is an example of a causal position operator.

On the Lie-Trotter theorems inL p -spaces

Abstract: In this paper we consider operatorsH0 andV possessing the following properties: (1) H0 is a positive self-adjoint operator acting inL2(M, γ) with γ a probability measure, so that exp(−tH0) is a contraction onL1(M, γ) for eacht>0. (2) V is a semibounded multiplicative operator acting inL2(M, γ) {fx379-1}

Nearest normal approximation for certain operators

Abstract: ABSTRACr. In this paper we show that binormal operators have nearest normal approximants. In fact, we exhibit nearest normals to such operators and, as a corollary, show that the hermitian part of a binormal operator with real spectrum is a nearest normal. We obtain further corollaries on nearest normal approximation to operators which are square roots of normal operators and then apply these results to perturbations of operator algebras.

On the noise operator approach to quantized dissipative systems

Abstract: Invoking complex classical coordinates and momenta a consistent Hamiltonian theory suitable for the quantization of dissipative systems has been developed previously. In another paper this formalism has been illustrated on the basis of a simple order parameter equation by means of density operator techniques. This quite naturally calls for a comparison with quantum noise operator techniques. The present paper is an attempt to satisfy these demands. Extensive use will be made of operator ordering techniques and quasi-classical Fokker-Planck equations. As before, a certain incompleteness in the extractable information is clearly exhibited. It will be observed that the two techniques do not produce similar results in a general dynamical state as a consequence of dissipation. However, in the stationary state and within certain approximations both methods do lead to identical conclusions for the order parameters statistics. It will be argued that within the present context in general noise operator techniques are to be favoured.

A note on the nature of the phonon collision operator

Abstract: The nature of a linearized phonon collision operator is examined. It is shown that such an operator is compact and completely continuous. The off-diagonal part is diagonal-compact. The essential spectrum of the full operator is absolutely continuous and covers the range ( mu , lambda ), where mu and lambda are, respectively, the minimum and maximum of row sums of a projected part of the full operator.

Matrix elements and their selection rules from ladder operator considerations

Abstract: Within the Schrodinger–Infeld–Hull factorization framework it is shown that, by introducing a parameter e in the quantization condition, that is, e(j–|m|)=integer ≥ 0, and, thus, considering “symmetrized” ladder operators, one can use the same formulas to handle both class I (e = +1) and class II (e = −1) problems. Starting from this unified point of view, after building up the associated angular momentum operators and their e-dependent eigenfunctions, one unique closed-form expression of the coupling coefficients is obtained. This expression embodies many sparse and known previous results, without being more intricate than any of them. The basic material, allowing the application of a Wignera–Eckart theorem to matrix elements of an operator on the basis of eigenfunctions of factorizable equations, and a quick determination of the associated selection rules are given. Some examples are treated as an illustration.

Some essentially self-adjoint Dirac type operators i (formulation and an extension of the rellich-kato theorem)

Abstract: It has been shown by Rellich, Weidman, and Gustafson-Rejto that the one-electron Dirac operator is essentially self-adjoint on the domain of infinitely differentiable functions with compact support, for atomic numbers less than or equal to 118. We state a double perturbation theorem which shows that the one-electron Dirac operator can admit another perturbation in addition to the Coulomb potential, which satisfies a mild Stummel type bound. In addition, the domain of the closure of the perturbed operator is the same as the domain of the closure of the unperturbed operator.