Showing papers on "Ladder operator published in 1970"

Canonically Conjugate Pairs, Uncertainty Relations, and Phase Operators

Abstract: Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.

Landau Diamagnetism from the Coherent States of an Electron in a Uniform Magnetic Field

Abstract: A complete set of coherent-state wave packets has been constructed for an electron in a uniform magnetic field. These states are nonspreading packets of minimum uncertainty that follow the classical motion. Use was made of the ladder operators that generate all the eigen-states of the Hamiltonian from any one energy eigenstate. The coherent states are the eigenstates of the two ladder operators that annihilate the zero-angular-momentum ground state. We have calculated the partition function, exploiting advantages of the coherent-state basis. The Landau diamagnetism and the de Haas-van Alphen oscillations are contained in the coherent-state framework.

Projection operators in nuclear structure calculations

Abstract: Projection operators are an important tool in nuclear structure theory, because in many circumstances it is useful to construct wave-functions ψ which are not eigenfunctions of some operator Λ, although it is apparent that the physical states must be eigenstates of that operator. Thus one first constructs ψ and then projects from it onto eigenfunctions of Λ. We discuss the cases of angular momentum, isospin, centre of mass energy, particle number and antisymmetry. We describe the integral projection operator, an expansion in shift operators, the product operator of Lowdin and another product operator (the cosine product). Certain methods which appear in the literature are seen to be equivalent to one or the other of these. We consider factors that influence the choice of an appropriate method. Projection occurs frequently in the context of a variational method (such as Hartree-Fock or BCS). We consider the question of projection before or after variation.

The twisting operator in multi-Veneziano theory

Abstract: Using the operatorial expression for the gauge conditions, we derive an explicit form for the twisting operator and the semi-twisting operator which takes theP states intoV states. The expression for the last operator facilitates the proof of theV states factorization previously found by us. We show that the twisting operator proposed in the literature is inconsistent with multiple factorization,i.e. factorization of amplitudes with extermal spinning particles. The correct twisting operator depends on the integration variables of the twisted line, and automatically satisfies double-twist invariance.

On the scattering operator for quantum fields

Abstract: We study quantum fields interacting by the interactions usually considered in the theory of elementary particles. That is we take the interaction density to be a polynomialP in the fields, and assume thatP=Pb+Py+Pw,wherePb is a fourth order polynomial in the boson fields only,Py is linear in the boson fields andPw is a polynomial in the fermi fields only. After introducing a space and momentum cut-off in the interaction we prove that the scattering operator exists for all values of the cut-off parameters. We then introduce the scattering operators of relativistic quantum fields as weak limit points of cut-off scattering operators as the cut-off is taken away.

Relativistic Kinematics for N-particle Scattering States II. Quantum Mechanical Treatment

Abstract: The generalization of grand angular momentum to N-particle states with relativistic energies achieved classically in a previous paper is now transcribed into quantum mechanical language. The grand angular momentum are the generators Lij of a O(3N - 3)-invariance group of the invariant mass operator for N free particles. A radial momentum is defined as well as O(3N - 3)-invariant operators conjugate to radial momentum or total mass. A construction is made of wave functions spanning irreducible representation spaces of O(3N - 3) which are labelled by the eigenvalues of togetherness Λ2 = ½ Σij Lij2. Finally the sum of all those spaces is shown to be irreducible under a group O(3N - 3, 1)/t (where t is time reversal) which is an invariance group of the mass operator and its canonical conjugate.

Operator reduction of three-body scattering equations.

Abstract: Operator multipliers of the Lippmann-Schwinger (LS) equation are used to obtain an uncoupled equation with calculable and unique solutions. The multiplier method is used as a basis for determining the relation between various formulations of the three-body problem. The formal but calculable solutions of the formulations of Faddeev, Lovelace, Rosenberg, Noble, and Newton are found to be identical in the sense that the different formulations are completely equivalent because they all use the same multiplier. They are also identical to the solution of the equation obtained by multiplying the L-S equation by an operator whose inverse exists. We also present an equation for the exact three-body bound-state wave function and put it into calculable form with the use of a multiplier. In a calculation that uses an incomplete set of basis states (such as in the shell model), we find on rigorous grounds that it is appropriate to use the $t$ operator of the residual interaction rather than the residual interaction itself. To indicate the wide usefulness of the multiplier method, the exact distorted-wave formulation is obtained and put into calculable form with the use of a multiplier.

On the Velocity Operator in Many-Boson System

Abstract: Garrison, Morrison and W ong,l> however, have recently shown that the commutation relation (1) is inconsistent with the nonnegative character of the density operator, and asserted that existence of the velocity operator which satisfies the relation (1) is not admissible. The point of their arguments is essentially equivalent to the wellknown theorem that canonical variables satisfying a canonical commutation relation have as eigenvalues all real numbers from + oo to oo (e.g. see reference 2)). Their assertion is quite correct if one introduces the velocity operator as a canonical conjugate of the density operator itself defined by p(x)=cf;*(x)c{;(x) or by p(x)=(1/N) X2]aJ(x-ra).3l It is however a hasty conclusion to assert non-existence of the velocity operator which satisfies the commutation relation (1). The interacting Bose system enclosed in a cubic box of volume Q is represented by the Hamiltonian h2 H= 2m itk2ak*ak

Asymptotic su(3) symmetry and baryon couplings.

Abstract: General broken-SU(3) sum rules for baryon transitions B′→(12)±+P (B′ denotes a nonet or octet baryon with arbitrary spin and parity) are derived. The approach is based on the use of a chiral SU(3)⊗SU(3) charge algebra, the hypothesis of partially conserved axial-vector current (PCAC) and, in particular, the assumption of asymptotic SU(3) symmetry formulated for the matrix elements of the vector charge VK. The VK is the SU(3) raising or lowering operator. The sum rules thus obtained are always compatible with the Gell-Mann-Okubo mass splittings of hadrons. They exhibit a simple modification of exact-SU(3) sum rules, but the effect is, in general, quite significant. As a specific application, the Y0*(1405) transition is discussed in some detail in order to compare with the recent result of Gell-Mann, Oakes, and Renner (GOR), based on a different approximation for broken SU(3) symmetry. It is shown that, in the absence of singlet-octet mixing, both approaches give the same result in this particular case, and that the GOR approximation can, in fact, be derived from our asymptotic SU(3) symmetry. Our asymptotic SU(3) symmetry, however, appears to be a more general and far-reaching prescription, useful in broken SU(3) symmetry when combined with the use of equal-time commutation relations involving the charge VK. A comment is also made about the hard-kaon and η-meson extrapolation resulting from the use of kaon and η-meson PCAC. The result is applied to the derivation of the values of the ΛNK and ΣNK couplings from the experimental information on the axial-vector semileptonic couplings of hyperons. The result is consistent with experiment.

Suggestive Approximate Dynamical Symmetry of an Electron Moving in the Field of Many Stationary Nuclei

Abstract: A group‐theoretical study has been employed in an effort to uncover the suggestive operator, if it exists, of the form L2 + O which commutes with the Hamiltonian of an electron moving in the many‐nucleus Coulomb field. The Casimir operator L2 may easily be identified as the square angular‐momentum operator, while the operator O may involve the internuclear separations and operators of the Lie algebra. Finally, however, the analysis has revealed the fact that such operator or dynamical invariant does not, in general, exist for any arbitrary stationary nuclei except in few special cases which, however, have their special geometrical symmetry.

Characteristic properties of strong integral and strong B-integral self-adjoint operators

Abstract: A proof that a strong integral (strong B-integral) self-adjoint operator in L2 (a, b) is a Hilbert-Schmidt operator (a kernel operator).

The spectrum of a selfadjoint compression of a self-adjoint operator.

Abstract: The relation between the spectrum of a selfadjoint operator and the spectrum of its compression is investigated. In particular, we show that the compression of a selfadjoint operator is essentially selfadjoint if and only if the spectrum of the closure of the compression is contained in the closed convex hull of the spec- trum of the operator. Relations between two conceptions of com- pressions or projections of operators are also considered.