Showing papers on "Ladder operator published in 2004"
TL;DR: In this article, it has been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm.
Abstract: It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate C is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method is used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT -symmetric quantum field theory.
235 citations
TL;DR: It is shown that an i phi(3) quantum field theory is a physically acceptable model because the spectrum is positive and the theory is unitary.
Abstract: In this Letter it is shown that an i phi(3) quantum field theory is a physically acceptable model because the spectrum is positive and the theory is unitary. The demonstration rests on the perturbative construction of a linear operator C, which is needed to define the Hilbert space inner product. The C operator is a new, time-independent observable in PT-symmetric quantum field theory.
134 citations
TL;DR: In this article, it was shown that the complementarity between position and momentum descriptions can be formulated rigorously as a claim about the existence of representations of the CCRs, and that in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa.
Abstract: We show that Bohr's principle of complementarity between position and momentum descriptions can be formulated rigorously as a claim about the existence of representations of the CCRs. In particular, in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa. Equivalently, if there are nonzero projections corresponding to sharp position values, all spectral projections of the momentum operator map onto the zero element.
94 citations
TL;DR: In this article, it is shown that a Hamiltonian that has an unbroken spacetime reflection symmetry also possesses a physical symmetry represented by a linear operator called C. The result is a new class of fully consistent complex quantum theories.
Abstract: The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of spacetime reflection (PT) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian quantum mechanical Hamiltonians are H = p 2 + ix 3 and H = p 2 - x 4 . The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry represented by a linear operator called C. Using C it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are positive, and the dynamics is governed by unitary time evolution. After a review of PT-symmetric quantum mechanics, new results are presented here in which the C operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.
91 citations
TL;DR: In this article, it was shown that all of them can be created via conventional fashion, i.e., the lowering operator eigenstate and the displacement operator, using the nonlinear coherent states approach.
Abstract: Considering some important classes of generalized coherent states known in literature, we demonstrated that all of them can be created via conventional fashion, i.e. the "lowering operator eigen-state" and the "displacement operator" techniques using the {\it "nonlinear coherent states"} approach. As a result we obtained a {\it "unified method"} to construct a large class of coherent states which already have been introduced by different prescriptions.
85 citations
TL;DR: In this article, it is shown that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real and the dynamics is governed by unitary time evolution.
Abstract: The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry represented by a linear operator called C. Using C it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are positive, and the dynamics is governed by unitary time evolution. After a review of PT-symmetric quantum mechanics, new results are presented here in which the C operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.
85 citations
TL;DR: In this paper, the eigenfunctions and eigenvalues of the Schrodinger equation with a ring-shaped non-spherical oscillator are obtained, and a realization of the ladder operators for the radial wave functions is studied.
64 citations
TL;DR: In this paper, a generalised Rayleigh functional is used that assigns to a vector x a zero of the function T(λ)x, x), where it is assumed that there exists at most one zero.
Abstract: Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T(λ) are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function T(λ)x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.
51 citations
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TL;DR: In this article, an analogue for Lip-normed matrix algebras of the second author's order-unit quantum Gromov-Hausdorff distance is introduced and it is shown that it is equal to the first author's complete distance.
Abstract: We introduce an analogue for Lip-normed operator systems of the second author's order-unit quantum Gromov-Hausdorff distance and prove that it is equal to the first author's complete distance. This enables us to consolidate the basic theory of what might be called operator Gromov-Hausdorff convergence. In particular we establish a completeness theorem and deduce continuity in quantum tori, Berezin-Toeplitz quantizations, and theta-deformations from work of the second author. We show that approximability by Lip-normed matrix algebras is equivalent to 1-exactness of the underlying operator space and, by applying a result of Junge and Pisier, that for n greater than or equal to 7 the set of isometry classes of n-dimensional Lip-normed operator systems is nonseparable. We also treat the question of generic complete order structure.
50 citations
TL;DR: In this paper, a generalized form of Gazeau-Klauder coherent states (CSs) is proposed and the ladder operator and displacement type can be obtained without employing the supersymmetric quantum mechanics (SUSYQM) techniques.
Abstract: Using the {\it analytic representation} of the so-called Gazeau-Klauder coherent states(CSs), we shall demonstrate that how a new class of generalized CSs namely the {\it family of dual states} associated with theses states can be constructed through viewing these states as {\it temporally stable nonlinear CSs}. Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the {\it supersymmetric quantum mechanics}(SUSYQM) techniques. Then, we have applied this method to some physical systems with known spectrum, such as Poschl-Teller, infinite well, Morse potential and Hydrogen-like spectrum as some quantum mechanical systems. Finally, we propose the generalized form of Gazeau-Klauder CS and the corresponding dual family.
49 citations
TL;DR: In this article, a partial differential operator depending on the coupling parameter α≥0 is considered and the spectral properties of the operator strongly depend on α. The operator was suggested in Smilansky (2003 Waves Random Media 14 S143−53) as a model of an irreversible physical system.
Abstract: A partial differential operator depending on the coupling parameter α≥0 is considered. The spectral properties of the operator strongly depend on α. The operator was suggested in Smilansky (2003 Waves Random Media 14 S143–53) as a model of an irreversible physical system.
TL;DR: In this paper, a subset of the kth-order supersymmetric partners of the harmonic oscillator admits third-order ladder operators, and provides a realization of second-order polynomial Heisenberg algebras.
TL;DR: Two recursion relations in terms of raising and lowering operators for only the principal and angular momentum quantum numbers through Laplace transforms are derived in this article, where the Laplace transform is used to define the principal operator.
TL;DR: In this article, a representation of angular momenta in terms of the operator relations corresponding to Gentile statistics in which one quantum state can be occupied by n particles is presented. And the result shows that there exist certain underlying connections between the operator realization of the Gentile statistic and the angular momentum (SU(2)) algebra.
Abstract: This paper seeks to construct a representation of the algebra of angular momentum (SU(2) algebra) in terms of the operator relations corresponding to Gentile statistics in which one quantum state can be occupied by n particles. First, we present an operator realization of Gentile statistics. Then, we propose a representation of angular momenta. The result shows that there exist certain underlying connections between the operator realization of the Gentile statistics and the angular momentum (SU(2)) algebra.
TL;DR: In this paper, a study of the one-loop dilatation operator in the scalar sector of $\mathcal{N} = 4$ SYM is presented, and an infinite tower of local conserved charges is constructed in this classical limit purely within the context of the matrix model.
Abstract: A study of the one-loop dilatation operator in the scalar sector of $\mathcal{N}=4$ SYM is presented. The dilatation operator is analyzed from the point of view of Hamiltonian matrix models. A Lie algebra underlying operator mixing in the planar large-N limit is presented, and its role in understanding the integrability of the planar dilatation operator is emphasized. A classical limit of the dilatation operator is obtained by considering a contraction of this Lie algebra, leading to a new way of constructing classical limits for quantum spin chains. An infinite tower of local conserved charges is constructed in this classical limit purely within the context of the matrix model. The deformation of these charges and their relation to the charges of the spin chain is also elaborated upon.
TL;DR: In this paper, the authors explore the structure of the bosonic analogues of the parafermion quantum Hall states and show that the many-boson wave functions of $k$-clustered quantum Hall droplets appear naturally as matrix elements of ladder operators in integrable representations of the affine Lie algebra.
Abstract: We explore the structure of the bosonic analogues of the $k$-clustered ``parafermion'' quantum Hall states. We show how the many-boson wave functions of $k$-clustered quantum Hall droplets appear naturally as matrix elements of ladder operators in integrable representations of the affine Lie algebra $\hat{su}(2)_k$. Using results of Feigin and Stoyanovsky, we count the dimensions of spaces of symmetric polynomials with given $k$-clustering properties and show that as the droplet size grows the partition function of its edge excitations evolves into the character of the representation. This confirms that the Hilbert space of edge states coincides with the representation space of the $\hat{su}(2)_k$ edge-current algebra. We also show that a spin-singlet, two-component $k$-clustered boson fluid is similarly related to integrable representations of $\hat{su}(3)$. Parafermions are not necessary for these constructions.
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TL;DR: In this article, the authors studied the properties of shifted vertex operator algebras, which are vertex algebraic structures derived from a given theory by shifting the conformal vector.
Abstract: We study the properties of shifted vertex operator algebras, which are vertex algebras derived from a given theory by shifting the conformal vector. In this way, we are able to exhibit large numbers of vertex operator algebras which are regular(rational and C_2-cofinite) and yet are pathological in one way or another.
TL;DR: In this article, it was shown that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation (Kraus representation) regardless of its initial condition and the path of its evolution in the state space.
Abstract: We show that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation (Kraus representation) regardless of its initial condition and the path of its evolution in the state space, and we provide a general expression of Kraus operators for arbitrary time-dependent density operator of an N-dimensional system. Moreover, applications of our result are illustrated through several examples.
TL;DR: In this article, it has been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm.
Abstract: It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition it is cumbersome to calculate C in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method can be used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.
TL;DR: In this paper, the authors considered a spin-1/2 charged particle in the plane under the influence of two idealized Aharonov-Bohm fluxes and showed that the Pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices.
Abstract: We consider a spin-1/2 charged particle in the plane under the influence of two idealized Aharonov–Bohm fluxes. We show that the Pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices. Further we explicitly construct a basis in the deficiency subspaces of the symmetric operator obtained by restricting the domain to functions with supports separated from the vortices. This construction makes it possible to apply the Krein’s formula to the Pauli operator.
TL;DR: In this paper, the normal ordering problem for (A,A) = [N+1] − [N]) bosonic ladder operators is solved for the case where A and A are one mode deformed.
Abstract: We solve the normal ordering problem for (A
†
A)
n
where A and A
† are one mode deformed ([A,A
†] = [N+1] − [N]) bosonic ladder operators. The solution generalizes results known for canonical bosons. It involves combinatorial polynomials in the number operator N for which the generating function and explicit expressions are found. Simple deformations provide examples of the method.
TL;DR: In this paper, the authors derived the quantum dynamics of a particle in the Modified Poschl-Teller potential from the group $SL(2,R)$ by applying a Group Approach to Quantization (GAQ).
Abstract: The quantum dynamics of a particle in the Modified Poschl-Teller potential is derived from the group $SL(2,R)$ by applying a Group Approach to Quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the $SL(2,R)$ group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU(2) where all the states are physical.
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TL;DR: In this article, the eigenvalue problem of the wavelet Galerkin operator associated to a wavelet filter was considered and the solution involves the construction of representations of the algebra AN, the C ⁄ -algebra generated by two unitaries U,V satisfying UV U i1 = V N introduced in (13).
Abstract: We consider the eigenvalue problem Rm0,m0h = ‚h, h 2 C(T), |‚| = 1, where Rm0,m0 is the wavelet Galerkin operator associated to a wavelet filter m0. The solution involves the construction of representations of the algebra AN — the C ⁄ -algebra generated by two unitaries U,V satisfying UV U i1 = V N introduced in (13).
TL;DR: In this article, a modularity on vertex operator algebras arising from semisimple primary vectors is studied, and it is shown that the internal automorphisms do not change the genus one twisted conformal blocks.
Abstract: In this article, using an idea of the physics superselection principal, we study a modularity on vertex operator algebras arising from semisimple primary vectors. We generalizes the theta functions on vertex operator algebras and prove that the internal automorphisms do not change the genus one twisted conformal blocks.
TL;DR: In this article, a multispecies model of Calogero type in D dimensions with harmonic, two-body and three-body interactions was defined, and the universal critical point at which the model exhibits singular behavior was detected.
TL;DR: In this article, the explicit form of the evolution operator of the Tavis-Cummings model with three and four atoms is given, which is an important progress in quantum optics or mathematical physics.
Abstract: In this letter the explicit form of the evolution operator of the Tavis–Cummings model with three and four atoms is given. This is an important progress in quantum optics or mathematical physics.
TL;DR: In this paper, the shape invariance symmetries with respect to two different parameters n and m were derived for the superpotentials A tanh ωy + B / A and − A cot ωθ + B csc ω θ, respectively.
TL;DR: In this article, the associated Laguerre functions in terms of two non-negative integers were introduced and simultaneously and separately realization of the laddering equations with respect to each of the integers by means of two pairs of ladder operators.
Abstract: Introducing the associated Laguerre functions in terms of two non-negative integers, we obtain simultaneously and separately realization of the laddering equations with respect to each of the integers by means of two pairs of ladder operators. Besides, two different types of shape-invariance symmetries are realized. This approach leads to a derivation of shape-invariance equations of third type which are realized by two simultaneous raising and lowering operators of two parameters.
TL;DR: In this article, the shape invariance condition in quantum mechanics is applied as an algebraic method to solve the Dirac-Coulomb problem and the ground state and excited states are investigated using new generalized ladder operators.