The non-commutative oscillator, symmetry and the Landau problem

The isotropic oscillator on a plane is discussed where the coordinate and momentum space are both considered to be non-commutative. We also discuss the symmetry properties of the oscillator for three separate cases when the non-commutative parameters Θ and for x and p-space, respectively, satisfy specific relations. We compare the Landau problem with the isotropic oscillator on non-commutative space and obtain a relation between the two non-commutative parameters and the magnetic field of the Landau problem.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

Semiclassical and quantum motions on the non-commutative plane

We study the canonical and the coherent state quantizations of a particle moving in a magnetic field on the non-commutative plane. Using a θ-modified action, we perform the canonical quantization and analyze the gauge dependence of the theory. We compare coherent states quantizations obtained through Malkin-Man'ko states and circular squeezed states. The relation between these states and the “classical” trajectories is investigated, and we present numerical explorations of some semiclassical quantities.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations

Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac’s symbols (ket versus bra, e.g., |q〉〈q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |〉〈| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac’s symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are presented.     

ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton-Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket-bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.     

Symplectic quantum mechanics

Using the notion of symplectic structure and Weyl (or star) product of non-commutative geometry, we construct unitary representations for the Galilei group and show how to rewrite the Schrödinger equation in phase space. This approach gives rise to a new procedure to derive Wigner functions without the use of the Liouville-von Neumann equation. Applications are presented by deriving the states of linear and nonlinear oscillators in terms of amplitudes of probability in phase space. The notion of coherent states is also discussed in this context.     

Solution of Coulomb system in momentum space

The solution of D-dimensional Coulomb system is solved in momentum space by path integral. From which the topological effect of a magnetic flux in the system is given. It is revealed that the flux effect represented by the two-dimensional field of Aharonov-Bohm covers any space-dimensions.     

Landau analog levels for dipoles in non-commutative space and phase space

We investigate the analog of Landau quantization, for a neutral polarized particle in the presence of homogeneous electric and magnetic external fields, in the context of non-commutative quantum mechanics. This particle, possessing electric and magnetic dipole moments, interacts with the fields via the Aharonov–Casher and He–McKellar–Wilkens effects. For this model we obtain the Landau energy spectrum and the radial eigenfunctions of the non-commutative space coordinates and non-commutative phase space coordinates. Also we show that the case of non-commutative phase space can be treated as a special case of the usual non-commutative space coordinates.     

Hamiltonian quantum dynamics with separability constraints

Schroedinger equation on a Hilbert space H, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space PH. Separable states of a bipartite quantum system form a special submanifold of PH. We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.     

Relativistic quantum particle in a homogeneous external field

The model of the relativistic quantum particle in a homogeneous external field is proposed. This model is realized in the one-dimensional relativistic configurational x-space and is described by the finite-difference equation. The momentum p-space in our case is the one-dimensional Lobachevsky space. We have found the wave functions and propagator for the model under study in both x- and p-representations.     

He-McKellar-Wilkens Effect in Noncommutative Space

The He-McKellar-Wilkens （HMW） effect in non-commutative （NC） space is studied. By solving the Dirac equations on NC space, we obtain topological HMW phase in NC space where the additional terms related to the space non-commutativity are given explicitly.     

Quantum anticentrifugal force for wormhole geometry

We show the existence of an anticentrifugal force in a wormhole geometry in R³. This counterintuitive force was shown to exist in a flat R² space. The role the geometry plays in the appearance of this force is discussed.     

An optimum Hamiltonian for non-Hermitian quantum evolution and the complex Bloch sphere

For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time τ. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.     

Regularization in the J-matrix method of scattering revisited

We present an alternative, but equivalent, approach to the regularization of the reference problem in the J-matrix method of scattering. After identifying the regular solution of the reference wave equation with the “sine-like” solution in the J-matrix approach we proceed by direct integration   to find the expansion coefficients in an L²$L^2$ basis set that ensures a tridiagonal representation of the reference Hamiltonian. A differential equation in the energy is then deduced for these coefficients. The second independent solution of this equation, called the “cosine-like” solution, is derived by requiring it to pertain to the L²$L^2$ space. These requirements lead to solutions that are exactly identical to those obtained in the classical J-matrix approach. We find the present approach to be more direct and transparent than the classical differential approach of the J-matrix method.     

Schwinger representation for the symmetric group: Two explicit constructions for the carrier space

We give two explicit constructions for the carrier space for the Schwinger representation of the group Sn. While the first relies on a class of functions consisting of monomials in antisymmetric variables, the second is based on the Fock space associated with the Greenberg algebra.     

CMB constraints on non-commutative geometry during inflation

We investigate the primordial power spectrum of the density perturbations based on the assumption that space is non-commutative in the early stage of inflation, and constrain the contribution from non-commutative geometry using CMB data. Due to the non-commutative geometry, the primordial power spectrum can lose rotational invariance. Using the k-inflation model and slow-roll approximation, we show that the deviation from rotational invariance of the primordial power spectrum depends on the size of non-commutative length scale L _s but not on sound speed. We constrain the contributions from the non-commutative geometry to the covariance matrix of the harmonic coefficients of the CMB anisotropies using five-year WMAP CMB maps. We find that the upper bound for L _s depends on the product of sound speed and slow-roll parameter. Estimating this product using cosmological parameters from the five-year WMAP results, the upper bound for L _s is estimated to be less than 10^?27 cm at 99.7% confidence level.     

The generalized uncertainty principle in (A)dS space and the modification of Hawking temperature from the minimal length

Recently, the Heisenberg's uncertainty principle has been extended to incorporate the existence of a large (cut-off) length scale in de Sitter or anti-de Sitter space, and the Hawking temperatures of the Schwarzshild–(anti) de Sitter black holes have been reproduced by using the extended uncertainty principle. I generalize the extended uncertainty to the case with an absolute minimum length and compute its modification to the Hawking temperature. I obtain a general trend that the generalized uncertainty principle due to the absolute minimum length “always” increases the Hawking temperature, implying “faster” decay, which is in conformity with the result in the asymptotically flat space. I also revisit the black hole-string phase transition, in the context of the generalized uncertainty principle.     

Coherent states of the inverted Caldirola-Kanai oscillator with time-dependent singularities

The coherent states for a system of time-dependent singular potentials coupled to inverted CK (Caldirola-Kanai) oscillator are investigated by employing invariant operator method and Lie algebraic approach. We considered Coulomb potential and inverse quadratic potential as singularities of the system. The spectrum of quantum states is discrete for λ < 0 while continuous for λ ? 0. The probability densities for both Fock state and coherent state are converged to the center as time goes by according to the dissipation of energy. We confirmed that the probability density in the coherent state oscillates back and forth like a classical wave packet.     

Divergence-free WKB theory

We present a divergence-free WKB theory, which is a new semiclassical theory modified by nonperturbative quantum corrections. Conventionally, the WKB theory is constructed upon a trajectory that obeys the bare classical dynamics expressed by a quadratic equation in momentum space. Contrary to this, the divergence-free WKB theory is based on a higher-order algebraic equation in momentum space, which represents a dressed classical dynamics. More precisely, this higher-order algebraic equation is obtained by including quantum corrections to the quadratic equation, which is the bare classical limit. An additional solution of the higher-order algebraic equation enables us to construct a uniformly converging perturbative expansion of the wavefunction. Namely, our theory removes the notorious divergence of wavefunction at a turning point from the WKB theory. Moreover, our theory is able to produce wavefunctions and eigenenergies more accurate than those given by the traditional WKB method. In addition, the divergence-free WKB theory that is based on the cubic equation allows us to construct a uniformly valid wavefunction for the nonlinear Schrödinger equation (NLSE). A recent short letter [T. Hyouguchi, S. Adachi, M. Ueda, Phys. Rev. Lett. 88 (2002) 170404] is the opening of the divergence-free WKB theory. This paper presents full formalism of this theory and its several applications concerning wavefunction and eigenenergy to show that our theory is a natural extension of the traditional WKB theory that incorporates nonperturbative quantum corrections.     

Trajectory interpretation of the uncertainty principle in 1D systems using complex Bohmian mechanics

Complex Bohmian mechanics is introduced to investigate the validity of a trajectory interpretation of the uncertainty principles ΔqΔp??/2 and ΔEΔt??/2 by replacing probability mean values with time-averaged mean values. It is found that the ?/2 factor in the uncertainty relation ΔEΔt??/2 stems from a quantum potential whose time-averaged mean value taken along any closed trajectory with a period T=2π/ω is proved to be an integer multiple of ?ω/2 for one-dimensional systems.