Schrodinger Picture Research Articles

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

AbstractThe relativistic effects have been widely reported to affect the chemical and physical properties of the heavy elements such as lanthanides (), actinides (), and transactinides (). This effect is definitely weakened by reducing the atomic number in lighter elements such as hydrogen‐like atoms (HLAs). The aim of present paper is to investigate the relativistic effects of electron motion in Bohr orbits on the chemical and physical properties of HLAs. The theoretical model is based on the Heisenberg (rather than Schrodinger) picture where the relativistic vibrational Hamiltonian (RVH) is expanded as a power series of harmonic oscillator Hamiltonian for the first time. By applying the first‐order RVH (correct to ) to the Heisenberg equation, a pair of coupled equations is obtained for the relativistic position and linear momentum of electron. A simple comparison of the first‐order relativistic and nonrelativistic equations reveals that the relativistic natural frequency of an HLA (like entropy) is slowly raised by increasing beyond . In general, RVH plays a fundamental role because it specifies the temporal relativistic variations of position, velocity, and linear momentum of the oscillating electron. The results are finally verified by demonstrating energy conservation.

Third quantization of the electromagnetic field

We consider an approach in which the usual wave function in the quadrature representation of mode j of the electromagnetic field is further quantized to produce a field operator. Since the electromagnetic field is already second quantized, this corresponds to an additional or third quantization. The third-quantization approach can be used to perform certain quantum optics calculations in the Heisenberg picture that could only be performed in the Schrodinger picture when using the conventional second-quantized theory. This approach also allows an interesting generalization of quantum optics and quantum electrodynamics that is analogous to symmetry breaking in elementary particle theory. The predictions of the generalized theory could be tested using a proposed photon scattering experiment.

Quantum Stochastic Cocycles and Completely Bounded Semigroups on Operator Spaces II

Quantum stochastic cocycles provide a basic model for time-homogeneous Markovian evolutions in a quantum setting, and a direct counterpart in continuous time to quantum random walks, in both the Schrodinger and Heisenberg pictures. This paper is a sequel to one in which correspondences were established between classes of quantum stochastic cocycle on an operator space or C*-algebra, and classes of Schur-action `global' semigroup on associated matrix spaces over the operator space. In this paper we investigate the stochastic generation of cocycles via the generation of their corresponding global semigroups, with the primary purpose of strengthening the scope of applicability of semigroup theory to the analysis and construction of quantum stochastic cocycles. An explicit description is given of the affine relationship between the stochastic generator of a completely bounded cocycle and the generator of any one of its associated global semigroups. Using this, the structure of the stochastic generator of a completely positive quasicontractive quantum stochastic cocycle on a C*-algebra whose expectation semigroup is norm continuous is derived, giving a comprehensive stochastic generalisation of the Christensen--Evans extension of the GKS&L theorem of Gorini, Kossakowski and Sudarshan, and Lindblad. The transformation also provides a new existence theorem for cocycles with unbounded structure map as stochastic generator. The latter is applied to a model of interacting particles known as the quantum exclusion Markov process, in particular on integer lattices in dimensions one and two.

W★ Dynamics of Infinite Dissipative Quantum Systems

We formulate the dynamics of an infinitely extended open dissipative quantum system , Σ, in the Schroedinger picture. The generic model on which this is based comprises a C★-algebra, A , of observables, a folium, ℱ , of states on this algebra and a one-parameter semigroup, τ , of linear transformations of ℱ that represents its dynamics and is given by a natural infinite-volume limit of the corresponding semigroup for a finite system. On this basis, we establish that the dynamics of Σ is given by a one-parameter group of completely positive linear transformations of the W★-algebra dual to ℱ . This result serves to extend our earlier formulation [ 1 ] of infinitely extended conservative systems to open dissipative ones.

Normal ordering normal modes

In a soliton sector of a quantum field theory, it is often convenient to expand the quantum fields in terms of normal modes. Normal mode creation and annihilation operators can be normal ordered, and their normal ordered products have vanishing expectation values in the one-loop soliton ground state. The Hamiltonian of the theory, however, is usually normal ordered in the basis of operators which create plane waves. In this paper we find the Wick map between the two normal orderings. For concreteness, we restrict our attention to Schrodinger picture scalar fields in 1+1 dimensions, although we expect that our results readily generalize beyond this case. We find that plane wave ordered n-point functions of fields are sums of terms which factorize into j-point functions of zero modes, breather and continuum normal modes. We find a recursion formula in j and, for products of fields at the same point, we solve the recursion formula at all j.

Paley_ Wiener Theorems for The Fourier Transform depending on the Heisenberg group

عرّفنا في بحثنا هذا زمرة هايزنبرغ، وهي الزمرة الأكثر شهرةً من زمر لي. ثمّ ناقشنا نظريّة التمثيل لهذه الزمرة، إضافةً إلى العلاقة بين نظريّة التمثيل لزمرة هايزنبرغ، ومؤثرات كميّة الحركة والموضع. وهذا ما يُبيّن لنا كيفية تحقيق الترابط بين زمرة هايزنبرغ والفيزياء. وسنأخذ في بحثنا هذا بوجهة نظر شرودنجر Schr dinger، التي تعتبر أنّ جميع التمثيلات التي نفرضها تتحقق على فضاء هيلبرت . ثمّ نُعرّف زمرة تحويل فورييه على زمرة هايزنبرغ. إضافةً إلى مفاهيم أخرى ذات أهميّة في الأبحاث المشابهة. وقد اعتمدنا في بحثنا هذا على صيغ جديدة لبعض المفاهيم الرياضيّة مثل تحويل فورييه وتحويل وايل. إنَ الهدف الرئيس من بحثنا هذا هو تقديم مبرهنة Paley_ Wiener لأجل تحويل فورييه على زمرة هايزنبرغ. وسوف نُبيّن أنّ مبرهنة Paley_ Wiener التقليديّة لتحويل فورييه التقليدي، تُميّز الدوال ذات الدعامة المتراصّة من خلال سلوك تحويل فورييه لها. وسنهتم بإنشاء النتائج لأجل زمرة تحويل فورييه، وتحويل وايل.

Measuring entangled neutrino states in a toy model QFT

Localized wave packet treatments of neutrino oscillations by various groups lead to mutually inconsistent predictions. The neutrino wave packet description arises as an approximate substitute for the evolution of an entangled state which is not localized. The disagreements arise from qualitative differences in the framework which are not specific to electroweak interactions, and so have analogues in simpler models. Therefore in this note we introduce a toy model which allows one to explicitly test these predictions while consistently keeping track of the entanglement of the neutrinos and the source particles. Our study is robust as we use only the Schrodinger picture evolution equations on the entangled state, which are solved explicitly without recourse to the wave packet approximation.

Multiple photon effective Hamiltonians in linear quantum optical networks

Abstract We give an alternative derivation for the explicit formula of the effective Hamiltonian describing the evolution of the quantum state of any number of photons entering a linear optics multiport. The description is based on the effective Hamiltonian of the optical system for a single photon and comes from relating the evolution in the Lie group that describes the unitary evolution matrices in the Hilbert space of the photon states to the evolution in the Lie algebra of the Hamiltonians for one and multiple photons. Our derivation complements previous results with a point of view which reminds of the Schrodinger picture. These group theory results allow us to prove that quantum optical linear networks obey different proportionality rules which relate the expected photon number at each input and output ports. In particular, we show that for a uniform photon input the average photon number at the output is preserved for any linear multiport.

Particle production and apparent decoherence due to an accelerated time delay

We study the radiation produced by an accelerated time-delay acting on the left moving modes. Through analysis via the Schrodinger picture, we find that the final state is a two-mode squeezed state of the left moving Unruh modes, implying particle production. We analyse the system from an operational point of view via the use of self-homodyne detection with broad-band inertial detectors. We obtain semi-analytical solutions that show that the radiation appears decohered when such an inertial observer analyses the information of the radiation from the accelerated time-delay source. We make connection with the case of the accelerated mirror. We investigate the operational conditions under which the signal observed by the inertial observer can be purified.

Schrödinger Picture Analysis of the Beam Splitter: an Application of the Janszky Representation

The Janszky representation constructs quantum states of a field mode as a superposition of coherent states on a line in the complex plane. We show that this provides a natural Schrodinger picture description of the interference between a pair of modes at a beam splitter.

Quadratic fermionic dynamics with dissipation

Gaussian solutions of the Cauchy problem for the GKS-L equation (in the Schrodinger picture) with quadratic fermionic generators are obtained. These Gaussian solutions are represented both as exponentials of quadratic forms in fermionic creation-annihilation operators and by their normal symbols. The coefficients of these forms are represented as algebraic functions of matrices.

Canonical bracket in quantum-classical hybrid systems

We study compound systems with a classical sector and a quantum sector. Among other consistency conditions we require a canonical structure, that is, a Lie bracket for the dynamical evolution of hybrid observables in the Heisenberg picture, interpolating between the Poisson bracket and the commutator. Weak and strong postulates are proposed. We explicitly construct one such hybrid bracket when the Hilbert space of the quantum sector is finite dimensional and show that it is unique if the strong postulates are enforced. The adjoint bracket for the Schrodinger picture version of the dynamics is also obtained. Unfortunately, preservation of the positivity of the density matrix under the evolution is not guaranteed. The case of a particle with classical position and momentum and quantum spin-$\frac{1}{2}$ is discussed and the spin-orbit dynamics is worked out.

Few-body quantum physics with strongly interacting Rydberg polaritons

We present an extension of our recent paper [Bienias et al., Phys. Rev. A 90, 053804 (2014)] in which we demonstrated the scattering properties and bound-state structure of two Rydberg polaritons, as well as the derivation of the effective low-energy many-body Hamiltonian. Here, we derive a microscopic Hamiltonian describing the propagation of Rydberg slow light polaritons in one dimension. We describe possible decoherence processes within a Master equation approach, and derive equations of motion in a Schroedinger picture by using an effective non-Hermitian Hamiltonian. We illustrate diagrammatic methods on two examples: First, we show the solution for a single polariton in an external potential by exact summation of Feynman diagrams. Secondly, we solve the two body problem in a weakly interacting regime exactly.

Time dependence of adiabatic particle number

We consider quantum field theoretic systems subject to a time-dependent perturbation, and discuss the question of defining a time dependent particle number not just at asymptotic early and late times, but also during the perturbation. Naively, this is not a well-defined notion for such a non-equilibrium process, as the particle number at intermediate times depends on a basis choice of reference states with respect to which particles and anti-particles are defined, even though the final late-time particle number is independent of this basis choice. The basis choice is associated with a particular truncation of the adiabatic expansion. The adiabatic expansion is divergent, and we show that if this divergent expansion is truncated at its optimal order, a universal time dependence is obtained, confirming a general result of Dingle and Berry. This optimally truncated particle number provides a clear picture of quantum interference effects for perturbations with non-trivial temporal sub-structure. We illustrate these results using several equivalent definitions of adiabatic particle number: the Bogoliubov, Riccati, Spectral Function and Schrodinger picture approaches. In each approach, the particle number may be expressed in terms of the tiny deviations between the exact and adiabatic solutions of the Ermakov-Milne equation for the associated time-dependent oscillators.

Spontaneous emission and the operation of invisibility cloaks

As a probe to explore the ability of invisibility cloaks to conceal objects in the quantum mechanics domain, we study the spontaneous emission rate of an excited two-level atom in the vicinity of an ideal invisibility cloaking. On this base, first, a canonical quantization scheme is presented for the electromagnetic field interacting with atomic systems in an anisotropic, inhomogeneous, and absorbing magnetodielectric medium which can suitably be used for studying the influence of arbitrary invisibility cloak on the atomic radiative properties. The time dependence of the atomic subsystem is obtained in the Schrodinger picture. By introducing a modified set of the spherical wave-vector functions, the Green tensor of the system is calculated via exact and discrete methods. In this formalism, the decay rate and as well the emission pattern of the aforementioned atom are computed analytically for both weak and strong coupling interaction, and then numerically calculations are done to demonstrate the performances of cloaking in the quantum mechanics domain. Special attention is paid to different possible orientations and locations of the atomic system near the spherical invisibility cloaking. Results in the presence and the absence of the invisibility cloak are compared. We find that the cloak works very well far from its resonance frequency to conceal a macroscopic object, whereas at near the resonance frequency the object is more visible than the situation where the object is not covered by the cloak.

Remarks on Nondeterministic Computation, Choices, and Formal Language

Based on the equivalence of the two different types of measurement protocols and the asymmetry between the Schrodinger and Heisenberg pictures, it has been previously proposed that negative sea fills the universe as a nondeterministic computation - a time-reversal process of the irreversible computations presented since the big bang. The goal of this paper is to extend the proposed subjective universe model, i.e., the universe as a quantum measurement: Motivated by the relationship between quantum theory and classical probability theory with continuity, it is argued that the frame of reference of the observer may be identified with classical probability theory where its choice, along with big bang singularity, should correspond to the quantum observable. That is, the physical version of singularity resolution corresponds to the case, where big bang singularity is equivalent to the continuity of the negative sea, or aether, filling the universe as a frame of reference of the observer. Moreover, based on the holographic principle, we identify the choice of the observer with the degrees of freedom proportional to the Planck area on the horizon. We also discuss that the continuity or infinity present in every formal language of choice acceptable in nondeterministic computation may be associated with the universal grammar proposed by Chomsky in linguistics.

Dirac’s Equation in R-Minkowski Spacetime

We recently constructed the R-Poincare algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the R-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac’s equation in the de Sitter spacetime using the Schrodinger picture.

Geometric phase carried by the observables and its application to quantum computation

We investigate geometric phases in terms of Heisenberg equation. We find that, equivalently to Schrodinger picture with a memory of its motion in terms of the geometric phase factor contained in th...

Foundations of Quantum Mechanics: On Rotations by $$4\pi $$ 4 π for Half-Integral Spin Particles

Rotations in Quantum Mechanics are a very well-known subject. When one is faced with rotations related to the SO(3) group, for instance, all the underlying operators are well-known and built from their classical counterparts. However, when it comes to represent rotations related to the SU(2) group, it is always argued that there is no classical counterpart from which the expressions for the quantum mechanical operators can be built. The approach is always done using matrix representation (and not by solving some Schrodinger equation). In the way of this aproach, one assumes that the operators related to SU(2) are pseudovectors (or antisymmetric rank two tensors) and end up with the weird result that only rotations by multiples of \(4\pi \) would bring the system to its original situation. In Olavo (Physica A 262:181–196, 1999) we have shown that any half-integral spin system can be represented in the Schrodinger picture by solving a Schrodinger equation with well defined expressions for the SU(2) differential operators. It turned out that two of these operators are symmetric tensors, not pseudovectors or antisymmetric tensors. In this paper we use these results to dismiss with the \(4\pi \) conundrum.

Some Statistical Properties for Interacting Between a Two Two-Level Atoms and the Electromagnetic Fields in Present of Converter Terms

Two two-level atoms interacting with a finite dimensional four-mode of radiation field is presented. The interaction of the field-field and the atom-field are considered. By using some canonical transformation, an exact solution of the wave function in the Schrodinger picture is obtained. The atomic inversion, the entropy squeezing, variance squeezing and scaled atomic Wehrl entropy phenomena are investigated. All theses phenomena are been controlled by finite state parameters and photon number operators.