Position In Quantum Mechanics Research Articles

Complex Quantum Hydrodynamics in Momentum Space with Broken Time-Reversal Symmetry

Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of criticism raised by Pauli against Bohmian mechanics is its missing symmetry between position and momentum that is present in classical phase space as well as in the quantum mechanical position and momentum representations. Both Madelung’s quantum hydrodynamics formulation and Bohmian mechanics are usually expressed only in position space. Recently, with the use of complex quantities, we were able to provide a hydrodynamic formulation also in momentum space. In this paper, we extend this formalism to include dissipative systems with broken time-reversal symmetry. In classical Hamiltonian mechanics and conventional quantum mechanics, closed systems with reversible time-evolution are usually considered. Extending the discussion to include open systems with dissipation, another form of symmetry is broken, that under time-reversal. There are different ways of describing such systems; for instance, Langevin and Fokker–Planck-type equations are commonly used in classical physics. We now investigate how these aspects can be incorporated into our complex hydrodynamic description and what modifications occur in the corresponding equations, not only in position, but particularly in momentum space.

Space and time transformations with a minimal length

Phenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics to introduce a minimal uncertainty in position in quantum mechanics. In the present work, we show the influence of space and time transformations in shaping quantities such as momentum, energy, and their relations with the generators of transformations. Thus, such an influence determines, among other aspects, the time evolution of a quantum system. In the exemplary case of Galilean transformations, the Schrödinger equation is identical to the ordinary case.

Imaging of Single-Photon Orbital-Angular-Momentum Bell States

Bell states are the most fundamental resource in realizing quantum information tasks and have a very unique position in quantum mechanics. Utilizing orbital angular momentum (OAM) to encode single-photon Bell states enables the realization of high-dimensional Hilbert space, which is crucial for the field of quantum information. In this paper, we design a single-photon OAM Bell-state evolution device based on the Sagnac interferometer, which can make one-to-one correspondence between the input Bell states and the output states. Moreover, we also develop a single-photon single-pixel imaging (SPI) technique, which improves spatial resolution and decreases acquisition time in the meantime, to capture interference images of output states. The results show that single-photon OAM Bell states can be fully recognized by comparing the differences of interference images. We innovatively utilize the SPI technique for single-photon OAM Bell-state recognition. This indicates that the SPI technique effectively promotes the study of quantum information based on OAM, while the quantum information based on OAM provides a clear application scenario for the SPI technique.

Local approaches for electric dipole moments in periodic systems and their application to real-time time-dependent density functional theory.

Within periodic boundary conditions, the traditional quantum mechanical position operator is ill-defined, necessitating the use of alternative methods, most commonly the Berry phase formulation in the modern theory of polarization. Since any information about local properties is lost in this change of framework, the Berry phase formulation can only determine the total electric polarization of a system. Previous approaches toward recovering local electric dipole moments have been based on applying the conventional dipole moment operator to the centers of maximally localized Wannier functions (MLWFs). Recently, another approach to local electric dipole moments has been demonstrated in the field of subsystem density functional theory (DFT) embedding. We demonstrate in this work that this approach, aside from its use in ground state DFT-based molecular dynamics, can also be applied to obtain electric dipole moments during real-time propagated time-dependent DFT (RT-TDDFT). Moreover, we present an analogous approach to obtain local electric dipole moments from MLWFs, which enables subsystem analysis in cases where DFT embedding is not applicable. The techniques were implemented in the quantum chemistry software CP2K for the mixed Gaussian and plane wave method and applied to cis-diimide and water in the gas phase, cis-diimide in aqueous solution, and a liquid mixture of dimethyl carbonate and ethylene carbonate to obtain absorption and infrared spectra decomposed into localized subsystem contributions.

Many-body electric multipole operators in extended systems

The quantum mechanical position operators, and their products, are not well-defined in systems obeying periodic boundary conditions. Here we extend the work of Resta who developed a formalism to calculate the electronic polarization as an expectation value of a many-body operator, to include higher multipole moments, e.g., quadrupole and octupole. We define $n$-th order multipole operators whose expectation values can be used to calculate the $n$-th multipole moment when all of the lower moments are vanishing (modulo a quantum). We show that changes in our operators are tied to flows of $n-1$-st multipole currents, and encode the adiabatic evolution of the system in the presence of an $n-1$-st gradient of the electric field. Finally, we test our operators on a set of tightbinding models to show that they correctly determine the phase diagrams of topological quadrupole and octupole models, capture an adiabatic quadrupole pump, and distinguish a bulk quadrupole moment from other mechanisms that generate corner charges.

An Application of the Sonine–Letnikov Fractional Derivative for the Radial Schrödinger Equation

The Sonine–Letnikov fractional derivative provides the generalized Leibniz rule and, some singular differential equations with integer order can be transformed into the fractional differential equations. The solutions of these equations obtained by some transformations have the fractional forms, and these forms can be obtained as the explicit solutions of these singular equations by using the fractional calculus definitions of Riemann–Liouville, Grünwald–Letnikov, Caputo, etc. Explicit solutions of the Schrödinger equation have an important position in quantum mechanics due to the fact that the wave function includes all essential information for the exact definition of a physical system. In this paper, our aim is to obtain fractional solutions of the radial Schrödinger equation which is a singular differential equation with second-order, via the Sonine–Letnikov fractional derivative.

Fully differential cross sections for the single ionization of helium by fast ions: Classical model calculations

Fully differential cross sections (FDCSs) have been calculated for the single ionization of helium by 1- and 3-MeV proton and 100-MeV/u ${\mathrm{C}}^{6+}$ ion impact using the classical trajectory Monte Carlo (CTMC) method in the nonrelativistic, three-body approximation. The calculations were made employing a Wigner-type model in which the quantum-mechanical position distribution of the electron is approximated by a weighted integral of the microcanonical distribution over a range of the binding energy of the electron. In the scattering plane, the model satisfactorily reproduces the observed shape of the binary peak. In the region of the peak the calculated FDCSs agree well with the results of continuum-distorted-wave calculations for all the investigated collisions. For 1-MeV proton impact the experimentally observed shift of the binary peak with respect to the first Born approximation is compared with the shifts obtained by different higher-order quantum-mechanical theories and the present CTMC method. The best result was achieved by CTMC, but still a large part of the shift remained unexplained. Furthermore, it was found that the classical theory failed to reproduce the shape of the recoil peak observed in the experiments, it predicts a much narrower peak. This indicates that the formation of the recoil peak is dominated by quantum-mechanical effects. For 100-MeV/u ${\mathrm{C}}^{6+}$ ion impact the present CTMC calculations confirmed the existence of the ``double-peak'' structure of the angular distribution of the electron in the plane perpendicular to the momentum transfer, in accordance with the observation, the prediction of an incoherent semiclassical model, and previous CTMC results. This finding together with wave-packet calculations suggests that the ``${\mathrm{C}}^{6+}$ puzzle'' may be solved by considering the loss of the projectile coherence. Experiments to be conducted using ion beams of anisotropic coherence are proposed for a more differential investigation of the ionization dynamics.

Uncertainty relations for characteristic functions

We present the uncertainty relation for the characteristic functions (ChUR) of the quantum mechanical position and momentum probability distributions. This inequality is more general than the Heisenberg Uncertainty Relation, and is saturated in two extremal cases for wavefunctions described by periodic Dirac combs. We further discuss a broad spectrum of applications of the ChUR, in particular, we constrain quantum optical measurements involving general detection apertures and provide the uncertainty relation that is relevant for Loop Quantum Cosmology. A method to measure the characteristic function directly using an auxiliary qubit is also briefly discussed.

Determination of the collective modes from the quantum-mechanical time-correlation functions

Theoretical characterization of vibrational spectra for large molecular systems often comes from the normal modes analysis derived from the quadratic approximation of the potential energy surface near its minimum. The normal modes of motion provide accurate representation of low-energy collective motion within a system near equilibrium, but they may be inadequate at high energies or for strongly anharmonic systems. In this article, the collective modes of motion are examined from the time-dependent perspective. It is shown that the imaginary part of the quantum-mechanical position–position correlation functions contains all the information about the collective modes of motion without the harmonic approximation on the potential energy of a system.

Probability Density of the CIR Model

We present an alternative derivation of the transition density in the Cox-Ingersoll-Ross (CIR) model. Applying methods developed in elementary quantum mechanics we show that the transition density can be determined from the eigenvalue problem of a second order differential operator with continuous spectrum. The operator turns out to be the stochastic analogue of the time-dependent quantum mechanical position operator in one dimension. Finally, to demonstrate the applicability of this technique to other one-dimensional diffusion problems, we re-derive the familiar transition densities of the Vasicek and Black-Scholes models and write down the equations of motion for the relevant operators in the Courtadon interest rate model.

High-efficiency manipulations of triply entangled states in neutron polarimetry

Entanglement occupies a peculiar position in quantum mechanics (QM). It occurs in quantum systems that consist of space-like separated parts or—more generally—in systems whose observables belong to disjoint Hilbert spaces. The latter is the case with single-neutron systems. Here, we report on a neutron polarimetric experiment, where a triply entangled Greenberger–Horne–Zeilinger state is exploited. The entanglement of spin, momentum and total energy degree of freedom is generated utilizing a suitable combination of radio-frequency and static magnetic fields. An average deviation of expectation values from theory—ideal circumstances—of 0.016(1) confirms the predictions of QM with high accuracy, demonstrating the high-efficiency manipulation of the entangled single-neutron system.

A connection between operator orderings and representations of the Lie algebra

We investigate special classes of polynomials in the quantum mechanical position and momentum operators arising from various operator orderings, in particular from the so-called μ-orderings generalizing well-known operator orderings in quantum mechanics such as the Weyl ordering, the normal ordering, etc. Viewing orderings as maps from the polynomial algebra on the phase space to the Weyl algebra generated by the quantum mechanical position and momentum operators we formulate conditions under which these maps intertwine certain naturally defined actions of the Lie algebra . These conditions arise via certain regularities in coefficients defining the orderings which can nicely be described in terms of some combinatorial objects called here ‘inverted Pascal diagrams’. At the end we establish a connection between radial elements in the Weyl algebra and certain polynomials of the ‘number operator’ expressible in terms of the hypergeometric function. This is related to another representation of , realized in terms of difference operators.

Classical and quantum study of the motion of a particle in a gravitational field

The classical and the quantum mechanical description of a one-dimensional motion of a particle in the presence of a gravitational field is thoroughly discussed. The attention is centered on the evolution of classical and quantum mechanical position probability distribution function. The classical case has been compared with three different quantum cases: (a) a quantum stationary case, (b) a quantum non-stationary zero approximation case, where the wave packet has the shape of the first eigenfunction, and (c) a quantum non-stationary general case, where the wave packet is a superposition of stationary states.

Time in quantum mechanics

Time is often said to play an essentially different role from position in quantum mechanics: whereas position is represented by a Hermitian operator, time is represented by a c-number. This difference is puzzling and has given rise to a vast literature and many efforts at a solution. It is argued that the problem is only apparent and that there is nothing in the formalism of quantum mechanics that forces us to treat position and time differently. The apparent problem is caused by the dominant role point particles play in physics and can be traced back to classical mechanics.

Poincare Covariance of Relativistic Quantum Position

A great number of problems of relativistic position in quantum mechanics are because of the use of coordinates that are not inherent objects of spacetime, cause unnecessary complications, and can lead to misconceptions. We apply a coordinate-free approach to rule out such problems. Thus it will be clear, for example, that the Lorentz covariance of position, required usually on the analogy of Lorentz covariance of spacetime coordinates, is not well posed and we show that in a right setting the Newton–Wigner position is Poincare covariant.

Quantum Mechanical Position Operator and Localization in Extended Systems

We introduce a fundamental complex quantity, ${z}_{L}$, which allows us to discriminate between conducting and nonconducting thermodynamic phases in extended quantum systems. Its phase can be related to the expectation value of the position operator, while its modulus provides an appropriate definition of a localization length. The expressions are valid for any fractional particle filling. As an illustration we use ${z}_{L}$ to characterize insulator to ``superconducting'' and Mott transitions in one-dimensional lattice models with infinite on-site Coulomb repulsion at quarter filling.

Aq-Schrödinger equation based on a Hopfq-deformation of the Witt algebra

In an earlier paper a q-Schrodinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q-deformation of the Witt algebra. This q-deformation is a deformation in the category of Lie algebras and hence the corresponding q-Witt algebra has a trivial Hopf algebra structure. In this paper, we extend the above algebra by the addition of a set of shift-type generators, which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics. The latter gives rise to a new kind of evolution equation and it is shown that in the limit q1 a specific class of Schrodinger equations is obtained from it. This specification of a particular class is a new phenomenon, because in earlier references, where a different q-deformation has been implemented or no deformation has been used at all, such a class could not be determined uniquely. The extended algebra used here has a nontrivial Hopf structure. The appearance of the shift-type generator in the q-deformed picture hence leads to a selection of a particular type of dynamics and delivers in the limit q1 new information for the characterization of the corresponding dynamics in the undeformed situation.

Quantum-Mechanical Position Operator in Extended Systems

The position operator (defined within the Schr\"odinger representation in the standard way) becomes meaningless when periodic boundary conditions are adopted for the wave function, as usual in condensed matter physics. I show how to define the position expectation value by means of a simple many-body operator acting on the wave function of the extended system. The relationships of the present findings to the Berry-phase theory of polarization are discussed.

Fourier transform identities in quantum mechanics and the quantum line

We give a quantum-mechanical interpretation of some modular identities ( LF) 3=λ L 2 arising in conformal field theory and the theory of quantum groups in relation to the mapping class group of a torus. The interpretation follows by evaluating the identity in the case of a non-standard group of the real line. The operation L takes the form of the Fourier transform of wave functions in L 2( R ) with length scale √ℏ τ/ m. We find that this can be factorised as λ L= exp(− iτ p 2/2mℏ) exp(− i x 2/2ℏτ) P exp(− iτ p 2/2mℏ), where x, p mare the usual quantum mechanical position and momentum operators , P is the parity and λ=(1−i)/√2 is a normalization constant determined by verifying the identity on wavepackets. We understand this further in terms of the observation that the Fourier transformation is realized on the wavefunctions in quantum mechanics as the evolution by a quantum harmonic oscillator of period 2 πτ for 1 4 of a cycle.

A geometric form of the canonical commutation relations

We develop here some aspects of a geometric approach to quantum theory, in which the quantum-mechanical position and momentum operators are represented by covariant derivatives. Here, the previously established formalism of Caianiello and his co-workers is extended to the case of an integrable almost complex Hermitian manifold. The general theory is then applied to the two-dimensional case, where the structure of the «quantum geometry” induced in the manifold by the quantum-mechanical CCR can be explicity determined.