Quantum Mechanical Wave Function Research Articles

Ground-state ferromagnetic transition in strongly repulsive one-dimensional Fermi gases

We prove that as a one-dimensional Fermi gas is brought across the resonance adiabatically from large repulsion to large attraction, the singlet ground state will give way to the maximum spin state, which is the lowest energy state among the states accessible to the system in this process. In the presence of tiny symmetry-breaking fields that destroy spin conservation, the singlet ground state can evolve to the ferromagnetic state or a spin segregated state. We have demonstrated these effects by exact calculations on fermion cluster relevant to current experiments, and have worked out the quantum-mechanical wave function that exhibits phase separation.

Complex cooridinate scaling and the Schrödinger equation

The complex rotation method (CRM) for the description of resonance states is critically analyzed by noting that quantum mechanical wave functions and properties are not affected by a change in spatial coordinates, complex or otherwise. It is shown by means of the Cauchy-Coursat Theorem that equivalent approximate solutions of the Schrodinger equation for a complex-rotated Hamiltonian H(θ) can be obtained without loss of accuracy by using the un-rotated Hamiltonian H(0) in its place. Despite the fact that the latter operator is hermitean, it is possible to obtain a complex symmetric matrix representation for it by following a few simple rules: (a) the square-integrable basis functions must have complex exponents, i.e. with non-zero imaginary components and (b) the symmetric scalar product must be employed to compute matrix elements of H(0). The approximate wave functions obtained by diagonalization of the latter matrix should satisfy the stationary principle as closely as possible. This objective can optimally be achieved by individually scaling the complex exponents in the basis functions. The nature of this approximation is investigated by means of explicit calculations which are based on diabatic RKR potentials for the B1Σ+-D′1Σ+ vibronic resonance states of the CO molecule.

Observing Anderson Co-Localization of Entangled Photon Pairs

In 1958, Philip Anderson suggested that a quantum mechanical wave function may undergo localization in a disordered lattice as a result of interference between different paths arising from multiple scattering. While direct evidence of this localization remains elusive in solid state physics, its optical analogue may be realized in random arrays of coupled waveguides.

Atoms — How small, and how large!

The spatial extent of a free or an isolated atom can be gauged by defining its radius, and there are several different ways in which this quantity can be defined. Four of them are considered in this article. Quantum mechanical wave functions are employed to estimate the most probable and the average radii of atoms. Atomic polarizability radius and van der Waals radius are also considered, and all the four radii are examined across the periodic table. Atomic ions are then briefly dwelt upon, and a big contrast is noted in the sizes of (free) Li+ ion and the exotic H− ion, both iso-electronic with helium atom. Brief mention is also made of large exotic species like the metastable atoms and Rydberg atoms. The article seeks to provide a quantitative glimpse of how small or large the atomic systems can be.

Ultracold neutron detectors based on 10B converters used in the qBounce experiments

Gravity experiments with very slow, so-called ultracold neutrons connect quantum mechanics with tests of Newton's inverse square law at short distances. These experiments face a low count rate and hence need highly optimized detector concepts. In the frame of this paper, we present low-background ultracold neutron counters and track detectors with micron resolution based on a 10B converter. We discuss the optimization of 10B converter layers, detector design and concepts for read-out electronics focusing on high-efficiency and low-background. We describe modifications of the counters that allow one to detect ultracold neutrons selectively on their spin-orientation. This is required for searches of hypothetical forces with spin–mass couplings.The mentioned experiments utilize a beam-monitoring concept which accounts for variations in the neutron flux that are typical for nuclear research facilities. The converter can also be used for detectors, which feature high efficiencies paired with high spatial resolution of 1–2μm. They allow one to resolve the quantum mechanical wave function of an ultracold neutron bound in the gravity potential above a neutron mirror.

Charge transport through interfaces: a tight-binding toy model and its implications

With the help of a tight-binding (TB) electronic-structure toy model we investigate the matching of parameters across hetero-interfaces . We demonstrate that the virtual crystal approximation, commonly employed for this purpose, may not respect underlying symmetries of the electronic structure. As an alternative approach we propose a method which is motivated by the matching of wave functions in continuous-space quantum mechanics. We show that this method obeys the required symmetries and can be applied in simple band to band transitions. Extension to multiple interfaces and to more sophisticated TB models is discussed.

Comparison of quantum and classical relaxation in spin dynamics.

The classical Landau-Lifshitz equation with a damping term has been derived from the time evolution of a quantum mechanical wave function under the assumption of a non-Hermitian Hamilton operator. Further, the trajectory of a classical spin (S) has been compared with the expectation value of the spin operator (Ŝ). A good agreement between classical and quantum mechanical trajectories can be found for Hamiltonians linear in Ŝ or S, respectively. Quadratic or higher order terms in the Hamiltonian result in a disagreement.

Use of Harriman’s construction in determining molecular equilibrium states

Information-theoretic description of the electron probabilities and currents in molecules is extended to cover the complex amplitudes (wave functions) of quantum mechanics. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. The previous one-electron development in such an entropic perspective on the molecular electronic structure is extended to cover N-electron states by adopting the Harriman-type framework of equidensity orbitals. This analysis emphasizes the phase part of electronic states, which generates the probability-current density and the associated non-classical entropy contributions, which allow one to distinguish the information content of states generating the same electron density and differing in their current composition. A complementary character of the Fisher and Shannon information measures is explored in the associated vertical (density-constrained) information principles, for determining the equilibrium state corresponding to the fixed ground-state electron density. It is argued that the lowest “thermodynamic” state generally differs from the true ground state of the system, by exhibiting the space-dependent phase and hence also the non-vanishing probability current, linked to the system electron distribution.

Entropic representation in the theory of molecular electronic structure

The entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrodinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.

The uncertainty product of position and momentum in classical dynamics

It is generally believed that the classical regime emerges as a limiting case of quantum theory. Exploring such quantum-classical correspondences provides a deeper understanding of foundational aspects and has attracted a great deal of attention since the early days of quantum theory. It has been proposed that since a quantum mechanical wave function describes an intrinsic statistical behavior, its classical limit must correspond to a classical ensemble—not to an individual particle. This idea leads us to ask how the uncertainty product of canonical observables in the quantum realm compares with the corresponding dispersions in the classical realm. In this paper, we explore parallels between the uncertainty product of position and momentum in stationary states of quantum systems and the corresponding fluctuations of these observables in the associated classical ensemble. We confine ourselves to one-dimensional conservative systems and show, with the help of suitably defined dimensionless physical quantities, that first and second moments of the canonical observables match with each other in the classical and quantum descriptions—resulting in identical structures for the uncertainty relations in both realms.

AN ENTROPIC PICTURE OF EMERGENT QUANTUM MECHANICS

Quantum mechanics emerges à la Verlinde from a foliation of ℝ3by holographic screens, when regarding the latter as entropy reservoirs that a particle can exchange entropy with. This entropy is quantized in units of Boltzmann's constant kB. The holographic screens can be treated thermodynamically as stretched membranes. On that side of a holographic screen where spacetime has already emerged, the energy representation of thermodynamics gives rise to the usual quantum mechanics. A knowledge of the different surface densities of entropy flow across all screens is equivalent to a knowledge of the quantum-mechanical wavefunction on ℝ3. The entropy representation of thermodynamics, as applied to a screen, can be used to describe quantum mechanics in the absence of spacetime, that is, quantum mechanics beyond a holographic screen, where spacetime has not yet emerged. Our approach can be regarded as a formal derivation of Planck's constant ℏ from Boltzmann's constant kB.

A self-assembled room temperature nanowire infrared photodetector based on quantum mechanical wavefunction engineering

A novel electrochemically self-assembled semiconductor nanowire infrared photodetector is demonstrated. Its operation is based on excitation of electrons from shallow trap levels in the bandgap into the conduction band—a process which prefers photon-induced excitations over phonon-induced excitations because of the nature of the initial and final state wavefunctions. This preference results in a reasonable signal-to-noise ratio (ratio of current under illumination to current in the dark) at room temperature. The signal-to-noise is further increased by integrating the nanowire with a tunnel barrier. The normal detector without the tunnel barrier has a normalized detectivity exceeding 107cm-Hz/W at room temperature (at 1V bias), while the tunnel detector has a ∼36 times smaller detectivity but also ∼7500 times smaller standby power dissipation and a slightly higher signal-to-noise ratio.

Diffeomorphism groups and nonlinear quantum mechanics

This talk is dedicated to my friend and collaborator, Prof. Dr. Heinz-Dietrich Doebner, on the occasion of his 80th birthday. I shall review some highlights of the approach we have taken in deriving and interpreting an interesting class of nonlinear time-evolution equations for quantum-mechanical wave functions, with few equations; more detail may be found in the references. Then I shall comment on the corresponding hydrodynamical description.

Screening properties and phase transitions in unconventional plasmas for Ising-type quantum Hall states

Utilizing large-scale Monte-Carlo simulations, we investigate an unconventional two-component classical plasma in two dimensions which controls the behavior of the norms and overlaps of the quantum-mechanical wavefunctions of Ising-type quantum Hall states. The plasma differs fundamentally from that which is associated with the two-dimensional XY model and Abelian fractional quantum Hall states. We find that this unconventional plasma undergoes a Berezinskii-Kosterlitz-Thouless phase transition from an insulator to a metal. The parameter values corresponding to Ising-type quantum Hall states lie on the metallic side of this transition. This result verifies the required properties of the unconventional plasma used to demonstrate that Ising-type quantum Hall states possess quasiparticles with non-Abelian braiding statistics.

Nonrelativistic limit of quantum field theory in inertial and noninertial frames and the principle of equivalence

We discuss the non-relativistic limit of quantum field theory in an inertial frame, in the Rindler frame and in the presence of a weak gravitational field, highlighting and clarifying several subtleties. We study the following topics: (a) While the action for a relativistic free particle is invariant under the Lorentz transformation, the corresponding action for a non-relativistic free particle is not invariant under the Galilean transformation, but picks up extra contributions at the end points. This leads to an extra phase in the non-relativistic wave function under a Galilean transformation, which can be related to the rest energy of the particle even in the non-relativistic limit. (b) We show how the solution to the generally covariant Klein-Gordon equation in a non-inertial frame, which has a time-dependent acceleration, reduces to the quantum mechanical wave function in the presence of an appropriate (time-dependent) gravitational field, in the non-relativistic limit. The extra phase acquired by the non-relativistic wave function in an accelerated frame actually arises from the gravitational time dilation and survives in the non-relativistic limit. (c) We provide a detailed description of the non-relativistic limit of the Feynman propagator in a weak gravitational field, and discuss related issues. [Abridged Abstract]

Origin of the different transport properties of electron and hole polarons in an ambipolar polyselenophene-based conjugated polymer

Understanding the mechanisms limiting ambipolar transport in conjugated polymer field-effect transistors (FETs) is of both fundamental and practical interest. Here, we present a systematic study comparing hole and electron charge transport in an ambipolar conjugated polymer, semicrystalline poly(3,3${}^{\ensuremath{'}\ensuremath{'}}$-di-$n$-decylterselenophene) (PSSS). Starting from a detailed analysis of the device characteristics and temperature/charge-density dependence of the mobility, we interpret the difference between hole and electron transport through both the Vissenberg-Matters and the mobility-edge model. To obtain microscopic insight into the quantum mechanical wave function of the charges at a molecular level, we combine charge modulation spectroscopy (CMS) measuring the charge-induced absorption signatures from positive and negative polarons in these ambipolar FETs with corresponding density functional theory (DFT) calculations. We observe a significantly higher switch-on voltage for electrons than for holes due to deep electron trap states, but also a higher activation energy of the mobility for mobile electrons. The CMS spectra reveal that the electrons that remain mobile and contribute to the FET current have a wave function that is more localized onto a single polymer chain than that of holes, which is extended over several polymer chains. We interpret this as evidence that the transport properties of the mobile electrons in PSSS are still affected by the presence of deep electron traps. The more localized electron state could be due to the mobile electrons interacting with shallow trap states in the vicinity of a chemical, potentially water-related, impurity that might precede the capture of the electron into a deeply trapped state.

Use of equivalent Hermitian Hamiltonian for PT-symmetric sinusoidal optical lattices

We show how the beam dynamics of non-Hermitian PT-symmetric sinusoidal optical lattices can be approached from the point of view of the equivalent Hermitian problem, obtained by an analytic continuation in the transverse spatial variable x. In this latter problem the eigenvalue equation reduces to the Mathieu equation, whose eigenfunctions and properties have been well studied. That being the case, the beam propagation, which parallels the time-development of the wave-function in quantum mechanics, can be calculated using the equivalent of the method of stationary states. We also discuss a model potential that interpolates between a sinusoidal and periodic square well potential, showing that some of the striking properties of the sinusoidal potential, in particular birefringence, become much less prominent as one goes away from the sinusoidal case.

Optimal scaling factors for CM1 and CM3 atomic charges in RM1‐based aqueous simulations

Scaling factors for atomic charges derived from the RM1 semiempirical quantum mechanical wavefunction in conjunction with CM1 and CM3 charge models have been optimized by minimizing errors in absolute free energies of hydration, ΔG(hyd) , for a set of 40 molecules. Monte Carlo statistical mechanics simulations and free energy perturbation theory were used to annihilate the solutes in gas and in a box of TIP4P water molecules. Lennard-Jones parameters from the optimized potentials for liquid simulations-all atom (OPLS-AA) force field were utilized for the organic compounds. Optimal charge scaling factors have been determined as 1.11 and 1.14 for the CM1R and CM3R methods, respectively, and the corresponding unsigned average errors in ΔG(hyd) relative to experiment were 2.05 and 1.89 kcal/mol. Computed errors in aniline and two derivatives were particularly large for RM1 and their removal from the data set lowered the overall errors to 1.61 and 1.75 kcal/mol for CM1R and CM3R. Comparisons are made to the AM1 method which yielded total errors in ΔG(hyd) of 1.50 and 1.64 kcal/mol for CM1A*1.14 and CM3A*1.15, respectively. This work is motivated by the need for a highly efficient yet accurate quantum mechanical (QM) method to study condensed-phase and enzymatic chemical reactions via mixed QM and molecular mechanical (QM/MM) simulations. As an initial test, the Menshutkin reaction between NH(3) and CH(3) Cl in water was computed using a RM1/TIP4P-Ew/CM3R procedure and the resultant ΔG(‡) , ΔG(rxn) , and geometries were in reasonable accord with other computational methods; however, some potentially serious shortcomings in RM1 are discussed.

Pre-service physics teachers comprehension of wave function and operator concepts in quantum mechanics

This qualitative study investigates students’ comprehension of wave function and operator concepts. The data of the study were collected by semi-structured interviews with 34 pre-service physics teachers and analyzed by using the content analysis method. As a result of qualitative analysis, different categories were determined towards the students’ comprehension of wave function and operator concepts. These results showed that (1) students did not have enough knowledge to define the concepts of wave function and operator properly, (2) they were influenced by the perspective of classical physics in making explanations related to the concepts of quantum mechanics, and (3) a great number of students who participated in the study made explanations by using examples frequently used in quantum mechanics. It is hoped that the results of this study will help the educators to recognize the students’ comprehensions in both modern physics and quantum mechanics courses. Key words: Comprehension, modern physics, operator, physics education, quantum mechanics, wave function.

Nonlocal phases of local quantum mechanical wavefunctions in static and time-dependent Aharonov–Bohm experiments

We show that the standard Dirac phase factor is not the only solution of the usual gauge transformation equations. The full form of a general gauge function (that connects systems that move in different sets of scalar and vector potentials), apart from Dirac phases (spatial or temporal integrals over potentials), also contains terms of classical fields that act nonlocally (in spacetime) on the local solutions of the time-dependent Schrödinger equation. As a result, the phases of wavefunctions in the Schrödinger picture are affected nonlocally by spatially and temporally remote magnetic and electric fields, in specific ways that are fully explored. These contributions go beyond the usual Aharonov–Bohm effects (magnetic or electric). (i) Application to cases of particles passing through full static magnetic or electric fields leads to cancellations of Aharonov–Bohm phases at the observation point; these cancellations are linked to behaviors at the semiclassical level (i.e. the old Werner and Brill experimental observations, or their ‘electric analogs’—or to more recent reports of Batelaan and Tonomura) but are shown to be far more general (true not only for narrow wavepackets but also for completely delocalized (spread-out) quantum states). By using these cancellations, certain previously unnoticed sign-errors in the literature are corrected. (ii) Application to time-dependent situations provides a remedy for erroneous results in the literature (concerning an uncritical use of Dirac phase factors) and leads to phases that contain an Aharonov–Bohm part and a field-nonlocal part: their competition is shown to recover relativistic causality in earlier ‘paradoxes’ (such as the van Kampen thought-experiment), while a more general consideration indicates that the temporal nonlocalities found here demonstrate in part a causal propagation of phases of quantum mechanical wavefunctions in the Schrödinger picture. This may open a new and direct way to address time-dependent double-slit experiments and the associated causal issues.