Quantum backflow is the unexpected effect that wavepackets consisting of only positive momentum components can apparently move in the negative direction. This is usually described in terms of the backflow constant, which is a dimensionless quantity describing least upper bound on the amount of probability that can flow backwards during a given time interval. Backflow is usually calculated for wavepackets that can be written as a sum of positive momentum plane waves. Here we present a calculation of the backflow constant using the localised free particle hermite wavefunctions where equal weights of positive and negative momentum eigenfunctions occur. The resulting backflow constant is substantially smaller than the accepted value. The reasons for this are discussed and finally we draw conclusions about the calculation of backflow more generally.

We apply the time-dependent supersymmetry methods of Bagrov, Samsonov and Shekoyan to the Schrödinger equation for a quantum bouncer. A new family of potentials, that correspond to the linear gravitational potential with a small oscillatory component superimposed upon it, is produced. Both the frequency and amplitude of the oscillatory part can be controlled and the corresponding eigenfunctions are found. These solutions are explored and basic observables are calculated. In particular we examine how motion in such a potential can be distinguished from motion in a linear gravitational field. We also point out some pedagogical aspects of this project.

If a wave function is written in polar form it becomes possible to write the Schrodinger equation of nonrelativistic quantum mechanics in a form analogous to the classical Hamilton-Jacobi equation with an extra term known as the quantum potential. Time-dependent supersymmetry is a procedure for finding new solutions of the Schrodinger equation if one solution is known. In this paper a time-dependent supersymmetry transformation is applied to a wave function in this polar form and it is shown that the classical potential plus the quantum potential is a conserved quantity under this transformation under certain circumstances. This leads to a modification of our view of the role of the quantum potential and also to a deeper appreciation of the function of a supersymmetry transformation.

We analyse the evolution of superoscillations in a relativistic wavepacket. A simple superoscillating wavepacket is set up and is allowed to evolve freely according to both the Klein–Gordon equation and the Dirac equation. The superoscillations evolve anisotropically and decay after a time. Both the lifetime and anisotropy can be understood in terms of the interaction of contributions to the wavepacket from components with strongly differing complex wavenumber. The analysis is supported by numerical calculations and the results are compared with the non-relativistic analysis. A potential experiment in which the significance of relativistic effects on superoscillations could be measured is proposed.

Here we report a project in which time-dependent supersymmetry has been employed to derive a new potential and eigenfunctions that satisfy the Schrödinger equation. The supersymmetry method is outlined and we apply it to a wavefunction obeying the free-particle Schrödinger equation. This leads to an exactly soluble model in which a quantum particle is seen to ‘surf’ on a time-dependent potential. The model can be solved and understood within both classical and quantum mechanics and the relationship between the two approaches is discussed. The mathematics of this formalism is accessible to a final year British undergraduate making supersymmetry derived Hamiltonians suitable as a final year theoretical physics research project.

In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is written down and it is shown that the backflow can be written as a function of a simple parameter which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Then backflow eigenvectors are fitted with some standard functions which lead to substantially higher backflow than has been found previously with fitting procedures, for some values of. In analysing the non-relativistic limit of the theory we show that this problem is one of those rare cases where the relativistic theory is intrinsically more simple than the non-relativistic theory.

In this paper we discuss quantum curl forces. We present both the classical and quantum theory of linear curl forces. The quantum theory is shown to reproduce the classical theory precisely if appropriate combinations of eigenfunctions are chosen. A series of examples are used to illustrate the theory and to demonstrate its limitations. Furthermore, we are able to point out an analogy between the quantum theory of curl forces and some of the squeezed light states of quantum optics.

Lipkin's zilches are a set of little-known conserved quantities in classical electromagnetic theory. Here we report a systematic calculation of the zilches for topologically non-trivial vacuum electromagnetic fields and their interpretation in terms of both the physical and mathematical properties of the fields. Several families of electromagnetic fields have been explored and examined computationally. In these cases it is found that the zilches can be written in terms of more familiar conserved quantities: energy, momentum and angular momentum. Furthermore we demonstrate that the zilches also contain information about the topology of the field lines for the fields we have examined, thus providing a previously unsuspected aspect to their interpretation. We conjecture that these properties generalise to all integrable fields.

Quantum-mechanical spin is often thought of in terms of classical angular momentum. In fact spin is defined by its commutation relations and the spin and orbital angular momentum operators are very different. Here we solve the Dirac equation in a rotating frame of reference and create a localized wave packet from the resulting wave functions to examine the consequences of the rotational motion. We highlight some unexpected effects in the properties of the wave packet and show that the spin operator and orbital angular momentum operator describe different aspects of the rotational properties of the wave packet. It is also observed that neither quantum-mechanical spin nor orbital angular momentum can be fully understood within an inertial frame.

We discuss null knotted solutions to Maxwell's equations, their creation through Bateman's construction, and their relation to the Hopf-fibration. These solutions have well-known, conserved properties, related to their winding numbers. For example: energy; momentum; angular momentum; and helicity. The current research has focused on Lipkin's zilches, a set of little-known, conserved quantities within electromagnetic theory that has been explored mathematically, but over which there is still considerable debate regarding physical interpretation. The aim of this work is to contribute to the discussion of these knotted solutions of Maxwell's equations by examining the relation between the knots, the zilches, and their symmetries through Noether's theorem. We show that the zilches demonstrate either linear or more complicated relations to the p-q winding numbers of torus knots, and can be written in terms of the total energy of the electromagnetic field. As part of this work, a systematic multipole expansion of the vector potential of the knotted solutions is being carried out.

The Dirac equation in a rotating frame of reference is derived from first principles within a linear approximation. This equation is employed to exhibit an equivalence between a particle in a Dirac oscillator potential and a free particle in a rotating frame of reference. A zero-point contribution to the energy of the particle, resulting from its spin, is also noted. Crown Copyright (C) 2016 Published by Elsevier B.V. All rights reserved.

A superoscillatory function is one in which the function oscillates faster than its fastest Fourier component. Superoscillations are now a well-established feature of certain types of wavepacket. Here we discuss quantum superoscillations in a wavepacket evolving according to the harmonic oscillator hamiltonian. The evolution of the wavepacket is investigated using an expansion in terms of eigenfunctions and in terms of the propagator using both an exact integration, and a saddle point approximation. The creation and decay of superoscillations as well as the barrier between them and normal oscillations is shown to depend on the behaviour of the saddle points. The work reported here is the result of a student project and we argue that super oscillations makes an excellent topic with which to introduce students to theoretical research.

In this Brief Report we discuss a solution of the free-particle Schrodinger equation in which the time and space dependence are not separable. The wave function is written as a product of exponential terms, Hermite polynomials, and a phase. The peaks in the wave function decelerate and then accelerate around t = 0. We analyze this behavior within both a quantum and a semiclassical regime. We show that the acceleration does not represent true acceleration of the particle but can be related to the envelope function of the allowed classical paths. Comparison with other "accelerating" wave functions is also made. The analysis provides considerable insight into the meaning of the quantum wave function.

We show how the two-dimensional Dirac oscillator model can describe some properties of electrons in graphene. This model explains the origin of the left-handed chirality observed for charge carriers in monolayer and bilayer graphene. The relativistic dispersion relation observed for monolayer graphene is obtained directly from the energy spectrum, while the parabolic dispersion relation observed for the case of bilayer graphene is obtained in the non-relativistic limit. Additionally, if an external magnetic field is applied, the unusual Landau-level spectrum for monolayer graphene is obtained, but for bilayer graphene the model predicts the existence of a magnetic field-dependent gap. Finally, this model also leads to the existence of a chiral phase transition.

We study the (2+1)-dimensional Dirac oscillator in the presence of an external uniform magnetic field ($B$). We show how the change of the strength of $B$ leads to the existence of a quantum phase transition in the chirality of the system. A critical value of the strength of the external magnetic field ($B_c$) can be naturally defined in terms of physical parameters of the system. While for $B=B_c$ the fermion can be considered as a free particle without defined chirality, for $BB_c$) the chirality is left (right) and there exist a net potential acting on the fermion. For the three regimes defined in the quantum phase transition of chirality, we observe that the energy spectra for each regime is drastically different. Then, we consider the $z$-component of the orbital angular momentum as an order parameter that characterizes the quantum phase transition.

We present experimental data and first-principles calculations of the x-ray resonant magnetic scattering (XRMS) line shapes of Er and Tm metals. Band structure calculations are presented that illustrate the existence of 4f/5d hybridization of unoccupied electron levels. This leads to spin and orbital features in the XRMS spectra that are not representative of the 5d occupied moments. General trends through the heavy rare earth series are discussed with reference to previous work. DOI: 10.1103/PhysRevB.87.165111

In this paper we demonstrate a surprising aspect of quantum mechanics that is accessible to an undergraduate student. We discuss probability backflow for an electron in a constant magnetic field. It is shown that even for a wavepacket composed entirely of states with negative angular momentum the effective angular momentum can take on positive values in some regions of space. This is also a demonstration of the azimuthal angle–angular momentum uncertainty relation. This example is chosen because the backflow motion persists indefinitely. However this work can easily be extended to other systems where the motion is notionally circular, such as the hydrogen atom.

We report first principles calculations of the geometry and electronic structure of 13 atom clusters of boron, aluminium, gallium and indium. These density functional theory calculations support the jellium model in the energy levels and molecular orbitals of the cluster and enable us to discuss the relevance of the superatom concept. We go on to examine a number of cluster symmetries in detail as a function of charge and comment on the successes and limitations of the jellium and superatom models in describing these clusters. In particular we find that the monovalent anionic cluster is the most stable and has the most symmetric structure. As charge changes the symmetry of the clusters decreases in a way that is dependent on symmetry and charge, but not atomic species.