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We rigorously investigate the rapidly oscillating contributions in the sinc-function representation of the Dirac delta function and the Fourier transform of the Coulomb potential. Beginning with a derivation of the standard integral representation of the Heaviside step function, we examine the representation of the Dirac delta function that contains a rapidly oscillating sinc function. By contour integration, we prove that the representation satisfies the properties of the Dirac delta function, although it is a function divergent at nonzero points. This is a good pedagogical example demonstrating the difference between a function and a distribution. In most textbooks, the rapidly oscillating contribution in the Fourier transform of the Coulomb potential into the momentum space has been ignored by regulating the oscillatory divergence with the screened potential of Wentzel. By performing the inverse Fourier transform of the contribution rigorously, we demonstrate that the contribution is a well-defined distribution that is indeed zero, even if it is an ill-defined function. Proofs are extended to exhibit that the Riemannâ€"Lebesgue lemma can hold for a sinc function, which is not absolutely integrable. [ABSTRACT FROM AUTHOR]

In this paper, we describe easy and cheap optics experiences for teaching undergraduate students the fundamental properties of Fourier transforms on an experimental basis. By exploiting the eye as the Fourier transforming device, a common magnifying lens, and quasi-coherent light from a small white LED, quantitative results can be easily obtained about the fundamental theorems used in Fourier analysis. The concept of coherence is also introduced in an elementary way. This approach has been successfully adopted to teach third-year students in physics, who operated in a completely autonomous way during the COVID-19 lockdown, without access to a laboratory. This proves the effectiveness of the method. Thanks to its experimental simplicity, it can easily be extended to teach and show the fundamental effects to a larger audience, including high-school students. [ABSTRACT FROM AUTHOR]

The problem of a solid sphere with uniform volume charge density is encountered in virtually all undergraduate calculus-based physics textbooks dealing with the topic of electromagnetism. This example illustrates well the use of Gauss's law and from there one can easily derive all the quantities of interest such as electrostatic field, potential, self-energy, and so on. Undergraduate physics majors are also well aware of the theory of Fourier transforms from having taken mathematics courses. Nevertheless, despite its great utility, the Fourier transform method is rarely mentioned as a powerful tool to solve physics problems at this level. To address this shortcoming, in this work we propose a possible scenario which may allow an instructor to introduce this powerful method to a proper undergraduate audience without any major pedagogical drawback. The case study that we choose is that of a solid sphere with uniform volume charge density. Specifically, we show the calculation of its electrostatic self-energy by using Fourier transform techniques. The main idea of this work is to draw reader's attention to the versatility of the approach that can, in principle, be applied to other more geometrically complicated bodies where Gauss's law does not lead to simple solutions. Concurrently, this work also provides instructional approaches that intertwine content-specific and pedagogical viewpoints that can be useful to all undergraduate students and teachers who wish to enhance their command of the subject. [ABSTRACT FROM AUTHOR]

We address an interesting problem in elementary quantum mechanics, namely how can one obtain the momentum state eigenfunctions for the infinite square well potential as a direct solution of the time-independent Schrödinger equation in momentum space. The conventional method for obtaining the momentum state eigenfunctions for the infinite square well potential is to take the Fourier transform of the coordinate space eigenfunctions. Unfortunately, the Schrödinger equation in momentum space for the infinite well potential is not well-defined. As such we must use a potential whose Fourier transform is well-defined and then take a limit in which the potential approaches that of the infinite well potential. We present three approaches to the problem: one based on dimensional analysis, one in which the infinite well potential is viewed as the limit of a finite well whose depth approaches infinity, and one in which the infinite well potential is viewed as the limit of a power law potential in the limit that the power approaches infinity. [ABSTRACT FROM AUTHOR]

Linear response laws and causality (the effect cannot precede the cause) are of fundamental importance in physics. In the context of classical electrodynamics, students often have a difficult time grasping these concepts because the physics is obscured by the intermingling of the time and frequency domains. In this paper, we analyse the linear response laws and causality in the time and frequency domains with the aim of pedagogical clarity. We will show that it is easy to violate causality in the frequency domain by making a vanishing absorption approximation. Furthermore, we will show that there can be subtle differences between Fourier transformingMaxwell equations and using a monochromatic source function. We discuss how these concepts can be obscured and offer some suggestions to improve the situation. [ABSTRACT FROM AUTHOR]

We analyse two-mirror resonators in terms of their fractional Fourier transform (FRFT) properties. We use the basic ABCD ray transfer matrix method to show how the resonator can be regarded as the cascade of two propagation-lens-propagation FRFT systems. Then, we present a connection between the geometric properties of the resonator (the g parameters) and those of the equivalent FRFT systems (the FRFT order and scaling parameters). Expressions connecting Gaussian beam q-transformation with FRFT parameters are derived. In particular, we show that the beam waist of the resonator's mode is located at the plane leading to two FRFT subsystems with equal scaling parameter which, moreover, coincides with the mode Rayleigh range. Finally we analyse the resonator's stability diagram in terms of the fractional orders of each FRFT subsystem, and the round trip propagation. The presented analysis represents an interesting link between two topics (optical resonators and Fourier optics) usually covered in optics and photonics courses at university level, which can be useful to teach and connect the principles of these subjects. [ABSTRACT FROM AUTHOR]

Fourier analysis plays a major role in the analysis and understanding of many phenomena in physics and contemporary engineering. However, students, who have often discovered this notion through numerical tools do not necessarily understand all the richness that can be derived from joint analysis in the temporal and spectral domains, particularly in the field of optics. As part of the second year of the master’s degree in Physics Lasers and Materials at the University of Burgundy, we have set up a series of experiments to highlight these concepts and to show, on a non-trivial example of periodic phase modulation, the precautions to be taken in the interpretation of the various experimentally accessible spectra. This hands-on session, made possible through the use of research infrastructure, is also an introduction to the use of standard optical telecommunications equipment. [ABSTRACT FROM AUTHOR]

It is well known that topics in physics which rely heavily on mathematics can present difficulties for graduate students; for example, problems involving the application of the Fourier transform. To overcome some related difficulties, an integrated approach where physics, mathematics and laboratory activities are strongly interconnected can be helpful. In this view, the present paper describes a device and a data analysis approach whose physics and mathematics facets are jointly taken into account. More specifically, we shall deal with a revised Rüchardt’s device, a topic that plays an important role in the physics curriculum, and we shall introduce the Fourier transform operator in order to extract the average oscillation frequency, thus avoiding computation of the average period of oscillations in the time space. Within this framework, it is shown how the use of the Fourier transform furnishes a straightforward measure of the experiment’s relevant physical quantities. [ABSTRACT FROM AUTHOR]

The Fourier transform method can be applied to obtain electromagnetic knots, which are solutions of Maxwell equations in a vacuum with non-trivial topology of the field lines and special properties. The program followed in this work allows us to present the main ideas and the explicit calculations at undergraduate level, so they are not obscured by a more involved formulation. We make use of the helicity basis for calculating the electromagnetic helicity and the photon content of the fields. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study double-slit diffraction at the nanometric scale. We treat the time evolution of a Gaussian wave packet in quantum mechanics traveling until it reaches a double-slit wall. A detailed investigation and analysis of the diffraction phenomenon and its progress is developed by numerically solving the Schrödinger equation. This is written using a potential intended to model the wall. It is solved by using finite differences in the time domain method, so that the probability amplitude, ψ, is obtained as a function of the spatial coordinates at discretized times. Then, the probability density, ρ, is found in a straightforward way and used to plot graphics or make videos. Furthermore, the time evolution of the wave packet is registered at several fixed points to acquire local information. Fourier transforms are calculated to complement the knowledge obtained. Students interested in didactic approaches, or in applications for nanodevices, could find interest in this approach. [ABSTRACT FROM AUTHOR]

Numerical differentiation is commonly used by a number of science students and researchers for data analysis. The differentiation of vectors of data points representing discrete samples of some underlying signal can be implemented in a computer using the central differencing scheme or the fast Fourier transform (FFT)-based approach. We point out that a simple extension of the continuous Fourier transform derivative identity to the discrete case, however, gives rise to results that are inconsistent with the central differencing scheme. In particular, for functions with step-like discontinuities, the numerical derivative computed with the FFT-based approach is corrupted by ringing artifacts. We describe the idea of a modified wave number for FFT-based numerical differentiation which leads to results that are consistent with the central differencing scheme. Modified wave number identities for numerical computation of both first and second order derivatives are described. The pitfalls of using a simplistic extension of the derivative identity for continuous Fourier transform to the discrete case and the methodology to avoid them are illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

In this work, the results of an analysis of the response of a near-resonant series resistance−inductance−capacitance (RLC) electric circuit with time-dependent forcing frequency by means of a wavelet cross-correlation approach are reported. In particular, it is shown how the wavelet approach enables frequency and time analysis of the circuit response to be carried out simultaneously—this procedure not being possible by Fourier transform, since the frequency is not stationary in time. A series RLC circuit simulation is performed by using the Simulation Program with Integrated Circuits Emphasis (SPICE), in which an oscillatory sinusoidal voltage drive signal of constant amplitude is swept through the resonant condition by progressively increasing the frequency over a 20-second time window, linearly, from 0.32 Hz to 6.69 Hz. It is shown that the wavelet cross-correlation procedure quantifies the common power between the input signal (represented by the electromotive force) and the output signal, which in the present case is a current, highlighting not only which frequencies are present but also when they occur, i.e. providing a simultaneous time-frequency analysis. The work is directed toward graduate Physics, Engineering and Mathematics students, with the main intention of introducing wavelet analysis into their data analysis toolkit. [ABSTRACT FROM AUTHOR]

In the present work, we develop a low-cost and simple experiment to visualise Fourier’s synthesis using a short, soft, and light plastic coiled spring oscillating in a horizontal plane, and a basic camera (120 fps). It is shown that the spring obeys a linear wave differential equation, as gravitational influence is neglected. A nonlinear criterion is evaluated to determine if magnitudes of the parameters in the initial conditions satisfy the linear wave equation. Our setup promotes some desirable characteristics that make Fourier’s synthesis experiments feasible, visual, and enlightening: (i) it requires few, common, and cheap resources, and the experiment can be carried out even in a high-school laboratory; (ii) since the spring’s tension is small (∼1 N, on average), the frequencies of normal modes are low (close to 2 Hz), and therefore, it is possible to record the oscillations just with the camera and extract a considerable number of position and time data in just one cycle; (iii) when the video is loaded in the Tracker free software, it can be reproduced in slow motion. Since the frequencies involved are low, an interesting and instructive temporal sequence of images of the spring displaying the typical trapezoidal shape appears clearly; (iv) the tools associated with the Tracker software tools can yield the relevant oscillation parameters, such as the damping constant, amplitudes, frequencies, and phases; and (v) it is possible to carry out superposition of a snapshot of the spring in Tracker at any time, and to draw the related Fourier synthesis graphs. The visual match between the shape of the spring and the theoretical graph is remarkable, and can be enhanced by adding the damping term. [ABSTRACT FROM AUTHOR]

The most important quantitative aspects of computational structural crystallography can be introduced in a satisfactory way using 1D truncated and periodic Gaussian functions to represent the atoms in a crystal lattice. This paper describes in detail and demonstrates 1D structural crystallography starting with the definition of such truncated Gaussians. The availability of the computer programme CRONE makes possible the repetition of the examples provided in the paper as well as the creation of new ones. [ABSTRACT FROM AUTHOR]

The present work highlights how, in order to characterize the motion of a variable mass pendulum, wavelet analysis can be an effective tool in furnishing information on the time evolution of the oscillation spectral content. In particular, the wavelet transform is applied to process the motion of a hung funnel that loses fine sand at an exponential rate; it is shown how, in contrast to the Fourier transform which furnishes only an average frequency value for the motion, the wavelet approach makes it possible to perform a joint time–frequency analysis. The work is addressed at undergraduate and graduate students. [ABSTRACT FROM AUTHOR]

Coherent sources of light are easily available to university undergraduate laboratory courses and the demonstration of electro-magnetic wave diffraction is typically made with light. However, the construction of arbitrary patterns for the study of light diffraction is particularly demanding due to the small linear scale needed when using sub-micrometer wavelengths, limiting the possibility to thoroughly investigate diffraction experimentally. We describe and test a simple and affordable method to develop arbitrary light diffraction patterns with first year undergraduate or last year high school students. This method is exploited to investigate experimentally the connection between diffraction and the Fourier transform, leading to the development of the concept of spectral analysis of a (2D) signal. We therefore discuss the possibility of building a teaching unit for first year undergraduate or last year high school students on the interdisciplinary topic of spectral analysis starting from an experimental approach to light diffraction. [ABSTRACT FROM AUTHOR]

The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation. [ABSTRACT FROM AUTHOR]

The literature on the exponential Fourier approach to the one-dimensional quantum harmonic oscillator problem is revised and criticized. It is shown that the solution of this problem has been built on faulty premises. The problem is revisited via the Fourier sine and cosine transform method and the stationary states are properly determined by requiring definite parity and square-integrable eigenfunctions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Magnetic recording head theory is based on the Karlqvist approximation to solve the Laplace equation over a polygonal domain that originates from a magnetostatic approach to describe the magnetic field produced by the read/write head in the recording medium. The approximation is reviewed and compared to various approaches dealing with solving the Laplace equation using different boundary conditions. The solution is obtained by the Green function, Fourier transform, Fourier series and finally by conformal mapping methods that allow us, on one hand, to comply with the Sommerfeld edge condition required at angular points and on the other, to obtain exact results. [ABSTRACT FROM AUTHOR]

We point out two interesting features of position-momentum uncertainty product: U = ΔxΔp. We show that two special (non-differentiable) eigenstates of the Schrödinger operator with the Dirac delta potential [V(x) = −V0δ(x)], V0 > 0, also satisfy Heisenberg’s uncertainty principle by yielding . One of these eigenstates is a zero-energy and zero-curvature bound state. [ABSTRACT FROM AUTHOR]