Mechanics Textbooks Research Articles

Quantum versus classical quenches and the broadening of wave packets

The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.

Research on the development and application of loose-leaf textbooks for soil mechanics

The research of this paper aims to develop a new form of a digital textbook for the course of "Soil Mechanics" to solve the existing problems in higher vocational education. The Soil Mechanics textbook is out of touch with the actual production of enterprises, and its content is outdated. In order to cultivate students to adapt to the digitization requirements, we have co-developed the loose-leaf textbooks for "Soil Mechanics" in conjunction with front-line companies, which break through the framework of traditional course syllabus and textbooks, emphasizing the work process learning of actual production projects. The textbooks comprehensively consider the educational role of the course in the aspects of knowledge and skills, learning process and methodology, emotions, attitudes and values, and integrate the teaching and implementation process with the work content and knowledge together.

A generalized scattering theory in quantum mechanics

In quantum mechanics textbooks, a single-particle scattering theory is introduced. In the present work, a generalized scattering theory is presented, which can be in principle applied to the scattering problems of arbitrary number of particle. In laboratory frame, a generalized Lippmann-Schwinger scattering equation is derived. We emphasized that the derivation is rigorous, even for treating infinitesimals. No manual operation such as analytical continuation is allowed. In the case that before scattering N particles are plane waves and after the scattering they are new plane waves, the transition amplitude and transition probability are given and the generalized S matrix is presented. It is proved that the transition probability from a set of plane waves to a new set of plane waves of the N particles equal to that of the reciprocal process. The generalized theory is applied to the cases of one- and two-particle scattering as two examples. When applied to single-particle scattering problems, our generalized formalism degrades to that usually seen in the literature. When our generalized theory is applied to two-particle scattering problems, the formula of the transition probability of two-particle collision is given. It is shown that the transition probability of the scattering of two free particles is identical to that of the reciprocal process. This transition probability and the identity are needed in deriving Boltzmann transport equation in statistical mechanics. The case of identical particles is also discussed.

Radial wave function of 2D and 3D quantum harmonic oscillator

One dimensional quantum harmonic oscillator is well studied in elementary textbooks of quantum mechanics. The wave function of one-dimensional oscillator harmonic can be written in term of Hermite polynomial. Due to the symmetry of the spring energy, the wave functions of two-dimensional and three-dimensional harmonic oscillators can be written as products of the one-dimensional case. Because of that, the wave functions of two- and three-dimensional cases are focused on cartesian coordinates. In this article, we utilize polar and spherical coordinates to describe the wave function of two- and three-dimensional harmonic oscillators, respectively. The radial part of the wave functions can be written in term of associated Laguerre polynomials.

Nanocracks in nature and industry.

Nanocracks in nature and industry.

A COMPLETE RE-DERIVATION OF THE HIDDEN VARIABLE EQUATION IN DAVID BOHM’S 1952 PAPERS

Schrodinger’s equation is a basic mathematical expression for a single particle in quantum mechanics. However, its physical meaning has never been assigned in any standard quantum mechanics text book. Here, a detailed complete re-derivation of hidden variable equation in David Bohm’s 1952 papers is given to assign some physical meanings to Schrodinger’s equation. This work shows some required detailed transformation of Schrodinger’s Equation into a form of Newtonian second law of motion with definite trajectory with some initial conditions given. All the quantum effects come from the single localized term called the quantum potential.

Compiling Textbook of Elastic-plastic Mechanics for Major of Oil and Gas Storage and Transportation Engineering

"Elastic-plastic mechanics" is a required course for engineering postgraduates in petroleum colleges and universities, such as students who major in oil and gas storage and transportation engineering. It is the basis for personnel engaged in structural safety assessment. In the past teaching process, the teaching effects of this course were unsatisfactory. There are many reasons for this phenomenon, such as the strong theoretical nature of this course, the need for a large number of formula derivation, the high requirements for students' mechanical foundation and mathematical foundation, the lack of appropriate teaching materials and the slackness of students' minds. Based on years of teaching experience, the teaching team has reformed the existing teaching materials of “Elastic-plastic mechanics” to meet the needs of petroleum colleges and universities. In this reform, we have referred to a large number of published textbooks of “Elasticplastic mechanics” and the experience of textbook construction at home and abroad. The newly compiled textbook of “Elastic-plastic mechanics” plays a certain role in improving the teaching quality of “Elastic-plastic mechanics” in petroleum colleges and universities.

Derivation of the differential continuity equation in an introductory engineering fluid mechanics course

The differential continuity equation is elegantly derived in advanced fluid mechanics textbooks using the divergence theorem of Gauss, where the surface integral of the mass flux flowing out of a finite control volume is replaced by the volume integral of the divergence of the mass flux within the control volume. To avoid the need for introducing the Gauss divergence theorem in an introductory fluid mechanics course, introductory textbooks in fluid mechanics have opted to use a more simple approach, which depends on the consideration of an infinitesimal control volume and the use of Taylor series expansion. This approach, however, involves a first order truncation of the Taylor series expansion and the use of approximate equality signs which may imply to undergraduate students that the derived continuity equation is an approximate equation. The present study proposes an alternative derivation of the differential continuity equation using a finite control volume and is based on the simple concept of the antiderivative function and the fundamental theorem of calculus. The proposed derivation eliminates the need to formally introduce the Gauss divergence theorem in an introductory engineering fluid mechanics course while avoiding the use of truncated Taylor series expansion and approximate equality signs, hence providing a more simple and sound understanding of the derivation of the differential continuity equation to undergraduate engineering students.

Video analysis of the fall and vertical downward launch of Styrofoam balls with air drag

The drag effect on a falling ball caused by air is a conventional subject in the most well-known textbooks of classical mechanics and fluid dynamics. Further, there are some papers that employ video analysis to track objects movements in the air making it possible to obtain position data as a function of time and its graphs. However, none of them addresses the case of a ball thrown downwards, maybe because this situation is neglected by textbooks or probably due to the experimental difficulty to perform this kind of launch manually. In this paper, both the fall and the vertical launch down of five different Styrofoam balls are filmed with an ordinary smartphone camera and analysed by using the free software Tracker. The position and velocities graphs depicted clearly the air drag effect on the ball’s movements. The entire raw data are compared with a theoretical model and therefore terminal and initial velocities can be extracted as adjusted parameters from simple mathematical fits. The Reynolds number and the drag coefficient calculated for each ball agree with those found in the literature.

A shadow of the repulsive Rutherford scattering and Hamilton vector

The fact that repulsive Rutherford scattering casts a paraboloidal shadow is rarely exploited in introductory mechanics textbooks. Another rarely used construction in such textbooks is the Hamilton vector, a cousin of the more famous Laplace–Runge–Lenz vector. We will show how the latter (Hamilton’s vector) can be used to explain and clarify the former (paraboloidal shadow).

Strategy on choice of layback spins in figure skating

Spin is an important component of figure skating, one of the most elegant events in the Winter Olympic Games. It is always presented as an example of the conservation of angular momentum in mechanics textbooks. However, the physics behind it in the actual operation is not that simple. Herein, we analyzed videos of an elite figure skater with open source video analysis. The moments of inertia of her body in six different layback positions were obtained. The average ice resistance during her spin was found to be about 26N and was put into consideration in the following calculation. Twenty-two different layback spins that score the same basic value were discussed. The initial angular momentum a skater needs when executing a spin is considered to be the largest contributor to its difficulty; the suggested easiest spin among the 22 was thus found by comparing their initial angular momentum. This paper presents a strategy that may help figure skaters achieve a high-scored layback spin efficiently, and the process itself will be an inspiring example of applying theory to practice for physics students.

A simpler graphical solution and an approximate formula for energy eigenvalues in finite square quantum wells

The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.

Detailed solution of the problem of Landau states in a symmetric gauge

Understanding the quantum problem of a free charged particle undergoing two-dimensional motion in a perpendicular uniform, constant magnetic field is necessary to the comprehension of some very important phenomena in physics. In particular, a grasp of the nature of the Landau states in a symmetric gauge is crucial to explain the underlying principles of quantum Hall effects. In this work we provide a step-by-step solution of this quantum problem in a pedagogical fashion that is easy to follow by an audience of undergraduate students and prospective physics teachers. This approach should enable undergraduate students to comprehend all the technical mathematical details involved in the process. Such details are routinely missing from mainstream quantum mechanics textbooks. In particular, this study allows a broad audience of students and teachers to gradually absorb knowledge not only on basic principles of quantum mechanics, but also on various special mathematical functions that are encountered in the process.

Conservação da velocidade do CM de duas partículas sob a ação de forças de atrito em movimento unidimensional (D = 1)

A força de atrito está presente no cotidiano dos alunos e a compreensão de sua ação sobre os corpos aproxima a Ciência de sua experiência. A lei de conservação do momento linear total, envolvendo sistemas de partículas sob a ação de forças dissipativas externas, não é geralmente discutida nos livros-texto de Mecânica Newtoniana utilizados nos ensinos médio e superior, nas instituições de ensino brasileiras. Discutimos neste trabalho o movimento mais simples que duas partículas podem ter: o movimento unidimensional em sentidos opostos, sob a ação de forças de atrito cinético externas. Mostramos que a conservação do momento linear do conjunto formado por essas partículas, sob a ação de duas forças de atrito cinético, ocorre apenas em partes do movimento unidimensional e quando elas se deslocam em sentidos opostos. Aproveitando o grande envolvimento dos alunos com aplicativos em celulares, apresentamos uma montagem experimental que verifica a conservação da velocidade do centro de massa (CM) desses dois corpos. O aparato experimental é simples e de baixo custo, podendo ser facilmente reproduzido em sala de aula como estratégia para estimular os estudantes a trabalharem com ferramentas matemáticas e métodos experimentais, de forma a discutirem situações físicas que incluem as forças de atritos presentes no seu dia a dia.

Calculating spherical harmonics without derivatives

The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates (under $\hat L^2$ and $\hat L_z$).

Objectivity in Quantum Measurement

The objectivity is a basic requirement for the measurements in the classical world, namely, different observers must reach a consensus on their measurement results, so that they believe that the object exists "objectively" since whoever measures it obtains the same result. We find that this simple requirement of objectivity indeed imposes an important constraint upon quantum measurements, i.e., if two or more observers could reach a consensus on their quantum measurement results, their measurement basis must be orthogonal vector sets. This naturally explains why quantum measurements are based on orthogonal vector basis, which is proposed as one of the axioms in textbooks of quantum mechanics. The role of the macroscopicality of the observers in an objective measurement is discussed, which supports the belief that macroscopicality is a characteristic of classicality.

Bernoulli’s theorem obtained directly from a classical mechanics textbook

In this paper we select a small element of fluid of a given mass and follow it as it moves about in space under the influence of the external forces exerted by the rest of the fluid and by gravity, using only ideas contained in any classical mechanics textbook. Then, without the need to consult a fluid mechanics textbook, we obtain the conditions for the energy conservation in a fluid and, from these conditions, we derive the standard form of Bernoulli’s theorem.

What is the uncertainty principle of non-relativistic quantum mechanics?

After more than ninety years of discussions over the uncertainty principle, there is still no universal agreement on what the principle states. The Robertson uncertainty relation (incorporating standard deviations) is given as the mathematical expression of the principle in most quantum mechanics textbooks. However, the uncertainty principle is not merely a statement of what any of the several uncertainty relations affirm. It is suggested that a better approach would be to present the uncertainty principle as a statement about the probability distributions of incompatible variables and the resulting restrictions on quantum states.

Limited accuracy of conduction band effective mass equations for semiconductor quantum dots

Effective mass equations are the simplest models of carrier states in a semiconductor structures that reduce the complexity of a solid-state system to Schrödinger- or Pauli-like equations resempling those well known from quantum mechanics textbooks. Here we present a systematic derivation of a conduction-band effective mass equation for a self-assembled semiconductor quantum dot in a magnetic field from the 8-band k · p theory. The derivation allows us to classify various forms of the effective mass equations in terms of a hierarchy of approximations. We assess the accuracy of the approximations in calculating selected spectral and spin-related characteristics. We indicate the importance of preserving the off-diagonal terms of the valence band Hamiltonian and argue that an effective mass theory cannot reach satisfactory accuracy without self-consistently including non-parabolicity corrections and renormalization of k · p parameters. Quantitative comparison with the 8-band k · p results supports the phenomenological Roth-Lax-Zwerdling formula for the g-factor in a nanostructure.

An impacting linear three body system

We study a system of three identical bodies that can move freely on a horizontal track. Initially one body moves and two are at rest. The moving body impacts with one of the resting bodies which then impacts with the third and so on. The impacts are assumed to be characterised by a coefficient of restitution. We investigate the total number of impacts, the final velocities of the bodies, and the final energy of the system in terms of the initial velocity and the coefficient of restitution. The problem, which originates from mechanics textbooks, can be analysed as a discrete dynamical system with three degrees of freedom. The full solution is more subtle that one might expect.