Quantum Mechanics Courses Research Articles

Requirements for future quantum workforce – a Delphi study

With new quantum technologies and new applications comes a new need for specialists, the new quantum workforce. This brings new challenges for education that the typical quantum mechanics courses for physicists do not address. Requirements for the future quantum workforce need to be collected and training programmes created. In between, there should be a European competence framework on which to build the training programmes. One goal of the European Flagship project QTEdu is to develop this framework for second-generation quantum technologies. The Delphi study presented here serves as a basis for this: The aim is to identify knowledge and competences in the field of quantum information technologies, which are partly already needed in industry today, but especially in the future.

A pedagogical approach to relativity effects in quantum mechanics

A simple but precise approach to relativistic quantum mechanics is presented. The approach is based on the use of a Schrödinger-like, little-known but well-studied quantum mechanics wave equation. Such formal similitude allows undergraduate students to quantitatively explore how the results corresponding to a typical non-relativistic quantum problem change when the particle is moving at relativistic speeds. No additional mathematical skills are required. We argue in favor of the academic use of this approach for including the implications of the special theory of relativity in introductory quantum mechanics courses.

Schrödinger’s equation from Snell’s law

A new derivation of Schrödinger’s equation is presented, based on Schrödinger’s original discussions on refraction and the optical-mechanical analogy, but adopting a much simpler formalism: Newtonian mechanics and some basic elements of classical wave theory (such as Snell’s law). We compare how particles and waves refract and show that the ‘law of particle refraction’ and the ‘law of wave refraction’ may become consistent if one assumes that a particle can be represented by a wave group. In this case, the differential equation whose solutions represent the waves forming such wave group is the Schrödinger equation. Due to the simplicity of the adopted mathematical formalism, we argue that this derivation can be used in quantum mechanics courses at introductory level to give students an idea of Schrödinger’s original path to his wave equation.

Challenges in addressing student difficulties with time-development of two-state quantum systems using a multiple-choice question sequence in virtual and in-person classes

Research-validated clicker questions as instructional tools for formative assessment are relatively easy to implement and can provide effective scaffolding when developed and implemented in a sequence. We present findings from the implementation of a research-validated clicker question sequence (CQS) on student understanding of the time-development of two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course for two consecutive years in virtual and in-person classes. The effectiveness of the CQS discussed here in both modes of instruction was determined by evaluating students’ performance after traditional lecture-based instruction and comparing it to their performance after engaging with the CQS.

Investigating and improving student understanding of the basics for a system of identical particles

We discuss an investigation of students' difficulties with systems of non-interacting identical particles. The investigation was carried out in advanced quantum mechanics courses at both the undergraduate and graduate levels by administering free-response and multiple-choice questions and conducting individual interviews with students. This research on student difficulties was used as a guide to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT) that helps students develop a functional understanding of the concepts necessary for determining the many-particle Hamiltonian and stationary state wavefunctions for a system of non-interacting identical particles.

Students’ difficulties with solving bound and scattering state problems in quantum mechanics

In upper level quantum mechanics courses, the typical process of devoting significant time to the example of a finite square potential well does not provide students with sufficient mastery of the relevant physics concepts and mathematical techniques to solve new problems.

Lagrange-multiplier regularization of eigenproblem for Jx

We solve the eigenproblem of the angular momentum J_x by directly dealing with the non-diagonal matrix unlike the conventional approach rotating the trivial eigenstates of J_z. Characteristic matrix is reduced into a tri-diagonal form following Narducci–Orszag rescaling of the eigenvectors. A systematic reduction formalism with recurrence relations for determinants of any dimension greatly simplifies the computation of tri-diagonal matrices. Thus the secular determinant is intrinsically factorized to find the eigenvalues immediately. The reduction formalism is employed to find the adjugate of the characteristic matrix. Improving the recently introduced Lagrange-multiplier regularization, we identify that every column of that adjugate matrix is indeed the eigenvector. It is remarkable that the approach presented in this work is completely new and unique in that any information of J_z is not required and only algebraic operations are involved. Collapsing of the large amount of determinant calculation with the recurrence relation has a wide variety of applications to other tri-diagonal matrices appearing in various fields. This new formalism should be pedagogically useful for treating the angular momentum problem that is central to quantum mechanics course.

The Model of Multi Limits of the Quantity of Electrons in Central Coulomb force Field

Based on the theory of periodic bifurcation of iterative equation, a conjectural model of periodic bifurcation of number of electrons in a central Coulomb force field is proposed. After that with the help of the methods Zeng’s The Course of Quantum Mechanics and Wu’s Methods of Mathematical Physics, [1], [2] the wave function of the electrons under the approximate state is solved in the central Coulomb force field. By using the method of separating variables for solving partial differential equations and some transformation and construction techniques, the strict mathematical solution of the Schrödinger equation for the electron in the field of central Coulomb force is obtained, and the iterative formula of the level of electron number is given theoretically. And using MATLAB, the multi-limit model of electron number is simulated under different initial value problems, to explore the change of the limit with the initial value and the factors affecting the limit number to a certain extent. Some potential research value of this model is also proposed.

Making relativistic quantum mechanics simple

The fundamentals of a quasi-relativistic wave equation, whose solutions match the Schrödinger results for slow-moving particles but are also valid when the particle moves at relativistic speeds, are discussed. This quasi-relativistic wave equation is then used for examining some interesting quantum problems where the introduction of relativistic considerations may produce remarkable consequences. We argue in favor of the academic use of this equation, for introducing students to the implications of the special theory of relativity in introductory quantum mechanics courses.

Scattering problems via real-time wave packet scattering

In this paper, we use a straightforward numerical method to solve scattering models in one-dimensional lattices based on a tight-binding band structure. We do this by using the wave packet approach to scattering, which presents a more intuitive physical picture than the traditional plane wave approach. Moreover, a general matrix diagonalization method that is easily accessible to undergraduate students taking a first course in quantum mechanics is used. Beginning with a brief review of wave packet transport in the continuum limit, comparisons are made with its counterpart in a lattice. The numerical results obtained through the diagonalization method are then benchmarked against analytic results. The case of a resonant dimer is investigated in the lattice, and several resonant values of the mean wave packet momentum are identified. The transmission coefficients obtained for a plane wave incident on a step potential and rectangular barrier are compared by investigating an equivalent scenario in a lattice. Finally, we present several short simulations of the scattering process, which emphasize how a simple methodology can be used to visualize some remarkable phenomena.

Accidental degeneracy of the hydrogen atom and its non-accidental solution in parabolic coordinates

The degeneracy associated with dynamical symmetry of a potential can be identified in quantum mechanics, by solving the Schrödinger equation analytically, using the method of separation of variables in at least two different coordinate systems, and in classical mechanics by solving the Hamilton–Jacobi equation. In the present pedagogical review, the notion of separability and superintegrability of a potential, with profound implications, is discussed. In an earlier tutorial paper, we addressed the n2-fold degeneracy of the hydrogen atom using the Casimir operators corresponding to the SO(4) symmetry of the 1/r potential. The present paper is a sequel to that work, in which we solve the Schrödinger equation for the hydrogen atom using separation of variables in the parabolic coordinate systems. In doing so, we take the opportunity to revisit some excellent works on symmetry and degeneracy in classical and quantum physics, if only to draw attention to these insightful studies, which unfortunately miss even a mention in most undergraduate and even graduate level courses in quantum mechanics and atomic physics.

The polarizability of a confined atomic system: an application of the Dalgarno–Lewis method

In this paper we give an application of the Dalgarno–Lewis method, the latter not usually taught in quantum mechanics courses. This is very unfortunate since this method allows one to bypass the sum over states appearing in the usual perturbation theory. In this context, and as an example, we study the effect of an external field, both static and frequency-dependent, on a model atom at fixed distance from a substrate. This can happen, for instance, when some organic molecule binds from one side to the substrate and from the other side to an atom or any other polarizable system. We model the polarizable atom by a short range potential, a Dirac-δ, and find that the existence of a bound state depends on the ratio of the effective ‘nuclear charge’ to the distance of the atom from the substrate. Using an asymptotic analysis, previously developed in the context of a single δ-function potential in an infinite medium, we determine the ionization rate and the Stark shift of our system. Using Dalgarno–Lewis theory we find an exact expression for the static and dynamic polarizabilities of our system valid to all distances. We show that the polarizability is extremely sensitive to the distance from the substrate, creating the possibility of using this quantity as a nanometric ruler. Furthermore, the line shape of the dynamic polarizability is also extremely sensitive to the distance from the substrate, thus providing another route to measure nanometric distances. The didactic value of the δ-function potential is widely accepted in teaching activities due to its simplicity, while still keeping the essential ingredients of a given problem.

Quantum mechanics for non-inertial reference frames

In quantum mechanics courses, the concept of a reference frame is frequently hidden. In this article, we present a basic perspective to introduce the transformations of a reference frame based on unitary transformations. These unitary transformations are inspired in canonical transformation for reference frame from the perspective of classical mechanics. The manner we deal with this change in reference frame allows us to discuss some fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. These unitary transformations are inspired in a reference-frame canonical transformation from the perspective of classical mechanics. The approach we take for this reference-frame change allows for a discussion on fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. We apply this unitary transformation in five examples.

Quantum composer: A programmable quantum visualization and simulation tool for education and research

Interactive simulations and visualizations augment the teaching and learning of quantum mechanics by making equations and concepts come to life. However, graphical visualizations are nearly always limited to a set of hard-coded functionalities. Text-based codes can offer a higher degree of flexibility, but only at the expense of steep learning curves or time investments. We introduce Quantum Composer, which allows the user to build, expand, or explore quantum mechanical simulations by interacting with graphically connectable nodes, each corresponding to a physical concept, mathematical operation, or visualization. Quantum Composer eliminates numerical and programming details while retaining accessibility, emphasis on understanding, and rapid feedback mechanisms. We illustrate its open-ended applicability through a series of examples in introductory and advanced quantum mechanics courses, student projects, and research environments.

Introducing SU(3) color charge in undergraduate quantum mechanics

We present a framework for investigating effective dynamics of SU(3) color charge. Two- and three-body effective interaction terms inspired by the Heisenberg spin model are considered. In particular, a toy model for a three-source “baryon” is constructed and investigated analytically and numerically for various choices of interactions. vpython is used to visualize the nontrivial color charge dynamics. The treatment should be accessible to undergraduate students who have taken a first course in quantum mechanics, and suggestions for independent student projects are proposed.

Ionization rate and Stark shift of a one-dimensional model of the hydrogen molecular ion

In this paper we study the ionization rate and the Stark shift of a one-dimensional model of the ion. The finding of these two quantities is reduced to the solution of a complex eigenvalue problem. We solve this problem both numerically and analytically. In the latter case we consider the regime of small external electrostatic fields and small internuclear distances. We find an excellent agreement between the ionization rate computed with the two approaches, even when the approximate result is pushed beyond its expected validity. The ionization rate is very sensitive to small changes of the external electrostatic field, spanning many orders of magnitude for small changes of the intensity of the external field. The dependence of the ionization on the internuclear distance is also studied, as this has a direct connection with experimental methods in molecular physics. It is shown that for large distances the ionization rate saturates, which is a direct consequence of the behavior of the energy eigenvalue with the internuclear distance. The Stark shift is computed and from it we extract the static polarizability of and compare our results with those found by other authors using more sophisticated methods. This work provides an extension of what is usually covered in a course of quantum mechanics where the double delta-potential is studied, while also establishing the connection to a concrete physics problem.

How do I introduce Schrödinger equation during the quantum mechanics course?

In this paper I explain how I usually introduce the Schrödinger equation during the quantum mechanics course. My preferred method is the chronological one. Since the Schrödinger equation belongs to a special case of wave equations I start the course with introducing the wave equation. The Schrödinger equation is derived with the help of the two quantum concepts introduced by Max Planck, Einstein, and de Broglie, i.e. the energy of a photon E = ћω and the wavelength of the de Broglie wave λ = h/p. Finally, the difference between the classical wave equation and the quantum Schrödinger one is explained in order to help the students to grasp the meaning of quantum wavefunction . A comparison of the present method to the approaches given by the authors of quantum mechanics textbooks as well as that of the original Nuffield A level is presented. It is found that the present approach is different from those given by these authors, except by Weinberg or Dicke and Wittke. However, the approach is in line with the original Nuffield A level one.

A Notable Quasi-Relativistic Wave Equation and Its Relation to the Schrödinger, Klein-Gordon, and Dirac Equations

An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schrödinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.

Effectiveness of guided group work in graduate level quantum mechanics

Implementing guided group work in a graduate level quantum mechanics course can be beneficial for students.

A polynomial approach to the spectrum of Dirac–Weyl polygonal Billiards

The Schrödinger equation in a square or rectangle with hard walls is solved in every introductory quantum mechanics course. Solutions for other polygonal enclosures only exist in a very restricted class of polygons, and are all based on a result obtained by Lamé in 1852. Any enclosure can, of course, be addressed by finite element methods for partial differential equations. In this paper, we present a variational method to approximate the low-energy spectrum and wave-functions for arbitrary convex polygonal enclosures, developed initially for the study of vibrational modes of plates. In view of the recent interest in the spectrum of quantum dots of two dimensional materials, described by effective models with massless electrons, we extend the method to the Dirac–Weyl equation for a spin-1/2 fermion confined in a quantum billiard of polygonal shape, with different types of boundary conditions. We illustrate the method’s convergence in cases where the spectrum is known exactly, and apply it to cases where no exact solution exists.