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<jats:p>Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.</jats:p>

<jats:p>Scattering processes are a standard topic covered in introductory courses on quantum mechanics and particle physics. Unfortunately, a full mathematical treatment tends to be overwhelming for undergraduate students. This article introduces some toy models that are easy to comprehend but still contain the essential features of quantum theory. We define a Hilbert space with state vectors and use creation/annihilation operators to construct transition matrices and S-matrices. We show how perturbation theory gives rise to Feynman diagrams and Feynman rules. We also discuss how we can use symmetry and group theory to restrict what interactions are possible.</jats:p>

<jats:p>Electromagnetic fields of relativistic charged particles have a broad frequency spectrum and a sophisticated spatial structure. Field lines offer a visual representation of this spatial structure. In this article, we derive a general set of equations for the field lines of any moving charged particle. The electric field lines are completely determined by the unit vector from the retarding point to the observation point. After proper transformations, the field line equations describe the rotation of this vector with an angular velocity coinciding with Thomas precession. In some cases, including all planar trajectories, the field line equations reduce to linear differential equations with constant coefficients. We present a detailed derivation of these equations and their general analytical solution. We then illustrate this method by constructing field lines for the “figure eight” motion of an electric charge moving under the influence of a plane wave, including complex field lines in three dimensions.</jats:p>

<jats:p>The quantum free energy of a system governed by a standard Hamiltonian is larger than its classical counterpart. The lowest-order correction, first calculated by Wigner, is proportional to ℏ2 and involves the sum of the mean squared forces. We present an elementary derivation of this result by drawing upon the Zassenhaus formula, an operator-generalization for the main functional relation of the exponential map. Our approach highlights the central role of non-commutativity between kinetic and potential energy and is more direct than Wigner's original calculation, or even streamlined variations thereof found in modern textbooks. We illustrate the quality of the correction for the simple harmonic oscillator (analytically) and the purely quartic oscillator (numerically) in the limit of high temperature. We also demonstrate that the Wigner correction fails in situations with sufficiently rapidly changing potentials, for instance, the particle in a box.</jats:p>

<jats:p>We consider a heavy external object moving in an ideal gas of light particles. Collisions with the gas particles transfer momentum to the object, leading to a force that is proportional to the object's velocity but in the opposite direction. In an ideal classical gas at temperature T, the force acting on the object is proportional to T. Quantum statistics causes a deviation from the T-dependence and shows that the force scales with T2 at low temperatures. At T = 0, the force vanishes in a Bose gas but is finite in a Fermi gas.</jats:p>

<jats:p>In this advanced instructional laboratory, students explore complex detection systems and nondestructive assay techniques used in the field of nuclear physics. After setting up and calibrating a neutron detection system, students carry out timing and energy deposition analyses of radiation signals. Through the timing of prompt fission neutron signals, multiplicity counting is used to carry out a special nuclear material (SNM) nondestructive assay. Our experimental setup is comprised of eight trans-stilbene organic scintillation detectors in a well-counter configuration, and measurements are taken on a spontaneous fission source as well as two (α,n) sources. By comparing each source's measured multiplicity distribution, the resulting measurements of the (α,n) sources can be distinguished from that of the spontaneous fission source. Such comparisons prevent the spoofing, i.e., intentional imitation, of a fission source by an (α,n) neutron source. This instructional laboratory is designed for nuclear engineering and physics students interested in organic scintillators, neutron sources, and nonproliferation radiation measurement techniques.</jats:p>