http://chineseinput.net/에서 pinyin(병음)방식으로 중국어를 변환할 수 있습니다.
변환된 중국어를 복사하여 사용하시면 됩니다.
Relativistic Analogue Hidden in Projectile Motion
Lim Jae Hoon,Jung Dong-Won,Kim U-Rae,Cho Sungwoong,Lee Jungil 한국물리학회 2020 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.77 No.8
We present a four-vector analogue Uμ≡U0,U)=(|V'|,V) that can be directly constructed from the velocity of a projectile. Here, V and V' are the velocities of a projectile at heights h+H and h, respectively. In the non-relativistic regime, the mathematical structure of this four-vector analogue has an exact correspondence to the relativistic counterpart four-velocity uμ=γ(1,β)c of a massive particle. Based on this observation, we illustrate the design of an introductory laboratory experiment to investigate the Lorentz invariance and its covariant nature by measuring the velocity of a projectile. The experiment may help students to acquire a concrete perspective of Lorentz covariance through their own measurements and analyses of a free-fall motion.
Physics analysis with Leibniz’s differential operators dn
Kim U-Rae,Cho Sungwoong,Lee Jungil 한국물리학회 2023 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.83 No.7
We introduce a systematic approach to represent Leibniz’s nth-order diferential operator dn as the ratio of an infnite product of infnitesimal diference operators to an infnitesimal parameter. Because every diference operator can be expressed as a diference of two shift operators that translate the argument of a function by fnite amounts, Leibniz’s diferential operator dn is eventually expressed as the infnite product of infnitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This flls the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of diferential equations.
The art of Schwinger and Feynman parametrizations
Kim U-Rae,Cho Sungwoong,Lee Jungil 한국물리학회 2023 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.82 No.11
We present a derivation of the Schwinger parametrization and the Feynman parametrization in detail and their elementary applications. Although the parametrizations are essential to computing the loop integral arising in relativistic quantum field theory, their detailed derivations are not presented in usual textbooks. Beginning with an integral representation of the unity, we derive the Schwinger parametrization by performing multiple partial derivatives and utilizing the analyticity of the gamma function. The Feynman parametrization is derived by the partial-fraction decomposition and the change of variables introducing an additional delta function. Through the extensive employment of the analyticity of a complex function, we show the equivalence of those parametrizations. As applications of the parametrizations, we consider the combinatorial factor arising in the Feynman parametrization integral and the multivariate beta function. The combinatorial factor corresponds to an elementary integral embedded in the time-ordered product of the Dyson series in the time-dependent perturbation theory. We believe that the derivation presented here can be a good pedagogical example that students enhance their understanding of complex variables and train the use of the Dirac delta function in coordinate transformation.
Solving an eigenproblem with analyticity of the generating function
Kim U-Rae,Jung Dong-Won,Kim Dohyun,Lee Jungil,Yu Chaehyun 한국물리학회 2021 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.79 No.2
We present a generating-function representation of a vector defined in either Euclidean or Hilbert space with arbitrary dimensions. The generating function is constructed as a power series in a complex variable whose coefficients are the components of a vector. As an application, we employ the generating-function formalism to solve the eigenproblem of a vibrating string loaded with identical beads. The corresponding generating function is an entire function. The requirement of the analyticity of the generating function determines the eigenspectrum all at once. Every component of the eigenvector of the normal mode can be easily extracted from the generating function by making use of the Schläfli integral. This is a unique pedagogical example with which students can have a practical contact with the generating function, contour integration, and normal modes of classical mechanics at the same time. Our formalism can be applied to a physical system involving any eigenvalue problem, especially one having many components, including infinite-dimensional eigenstates.
Bra-Ket Representation of the Inertia Tensor
Kim U-Rae,Kim Dohyun,Lee Jungil 한국물리학회 2020 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.77 No.11
We employ Dirac's bra-ket notation to define the inertia tensor operator that is independent of the choice of bases or coordinate system. The principal axes and the corresponding principal values for the elliptic plate are determined only based on the geometry. By making use of a general symmetric tensor operator, we develop a method of diagonalization that is convenient and intuitive in determining the eigenvector. We demonstrate that the bra-ket approach greatly simplifies the computation of the inertia tensor with an example of an N-dimensional ellipsoid. The exploitation of the bra-ket notation to compute the inertia tensor in classical mechanics should provide undergraduate students with a strong background necessary to deal with abstract quantum mechanical problems.
Inertia tensor of a triangle in barycentric coordinates
Kim U-Rae,정동원,Yu Chaehyun,Han Wooyong,Lee Jungil 한국물리학회 2021 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.79 No.7
We employ the barycentric coordinate system to evaluate the inertia tensor of an arbitrary triangular plate of uniform mass distribution. We find that the physical quantities involving the computation are expressed in terms of a single master integral over barycentric coordinates. To expedite the computation in the barycentric coordinates, we employ Lagrange undetermined multipliers. The moment of inertia is expressed in terms of mass, barycentric coordinates of the pivot, and side lengths. The expression is unique and the most compact in comparison with popular expressions that are commonly used in the field of mechanical engineering. A master integral that is necessary to compute the integral over the triangle in the barycentric coordinate system and derivations of the barycentric coordinates of common triangle centers are provided in appendices. We expect that the barycentric coordinates are particularly efficient in computing physical quantities like the electrostatic potential of a triangular charge distribution. We also illustrate a practical experimental design that can be immediately applied to general-physics experiments.
Generating-function representation for scalar products
Kim U-Rae,Jung Dong-Won,Kim Dohyun,Lee Jungil,Yu Chaehyun 한국물리학회 2021 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.79 No.5
We employ the generating-function representation for an n-dimensional vector in Euclidean or Hilbert space to evaluate scalar products. The generating function is constructed as a power series in a complex variable weighted by the components of a vector. The scalar product is represented by a convolution of the generating functions for the vectors integrated over a closed contour in the complex plane. The analyticity of the generating functions associated with the Laurent theorem reduces the evaluation of the scalar product into counting combinatoric multiplicity factors. As applications, we provide two exemplary computations: the sum of the squares of integers and the normalization of normal modes in a vibrating loaded string. As a byproduct of the latter example, we find a new alternative proof of a famous trigonometric identity that is essential for Fourier analyses.
KyungTae Kim,June-Haak Ee,Kyoung Hoon Kim,U-Rae Kim,이정일 한국물리학회 2020 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.76 No.4
We investigate the motion of a massive particle constrained to move along a path consisting of two line segments on a vertical plane under an arbitrary conservative force. By fixing the starting and end points of the track and varying the vertex horizontally, we find the least-time path. We define the angles of incidence and refraction similar to the refraction of a light ray. It is remarkable that the ratio of the sines of these angles is identical to the ratio of the average speeds on the two partial paths as long as the horizontal component of the conservative force vanishes.