Table of Content

01 March 2024 Volume 43 Issue 01

  

Uncertainty principles based on Fourier Transform

YU Hai-yan1, ZHENG Shen-zhou2

College Physics. 2024, 43(01):  1.  doi:10.16854/j.cnki.1000-0712.230075

Abstract ( 448 )   PDF (797KB) ( 411 )  

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Taking the square of normalized modulus for the wave function of quantum mechanics as the density function of probability, the Heisenberg inequality with positive lower bound for a product of the variance of displacement function and velocity function is described. It is proved by the derivative property of Fourier transform, Plancherel lemma and Cauchy-Schwarz inequality. Hardy uncertainty principle shows that an integrable function and its Fourier transform can not rapid attenuation at the same time. The Gauss function with a negative power is to achieve the best way for the Hardy uncertainty. We apply the Phragmen-Lindelof theorem (unbounded region on the maximum modulus principle) and the argument of complex analysis to prove the Hardy uncertainty principle. In addition, we also provide some generalizations of Hardy inequality, such as the Morgan inequality and the Beurling uncertainty principle.



An explanation of the 1st-order WKB method in the applied research of the black hole quasinormal modes

XU Xiao-bao

College Physics. 2024, 43(01):  5.  doi:10.16854/j.cnki.1000-0712.230046

Abstract ( 189 )   PDF (799KB) ( 131 )  

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An important method to compute quasinormal modes of black holes is the WKB method. The main principle of this method is to match the two asymptotic solutions satisfying the boundary conditions with the approximate solution at the top of the effective potential across two turning points, so as to determine quasinormal modes. However, many literatures do not explain how the asymptotic solutions are matched. In this paper, a more reasonable explanation for this problem is given.

A new teaching method of commutation relation from Noether^,s theorem

WANG Feng, WANG Wei, FENG Xiu-rong

College Physics. 2024, 43(01):  10.  doi:10.16854 /j.cnki.1000-0712.230016

Abstract ( 283 )   PDF (719KB) ( 253 )  

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The commutation relation of operators in quantum mechanics is of great significance for students to understand quantum mechanics, and is also directly related to Heisenberg^,s uncertainty principle. In undergraduate courses, the commutation relation of operators is usually directly given in theory, or its derivation depends on the specific representation of operators under the selected concrete representation. The introduction process of the commutation relation formula of operators has no organic connection with classical physics, and the lack of universality is easy to be misunderstood by students. The idea is based on the universally established Noether^,s theorem in both classical and quatum physics, which corresponds to the consevation of quantities and symmetries in the physical system.  This paper aims to introduce a more enlightening teaching method for the derivation of operator commutation relations, providing students with a simple and natural way of understanding.



Hyperspherical adiabatic potential of Helium

REN Zhen-zhong

College Physics. 2024, 43(01):  13.  doi:10.16854/j.cnki.1000-0712.230043

Abstract ( 210 )   PDF (813KB) ( 181 )  

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Based on B spline and two particle angular function, the hyperspherical adiabatic Hamiltonian equation of Helium atom is solved. The hyperspherical potential curves of  1S and 1P  state are presented, the channel functions are analyzed. The result is confirmed with the previous data. The calculation procedure can be used for the high angular momentum problems of three body system.



Discussion on the experimental principle of sound velocity measurement

SU De-zhi, ZHANG Ji-lei, SUN Xiao-wei, DUAN Jin-peng, MA Shuai-qi

College Physics. 2024, 43(01):  16.  doi:10.16854/j.cnki.1000-0712.230031

Abstract ( 445 )   PDF (1113KB) ( 1827 )  

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Aiming at the problem that the theoretical derivation of the change of sound pressure at the receiving end in the sound velocity measurement experiment is not perfect, especially the unclear explanation of the principle for the phase comparison method, a multiple reflection model based on sound pressure equation is established from the relationship between the amplitude and the sound pressure. In the model the plane simple harmonic wave function, the formula of sound pressure reflection coefficient and the principle of wave superposition are used to derive the formula of the change of the amplitude and phase of sound pressure at the receiver with distance, and the principle of the standing wave method and the phase comparison method are explained. The simulation results show that the amplitude and phase of sound pressure at the receiver are not strictly periodic functions, and the measurement accuracy is related to factors such as the sound absorption coefficient and the sound pressure reflection coefficient. Finally, the experimental verification is carried out, and the experimental measurement results are in good agreement with the calculation results of this model. This study provides theoretical guidance for correctly understanding the physical nature of the experiment.


Exploration and practice of integrating scientific research into solid state physics teaching

LUO Yu-bo

College Physics. 2024, 43(01):  22.  doi:10.16854/j.cnki.1000-0712.230028

Abstract ( 236 )   PDF (917KB) ( 254 )  

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 The integration of cutting-edge science and technology into undergraduate teaching is an important route to realize professional knowledge from learning, application to innovation. In this paper, the scientific research of thermoelectric materials is integrated into the teaching of solid physics, in which the research frontier of thermoelectric materials can promote the teaching and thinking of basic knowledge in solid state physics, such as crystal structure, crystal combination, crystal thermal properties, free electron theory and energy band theory. It aims to strengthen students^, understanding of the basic knowledge of solid state physics, stimulate learning interest, expand the level of knowledge, cultivate students^, scientific thinking and ability to use the basic knowledge to solve practical problems, and effectively improve the teaching quality of solid state physics course.



New cases of applications and experience of 3D Max  modeling in medical physics course

HUANG Cui-ying, LI Fa-bo, CHEN Xiao-qing, ZHENG Hai-bo

College Physics. 2024, 43(01):  26.  doi:10.16854/j.cnki.1000-0712.230130

Abstract ( 190 )   PDF (962KB) ( 318 )  

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 Understanding obscure concepts and calculating complex mathematical equations are the essential issues and challenges in the courses of medical physics. To enhance the effectiveness of lectures, it is crucial to display spatial structures and dynamic processes visually. In this study, Young's double slit interference, simple harmonic vibration equation, refractive error, and correction of the eye are selected by questionnaire, and 3D Max is employed to construct their 3D dynamic models. The corresponding courses are then designed and assisted by using virtual experiment teaching. This approach enriches the expression form of abstract physical processes, innovates classroom knowledge presentation and provides references for updating the teaching mode of medical physics and deepening curriculum reform.



Theoretical and experimental research of vertical vibration standing-waves of one-dimensional horizontal string under resonant exciting and gravitational action

CAI Wei-rui1, LI Zhi-chao1, SHEN Han1, 2, WANG Meng1, 2, FANG Yi-zhong1, 2

College Physics. 2024, 43(01):  31.  doi:10.16854/j.cnki.1000-0712.230077

Abstract ( 205 )   PDF (931KB) ( 232 )  

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By utilizing the eigen function expansion method and Laplace transformation, under gravitational action, the wave equation of vertical vibration of one-dimensional string which one end is fixed and the other end oscillates harmonically with a small amplitude is solved strictly. The explicit solution is acquired, and the resonant eigen frequencies that born the standing-waves of string (or called elastic rope) is got. It is also represented that the eigen frequencies and the standing-waves antinodes are all independent to gravity action. The linear changing relationship between the standing-waves loop numbers and the resonant frequency is observed and computed experimentally. Using the wave velocity expression in the wave equation derived theoretically together with the tension force and the length of elastic rope which measured while vibrating, the line mass density of rope is figured out. The result agrees well to the experimental value. From another angle, these works are well verified the correctness of wave equation of vertical vibration of one-dimensional string which elucidated by theory. 






Analytical calculation and numerical simulation of  the dynamics of water-entry of a slender rod

ZHANG Yu-feng1, XIE Dong2, WEI Xiang-zhong1, SHANGGUAN Wang-zuo1

College Physics. 2024, 43(01):  37.  doi:10.16854/j.cnki.1000-0712.220549

Abstract ( 248 )   PDF (894KB) ( 216 )  

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Study on water-entry of heavy objects is of great importance to practical application and theoretical modeling, as it is widely present in daily life, and many fields of engineering and science as well. Combining theoretical derivation with numerical simulations, we study the impact of the damping coefficient of water and the density of the rod on the water-entry dynamics of a uniformly distributed rod at low velocity. The dynamic differential equation is established according to Newton's second law of motion and the analytical solution is obtained, based on which numerical simulations are carried out and the motion characteristics of the rod is analyzed. Numerical results show that a rod with greater density than that of water will eventually reach a stable maximum sinking speed, which is proportional to the density difference between the rod and the water, and inversely proportional to the damping coefficient of the water. The greater the density difference, the longer it takes to reach the maximum sinking speed. However, when the density difference is small, the velocity of the rod reaches a peak at first, and then it decelerates and gradually approaches the final sinking speed. The formation of velocity peak is reasonably accounted based on the forces acting on the rod and thus its acceleration variations. On the contrary, if the density of the rod is less than that of the water, it will eventually suspend in the water at still. Results also show that the smaller the damping coefficient of water, the longer it takes for the rod to reach the maximum sinking speed, and vice versa. The effect


Research on the marangoni effect in droplet explosions

Xu Yawen, Sun Shiran, Zhang Jiachao, Pan gang

College Physics. 2024, 43(01):  44.  doi:10.16854/j.cnki.1000-0712.230011

Abstract ( 255 )   PDF (1011KB) ( 162 )  

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 This paper demonstrates the instability when a mixture of water and ethanol is deposited on vegetable oil. The droplets temporarily diffuse and spontaneously disintegrate into thousands of tiny droplets to form a droplet explosion phenomenon. This paper demonstrates through theoretical analysis and experimental investigation that this phenomenon is produced by the interaction of the wetting transition of droplets and the Marangoni effect. The fragmentation phenomenon is measured in two dimensions: total droplet explosion time and diffusion radius, and the parameters affecting droplet explosion fragmentation and affecting final droplet size parameters are obtained.









Precession and nutation of the frozen Earth

LIU Wen-liang, OU Jian-wen, LIU Yi-min, ZHUANG Ni-ya, LI Yi-hao, Long Guang-bo

College Physics. 2024, 43(01):  51.  doi:10.16854/j.cnki.1000-0712.230120

Abstract ( 224 )   PDF (838KB) ( 241 )  

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Based on the Dzhanibekov effect, some opinions have been circulated in the community that the Earth flip occurs when a reduction or increase of ice at the geographic poles as a weight imbalance can be caused during the melt or freeze of the Earth. In this paper, we investigate the changes in the Earth's freezing latitude that lead to changes in the global moment of inertia of the Earth, and analyze the Earth's precession and nutation. Both the Poinsot case without external force and the case of including tidal force in the calculation of the rotational motion of the triaxial rigid Earth indicate that the rotation of the Earth is most unstable when the freezing latitude is at 35°. Even if frozen to a depth of 33 km, the moment of inertia of the Earth's rotational axis does not change to an intermediate size, and the Dzhanibekov effect does not occur on the Earth. Therefore, our Earth will not be destroyed by the effect of Dzhanibekov flip. The simple calculation in this paper is helpful to understand the Earth's precession and nutation.






Design and research of impedance transformer based  on rectangular waveguide

HUANG Yong, ZHAO Teng, WANG Xiu-fang

College Physics. 2024, 43(01):  57.  doi:10.16854 /j.cnki.1000-0712.230010

Abstract ( 436 )   PDF (874KB) ( 480 )  

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Impedance mismatch is an important problem in the design of microwave RF circuits, which affects the size of power transmission and can cause microwave circuits to fail to work properly when severe. In order to eliminate reflections and improve the transmission efficiency of power, it is necessary to insert a matching network between microwave components to achieve impedance matching. The main study in this paper is an impedance matching network that matches pure resistive loads, called impedance transformer. In this paper, we propose to design a stepwise impedance transformer for a rectangular waveguide directional coupler with a coupling degree of C of about 30 dB, a directionality D ≥ of 20 dB, S11 <-20 dB, a frequency bandwidth of ± 0.2 GHz and a center frequency of 5.8 GHz, so that it matches the BJ70 interface. Three impedance transformer, namely one-quarter wavelength impedance transformer, binomial impedance transformer and Chebyshev impedance transformer, were designed and studied. Summarizing the applicable occasions of the three types of impedance transformer, the comprehensive index of the quarter-wavelength impedance transformer is best obtained by comparison, which is most suitable for the matching network of the directional coupler designed in this paper.



Analytical solution of the intensity distribution of Gaussian  beam diffracted by a circular aperture 

MA Huan, FU Ke, SUI Lin-hong, WU Yan-ling, LI Rui-fang, LI Xi-bin

College Physics. 2024, 43(01):  64.  doi:10.16854/j. cnki.1000-0712.230006

Abstract ( 316 )   PDF (913KB) ( 227 )  

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Based on the far-field Fresnel Kirchhoff diffraction formula and combined with the properties of Bessel function and other special functions,the diffraction intensity distribution of a circular aperture with normal incidence Gaussian beam is calculated. The intensity distribution near the focus is expressed in the form of Lommel function. Then three special regions,namely geometric focal plane,optical axis and geometric shadow region boundary,are selected,and the analytical expression and numerical results of light intensity distribution are given theoretically and numerically. It is found that the smaller the waist radius of Gaussian beam,the larger the radius of the Airy spot,and the greater the focal depth.Finally,the integral intensity of light intensity on the focal plane is calculated. The result indicates that the smaller the waist radius is,the more concentrated the light intensity is to the center. Compared with the diffraction of plane wave,the Gaussian beam will not change the overall contour of the light intensity distribution,but will make the light intensity concentrate to the focus.



Weighted imaging and fat suppression techniques based on magnetic resonance relaxation time

YIN Chao-yang, SUN Jun-li, YAO Hong-ying

College Physics. 2024, 43(01):  70.  doi:10.16854 /j.cnki.1000-0712.230025

Abstract ( 158 )   PDF (949KB) ( 163 )  

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Magnetic resonance imaging is a new imaging technology based on the resonance characteristics of biological magnetic nuclei in magnetic field, in which relaxation time is an important physical quantity to realize and control imaging. In this paper, the longitudinal relaxation time  T1 and the transverse relaxation time T2 of relevant samples were measured by magnetic resonance imaging instrument. T1 and T2 weighted images were achieved by soft pulse spin echo sequences based on different relaxation times. Fat suppression was realizes by magnetic resonance imaging with reverse recovery imaging sequences. 



Theoretical and experimental study on the paraxial invisibility condition of multi-lens System

YU Lin-xuan1, LI Lu-yang1, WANG Juan-juan1, XIA Lin2, Lu Li-ping1, Ge Meng-yan1

College Physics. 2024, 43(01):  74.  doi:10.16854 /j.cnki.1000-0712.230009

Abstract ( 175 )   PDF (875KB) ( 192 )  

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Based on the paraxial invisibility condition of multiple lens system, the invisibility distances of four-lens system and five-lens system are calculated by using the image intersection method, and the lens system experiment is built according to the calculation results. The invisibility range is measured and discussed to verify the calculation results. Finally, COMSOL is used to simulate the geometric optical path of multiple lens system in parallel light, and the invisibility range between different lenses is further discussed. The results are in good agreement with the experimental measurements and theoretical calculations.