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AP® Physics C2005 Free response QuestionsThese materials were produced by Educational Testing Service® (ETS®), which develops and administers the examinations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and theirprograms, services, and employment policies are guided by that principle.The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity.Founded in 1900, the association is composed of more than 4,200 schools, colleges, universities, and other educational organizations. Each year, theCollege Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services incollege admission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of equity andexcellence, and that commitment is embodied in all of its programs, services, activities, and concerns.Copyright © 2005 by College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board. APIEL is a trademark owned by the College Entrance Examination Board. PSAT/NMSQT is a registered trademark jointly owned by the College Entrance Examination Board and the National Merit Scholarship Corporation.Educational Testing Service and ETS are registered trademarks of Educational Testing Service.2005M1. A ball of mass M is thrown vertically upward with an initial speed of v o. It experiences a force of air resistance given by F = -kv, where k is a positive constant. The positive direction for all vector quantities is upward. Express all algebraic answers in terms of M, k, v o, and fundamental constants.a. Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball movesupward?increases decreases remains the sameJustify your answer.b. Write, but do NOT solve, a differential equation for the instantaneous speed v of the ball in terms of time t as theball moves upward.c. Determine the terminal speed of the ball as it moves downward.d. Does it take longer for the ball to rise to its maximum height or to fall from its maximum height back to theheight from which it was thrown?longer to rise longer to fallJustify your answer.e. On the axes below, sketch a graph of velocity versus time for the upward and downward parts of the ball's flight,where t f is the time at which the ball returns to the height from which it was thrown.2005M2. A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to use the data to determine the mass M S of Saturn. Assume the orbits of these moons are circular.a. Write an algebraic expression for the gravitational force between Saturn and one of its moons.b. Use your expression from part (a) and the assumption of circular orbits to derive an equation for the orbitalperiod T of a moon as a function of its orbital radius R.c. Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn'smass?d. Complete the data table by calculating the two quantities to be graphed. Label the top of each column, includingunits.e. Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers to indicate thescale.f. Using the graph, calculate a value for the mass of Saturn.2005M3. A system consists of a ball of mass M 2 and a uniform rod of mass M 1 and length d. The rod is attached toa horizontal frictionless table by a pivot at point P and initially rotates at an angular speed ω, as shown above left. The rotational inertia of the rod about point P is 31 M 1d2 . The rod strikes the ball, which is initially at rest. As a result of this collision, the rod is stopped and the ball moves in the direction shown above right. Express all answers in terms of M 1, M 2, ω, d, and fundamental constants.a. Derive an expression for the angular momentum of the rod about point P before the collision.b. Derive an expression for the speed v of the ball after the collision.c. Assuming that this collision is elastic, calculate the numerical value of the ratio M 1 / M 2d. A new ball with the same mass M 1 as the rod is now placed a distance x from the pivot, as shown above. Againassuming the collision is elastic, for what value of x will the rod stop moving after hitting the ball?2005E1. Consider the electric field diagram above.a. Points A, B, and C are all located at y = 0.06 m .i. At which of these three points is the magnitude of the electric field the greatest? Justify your answer.ii. At which of these three points is the electric potential the greatest? Justify your answer.b. An electron is released from rest at point B.i. Qualitatively describe the electron's motion in terms of direction, speed, and acceleration.ii. Calculate the electron's speed after it has moved through a potential difference of 10 V.c. Points B and C are separated by a potential difference of 20 V. Estimate the magnitude of the electric fieldmidway between them and state any assumptions that you make.d. On the diagram, draw an equipotential line that passes through point D and intersects at least three electric fieldlines.2005E2. In the circuit shown above, resistors 1 and 2 of resistance R1 and R2 , respectively, and an inductor of inductance L are connected to a battery of emf ε and a switch S. The switch is closed at time t = 0. Express all algebraic answers in terms of the given quantities and fundamental constants.a. Determine the current through resistor 1 immediately after the switch is closed.b. Determine the magnitude of the initial rate of change of current, dI/dt , in the inductor.c. Determine the current through the battery a long time after the switch has been closed.d. On the axes below, sketch a graph of the current through the battery as a function of time.Some time after steady-state has been reached, the switch is opened.e. Determine the voltage across resistor 2 just after the switch has been opened.2005E3. A student performs an experiment to obtain the value of μ0, the magnetic permeability of vacuum. Shemeasures the magnetic field along the axis of the long, 100-turn solenoid PQ shown above. She connects ends P and Q of the solenoid to a variable power supply and an ammeter as shown. End P of the solenoid is taped at the 0 cm mark of a meterstick. The solenoid can be stretched so that the position of end Q can be varied. The student then positions a Hall probe* in the center of the solenoid to measure the magnetic field along its axis. She measures the field for a fixed current of 3.0 A and various positions of the end Q. The data she obtains are shown below.a. Complete the last column of the table above by calculating the number of turns per meter.* A Hall Probe is a device used to measure the magnetic field at a point.b. On the axes below, plot the measured magnetic field B versus n . Draw a best-fit straight line for the data points.c. From the graph, obtain the value of μ0, the magnetic permeability of vacuum.d. Using the theoretical value of μ0 = 4π x 10-7 TM/A, determine the percent error in the experimental value of μ0computed in part (c).。
ensemble报错英[ɒn'sɒmb(ə)l]美[ɑn'sɑmbl]跟读口语练习∙n. 全体；总效果；全套服装；全套家具；合奏组∙adv. 同时∙oxide-assistant growth氧化物辅助生长penetration[pen·e·tra·tion || ‚penɪ'treɪʃn]n.穿透; 穿透深度; 穿透能力; 渗透electrode[e·lec·trode || ɪ'lektrəʊd]n.电极weak[wɪːk]adj.不牢固的, 虚弱的, 弱的trivial[ˈtriviəl]a.琐碎的，不重要的type[taɪp]n.类型, 模范, 典型hybrid['haibrid]n.混血儿, 杂种, 混合物a.混合的, 杂种的, 混合语的potential barrier penetration 势垒贯穿linear harmonic oscillator 线性谐振子zero proint energy 零点能Coulomb field 库仑场operator 算符angular momentum operator 角动量算符eigen value 本征值secular equation 久期方程dipole transition 偶极子跃迁filter[fil·ter || 'fɪltə(r)]n.过滤器, 用于分隔过滤的东西;v.过滤, 走漏, 渗透; 滤过; 走漏; 渗入thermionic[,θә:mi'ɒnik]a.热离子的[电]热离energy barrier2能障(能量位垒)trade off物物交换, 交换; 权衡, 让步parameter[pa·ram·e·ter || pə'ræmɪtə(r)]n.参数, 参量; 界限; 因素, 特征;external[ex'ter·nal || -nl]n.外部; 外面adj.外部的, 表面的, 客观的finite[fi·nite || 'faɪnaɪt]adj.有限的; 限定的; 有穷的an idealized quantum wirecomplementaryadj.补充的; 补足的seebeck coefficient[电]塞贝克系数generic[ge·ner·ic || dʒɪ'nerɪk]adj.属的; 一般的; 类的tuning[tun·ing || 'tuːnɪŋ/'tju-]n.调音, 调律, 调整coefficient[co·ef·fi·cient || ‚kəʊɪ'fɪʃnt]n.系数power factor[化]功率因数evaluate[e·val·u·ate || ɪ'væljʊeɪt]v.评估, 赋值, 评价open circuit[化]开路; 断路[医]断路, 切断电路operatinga.操作的, 工作的, 营业上的, 业务的, 关于收支的, 关于损益的, 外科手术的approach[əˈprəutʃ]v.靠近n.接近,临近;途径;方式,方法chemical potential化学势mechanism[mech·a·nism || 'mekənɪzm]n.机械, 结构, 机构be independent of temperature.compare our results across the three systems configuration[con·fig·u·ra·tion || kən‚fɪgjʊ'reɪʃn] n.结构; 形态; 表面配置; 行星的方位for simplicity为简单起见symmetrically[si'metikli]ad.匀称地, 对称地equilibrium[,i:kwi'libriәm]n.平衡, 平静, 均衡[化]平衡[医]平衡at equilibriumnonlinear operationnonlinear[ˈnɔnˈliniə]非线性; 非线性的; 非直线型的非线性运算division[di·vi·sion || dɪ'vɪʒn]n.区分, 分开, 除法; 部门magnetotransport磁运输measurement['meas·ure·ment || 'meʒə(r)mənt] n.测量法; 尺寸; 度量presence[pres·ence || 'prezns]n.出席, 存在, 到场the orbital motion of electrons电子轨道运动preserveda.喝醉的[计]保留的spin-orbit interaction[化]自旋轨道偶合interaction[ˌintəˈrækʃən]n.相互作用，相互影响fluctuation[fluc·tu·a·tion || ‚flʌktʃʊ'eɪʃn]n.波动; 动摇; 变动deduce[diˈdju:s]vt.推论，推断，演绎ground state基态first excited statea fixed number of electronspromising['prom·is·ing || 'prɑmɪsɪŋ/'prɒm-] adj.有希望的, 前途有望的spintronics自旋电子学effective mass[电]有效质量magnetic moment[化]磁矩concerning[con'cerning || -nɪŋ]prep.关于spin[spin]n.旋转, 自旋, 疾驰, 情绪低落fabrication[fab·ri·ca·tion || ‚fæbrɪ'keɪʃn] n.制作; 伪造物; 构成fabrication[fab·ri·ca·tion || ‚fæbrɪ'keɪʃn] n.制作; 伪造物; 构成epitaxial[ˏepɪˋtæksɪəl]外延的render[ˈrendə]vt.使得，致使；给予，提供；翻译magnetic field磁场spectrum[spec·trum || 'spektrəm]n.谱; 频谱; 光谱; 射频频谱the quenching of the orbital momentum轨道动量的淬灭quantum devices量子器件nanomaterialparticle[par·ti·cle || 'pɑrtɪkl /'pɑːt-]n.粒子, 极小量, 点A strongreduction in the effective electron or hole g factor toward 2 ultra[ul·tra || 'ʌltrə]adj.过度的, 极端的, 过激的heterostructure[ˏhetərəʊˋstrʌktʃə(r)]异晶结构; 异质结构exceed[ex·ceed || ɪk'siːd]v.超过, 胜过, 超越magnetic field splitting磁场劈裂complementaryadj.补充的; 补足的blockade[block·ade || blɒ'keɪd]n.封锁; 障碍物, 阻碍物; 道路阻塞v.封锁; 挡住; 阻塞chargestabilitydiagram电荷稳定性图conductance ridges电导脊fabricate['fab·ri·cate || 'fæbrɪkeɪt]v.制造; 伪造; 组装; 杜撰reactor[riˈæktə]n.反应堆aerosol['єәrәusɒl]n.烟, 雾[化]气溶胶; 烟雾剂diameter[di·am·e·ter || daɪ'æmɪtə]n.直径substraten.底层, 地层, 衬底, 基底, 基质, 酶作用物, 酶解物[计]衬底; 基片[化]反应物nucleation[化]成核作用with respect to关于，至于indiumn.铟antimony[an·ti·mo·ny || 'æntɪmənɪ]n.锑blende[blend]n.闪锌矿stacking faults,twin boundarieswithatomically flat {110}doped[dəupt]掺杂的; 掺杂过的thick[θɪk]adj.厚的, 浓的, 粗壮的adv.厚厚地; 浓浓地; 密集地; 强烈地capping layer覆盖层lithography[li'θɒgrәfi]n.平版印刷术electron beam.电子束etch[etʃ]v.蚀刻, 蚀镂; 施行蚀刻法solution[so·lu·tion || sə'luːʃn]n.解决, 溶液, 解答evaporation[e·vap·o·ra·tion || ɪ‚væpə'reɪʃn]n.蒸发; 消失; 发散rinse[rɪns]n.冲洗; 染发; 漂洗; 染发剂v.以清水冲洗, 漂清, 漱arebriefly etched in a (NH4)2S x solution followed by a rinse inH2Ofabricate['fab·ri·cate || 'fæbrɪkeɪt]v.制造; 伪造; 组装; 杜撰probe[prəʊb]n.探针, 探测针, 调查v.用探针测, 详细调查; 用探针探查, 探测; 调查; 查究probe station探针台cryostat['kraiәstæt]n.低温恒温器versus[ˈvə:səs]prep.以…为对手，对；与…相比之下source-drain voltageback gate voltage, V bgcontact separation[电]接触分离diamond-shaped[ˋdaɪəməndˋʃeɪpt]菱形even['i:vәn]a.平坦的, 相等的, 连贯的, 均等的, 公平的, 偶数的, 平均的, 平衡的, 恰好的vt.使平坦, 使相等vi.变平, 成为相等ad.甚至, 实际上, 完全, 十分odd[ɑd /ɒd]adj.奇数的, 剩余的, 古怪的alternate['ɒ:ltәnәt]a.交替的, 轮流的, 间隔的vt.轮流, (使)交替vi.轮流, (使)交替alternating交替的; 更迭的spin degeneracy[化]自旋简并性degeneracy[diˈdʒenərəsi]蜕化; 退化; 简并度; 简并性lateral[lat·er·al || 'lætərəl]n.侧部; 边音; 支线adj.侧面的, 旁边的oscillation[os·cil·la·tion || ɑsɪ'leɪʃn /ɒs-] n.摆动; 振动评击]wurtzite[ˈwəːtsait]纤维锌矿; 纤锌矿in thewurtzite phase,in the zinc blende phasedetachment[diˈtætʃmənt]after detachment from the growthsubstratequantized量子化的topological拓扑的helical[ˈhelikəl]螺线; 螺旋; 螺旋形plateau[ˈplætəu]n.高原；(上升后的)稳定时期(或状态)subband次能带spin−orbit coupling自选轨道耦合in the presence of a strong spin−orbit coupling由于一个强的自旋轨道耦合quasi['kwɑ:zi:]a.类似的, 准的computation[com·pu·ta·tion || ‚kɒmpjuː'teɪʃn]n.计算, 计量, 计数platform[plat·form || 'plætfɔrm /-fɔːm]n.月台, 坛, 讲台polarization[po·lar·i·za·tion || ‚pəʊləraɪ'zeɪʃn]n.产生极性; 偏振; 极化; 对立, 给予两个相反的倾向;hybrid[ˈhaibrid]桥接; 桥接岔路; 杂化; 杂种; 间生; 混合; 混合; 混合物; 混频环; 混杂的fulfillment[ful'fill·ment || fʊl'fɪlmənt]n.完成, 履行; 满足, 成就; 实现extent[ikˈstent]n.程度，范围，限度；广度，宽度，大小signature['signәtʃә]n.签字, 识别标志, 调号diffusive[di'fju:siv]a.散播的, 冗长的, 扩散的The formation of subbandsspin-degenerate自旋简并suppression[sup'pres·sion || sə'preʃn]n.压制; 禁止; 镇压; 抑制micrometer[mai'krɒmitә]n.测微计[医]微米imperfection[im·per·fec·tion || ‚ɪmpə(r)'fekʃn]n.不完美; 瑕疵; 缺点scatteringn.分散; 散落backscattering[ˋbækˏskætərɪŋ]geometry[ge'om·e·try || dʒɪ'ɑmɪtrɪ/-'ɒm-]n.几何学constriction[kәn'strikʃәn]n.压缩, 收缩, 紧压的感觉trajectory[trəˈdʒektəri]n.(抛射物)弹道轨道indication[in·di·ca·tion || ‚ɪndɪ'keɪʃn]n.指出; 迹象; 指示quasi['kwɑ:zi:]a.类似的, 准的spectroscopy[spek'trɒskәpi]n.光谱学lock(-)in同步intermediate[,in·ter'me·di·ate || ‚ɪntə(r)'mɪːdɪeɪtd] n.中间物, 调停者v.作中间人; 干预adj.中间的, 中级的fraction[frac·tion || 'frækʃn]n.分数, 破片, 小部分confirmation[con·fir·ma·tion || ‚kɒnfə'meɪʃn]n.证实, 批准, 巩固; 确认derivative[diˈrivətiv]变型; 导数; 从变量; 派生的; 派生物; 衍化物; 衍生物; 诱导剂The derivative of conductance to gate voltage电导对于栅电压的导数transconductance[计]跨导intersect[ˌintəˈsekt ; -ər-]交叉; 相交schematicallyadv.纲要性地; 图表式地pronounced[pro'nounced || prə'naʊnst]adj.显著的, 明白的, 断然的dominate['domi·nate || 'dɑmɪneɪt /'dɒm]v.支配, 控制, 统治; 高耸于dominated中等的; 中等木obscure[əbˈskjuə]a.不著名的；费解的vt.使变模糊，掩盖enabling observation of the 0.5g Qplateau for B ≥2 T and the 1.0g Q plateau for B ≥3 T.schematic[sche·mat·ic || skɪː'mætɪk]adj.概要的, 图解的macroscopic英[,mækrə(ʊ)'skɒpɪk]美[,mækrə'skɑpɪk]adj. 宏观的；肉眼可见的coherent transport line for microscopic quantum system 微观量子系统的相干输运过程intersection英[ɪntə'sekʃ(ə)n]美[,ɪntɚ'sɛkʃən]∙n. 交叉；十字路口；交集；交叉点The schematic of Ohm’s law of classical transport for macroscopic system 对于宏观系统经典输运欧姆定律的图解discharge英[dɪs'tʃɑːdʒ]美[dɪs'tʃɑrdʒ]∙vt. 解雇；卸下；放出；免除∙vi. 排放；卸货；流出∙n. 排放；卸货；解雇放电the microstructure evolution of silicon nanowires during the charge/discharge process as severing for the anodematerialforlithium-ion batteries;thecharge-dischargecharacteristic curves.充放电特征曲线锂电池充放电过程中硅纳米线作为阳极的演化规律anode英['ænəʊd]美['ænod]∙n. [电子] 阳极，[电] 正极。
论文题目：ˆL算符及其本征函数量子力学中2（理工类）ˆL算符及其本征函数1量子力学中2摘要角动量算符是量子力学中一个很重要的力学量，本论文分别对2ˆL的定义、意义、性质以及作用做了阐述，给出了2ˆL算符在球坐标系中的表示式，并用经典坐标变换以及对易关系进行了推导，2ˆL是描述旋转运动及原子分子状态的一个重要的物理量，因此对2ˆL 的研究将有助于理解量子力学中的诸多问题。
本论文将采取理论分析，并结合数学推导的方法，在掌握大量材料的基础上，作出自己的见解，把理论模型建立在合理的体系上，立足实际情况对它们进行深入的分析和研究。
关键词角动量算符；空间转子；角量子数；自旋The 2ˆL in the Quantum Mechanics and Its EigenfunctionAbstractAngular momentum operator is a very important mechanics in quantum mechanics ,this paper definite the definition, significance, as well as the nature of the2ˆL operator , and gives the expression of 2ˆL operator in spherical coordinates .And according with classic and easy to transform the relationship between the derivation. The 2ˆL operator is a very important mechanics which describe rotary movement and the state of Atomic and Molecular, so it will help to understand lots of questions of quantum mechanics. This paper will take theoretical analysis, and mathematical derivation of the method, the availability of large on the basis of material to make their own opinion, the theoretical model based on a reasonable system, based on the actual situation on their conduct in-depth analysis and research.Keywordsangular momentum operator；Spatial rotor；Azimuthal quantum number；Spinning1作者简介：王慧1986年10月出生，女汉族河南兰考人，郑州大学物理工程学院凝聚态物理专业硕士研究生一年级，主要研究方向为陶瓷功能材料。
物理化学测验 April, 2003(Examination of Physical Chemistry)系别(department)： 学号(register number)：姓名(name)： 成绩(score)：This is a closed book exam. Use of a calculator is permitted. Show all of your work. Don’t give help to, or get help from others. Some useful constants and results are:Js h 341063.6-⨯= kg m e 311011.9-⨯= C e 191060.1-⨯=18109989.2-⨯=ms c eV mol kJ 210036.1/1-⨯= 231002.6⨯=A Nλν/hc h E == mv h =λ h p x ≥∆∆ h t E ≥∆∆ π2hn M =⎪⎪⎭⎫ ⎝⎛-=λ2221111n n R φθcos sin r x = φθsin sin r y = θcos =z φθθθθθ2222222sin 1sin sin 11∂∂+∂∂∂∂+∂∂∂∂=∇r r r r r r ωI M =l x n l x πψsin 2)(=2228mlh n E n = ()νh v E v 2/1+= μπk v 21= )1(822+=J J I h E J π 221n E -=(atomic unit) ())2sin(4121sin 2ax a x ax dx -=⎰ 2cos 2sin 2sin sin βαβαβα-+=+ 2sin 2cos 2sin sin βαβαβα-+=-θcos 2222ab b a c ++=1. Write down the Hamiltonian for the lithium atom in atomic unit and state the physical meaning of each term in it. Give the complete wave function for lithium atom in ground state as Slater determinant. If the ionization energy is 5.38eV , estimate the effective nuclear charge felt by the electron in lithium atom. (6 points)2. The operator xˆ is the one-dimension position operator, and ()()x i h ∂∂/2/π is theone-dimension momentum operator. Do these operators commute? Show your conclusion in one-dimension box. (5 points)3. If the average energy associated with a oscillator of frequency ν in a cavity is()1/exp 832-=T k hv h c E B νπν Show that the expression for the low-frequency limit of the average energy is just the Rauleigh’s equation:328cT k E B πν= (4 points) 4. The threshold frequency for emission of photoelectrons from metallic sodium was113105.5-⨯s . Calculate the work function for sodium in unit eV . (4 points)5. Calculate the value of the de Broglie wavelength associated with an oxygen molecule moving with a speed of 4251-⋅s m at 0︒C. (5 points)6. Assume that three real functions 1ψ, 2ψ and 3ψ are normalized and orthogonal. Normalize the function 3212321ψψψ+-.(5 points)7. A wavefunction of hydrogen atom is:θπψcos 323241218103/002/30a r e a r a r a -⎪⎪⎭⎫ ⎝⎛-⎪⎪⎭⎫ ⎝⎛=Please give the eigenenergy and angular momentum of this atomic orbital. (6 points)8. If a rigid rotates in the XY plane, about the Z axis, the angular momentum operator isφπ∂∂=i h L 2ˆ If the moment of inertia (转动惯量) is I , what is the energy operator? (5 points)9. Unsold’s theorem states that, for a given value l , the sum of the values of()()[]2,∑ΦΘmmm l φθ is independent of θ and φ. Please validate this theorem for all p 2 orbitals. (4 points)10. For a F 2 term, calculate the angle between the angular momentum of orbital and the angular momentum of spin. (6 points)11. Give an example of a quantum-mechanical system for which the spacing between energy levels: (a) increases as E increases; (b) remains the same as E increases; (c) decreases as E increases. (6 points)12. How many states are there in the each of the following terms? (a) P 2, (b) S 3, (c) ∑3, (d) ∏2 (4 points)13. Let the line between atom A and atom B of a polyatomic molecule is the z axis. For each of the following atom A atomic orbitals, state whether it will contribute to a σ, π, or δ localized MO in the molecule. (a) s ; (b) x p ; (c) 2222y x zd --; (d) z p (4points) 14. Write down the Hamiltonians of hydrogen molecule in atomic unit with or without Born-Oppenheimer approximation, respectively. And express the complete wavefunction for H 2 as Slater determinants. (6 points)15. What are the degeneracies of the following orbitals for hydrogen-like atoms: 1) 1=n , 2)2=n , 3) 3=n ? (3 points)16. Write down the 2sp hybrid orbitals that lie in the xy plane and one makes an angle of 120︒ to the -x axis. Give the values of angular momentum for these orbitals. (6 points)17. Predict the magnetism of B 2 by using MO and VB theory, respectively.(4 points)18. This problem concerns a one-dimension box of length l , filled with two particles.1) Assume two particles are not identical, and have unequal masses 1m and 2m . Write the wavefunction ),(21x x ψ describing the two-particle ground state and give the energy of that state (note: ),(21x x ψ can be written as in form ),(21x x ψ )()(2211x x ψψ=). (4 points)2) Now assume the masses are equal, i.e., m m m ==21, give the wavefunction for the first three two-particles states and indicate any degeneracies. (6 points)3) Now assume the particles in the box have spin 2/1=s , the same as electron. Write the total 2-particle ground state wavefunction, including spatial and spin parts. Don’tforget normalize your function, including spin part. (2 points)19. If a one-dimension box has the following potential energy profile:0)(=x V )4/3 , 4/0(l x l l x ≤<<≤;∞=)(x V ) , 0(l x x ><;0)(V x V = ()4/34/l x l ≤≤Assume the value of 0V is small, use the perturbation theory to calculate the energy of a particle with mass m in this box. (5 points)。
各种量子数之间的关系Quantum numbers play a crucial role in understanding the behavior and properties of an atom. These numbers describe the exact energy levels of electrons and their spatial orientation within an atom. The four quantum numbers are the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms).量子数在理解原子的行为和性质中起着至关重要的作用。
这些数描述了电子的精确能级以及它们在原子内的空间取向。
主要的四个量子数是主量子数（n）、方位量子数（l）、磁量子数（ml）和自旋量子数（ms）。
The principal quantum number (n) determines the energy level of an electron and correlates with the size of the orbital. It can have integer values starting from 1, with higher values indicating higher energy levels and larger orbitals. As n increases, the electron is located further away from the nucleus, which also increases the energy of the electron.主量子数（n）决定了电子的能级，并与轨道的大小相关。
