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量子力学考试题量子力学考试题（共五题，每题20分）1、扼要说明：（a ）束缚定态的主要性质。
（b ）单价原子自发能级跃迁过程的选择定则及其理论根据。
2、设力学量算符（厄米算符）∧F ，∧G 不对易，令∧K ＝i （∧F ∧G -∧G ∧F ），试证明：（a ）∧K 的本征值是实数。
（b ）对于∧F 的任何本征态ψ，∧K 的平均值为0。
（c ）在任何态中2F +2G ≥K3、自旋/2的定域电子（不考虑“轨道”运动）受到磁场作用，已知其能量算符为S H ??ω=∧H ＝ω∧z S +ν∧x S （ω，ν>0，ω?ν）（a ）求能级的精确值。
（b ）视ν∧x S 项为微扰，用微扰论公式求能级。
4、质量为m 的粒子在无限深势阱（0<x</x5、某物理体系由两个粒子组成，粒子间相互作用微弱，可以忽略。
已知单粒子“轨道”态只有3种：a ψ(→r )，b ψ(→r )，c ψ(→r )，试分别就以下两种情况，求体系的可能（独立）状态数目。
（i ）无自旋全同粒子。
（ii ）自旋 /2的全同粒子（例如电子）。
量子力学考试评分标准1、（a ），（b ）各10分（a ）能量有确定值。
力学量（不显含t ）的可能测值及概率不随时间改变。
（b ）（n l m m s ）→（n’ l’ m’ m s ’）选择定则：l ?＝1±，m ?＝0,1±，s m ?＝0 根据：电矩m 矩阵元-e →r n’l’m’m s ’，n l m m s ≠0 2、（a ）6分（b ）7分（c ）7分（a ）∧K 是厄米算符，所以其本征值必为实数。
（b ）∧F ψ＝λψ，ψ∧F ＝λψ K ＝ψ∧K ψ＝i ψ∧F ∧G -∧G ∧F ψ ＝i λ｛ψ∧G ψ-ψG ψ｝＝0 （c ）(∧F +i ∧G )(∧F -i ∧G )=∧F 2+∧G 2-∧Kψ(∧F +i ∧G )(∧F -i ∧G )ψ=︱(∧F -i ∧G )ψ︱2≥0 ∴<∧F 2+∧G 2-∧K >≥0，即2F +2G ≥K 3、(a)，(b)各10分(a) ∧H ＝ω∧z S +ν∧x S ＝2 ω［1001-］+2 ν［0110］＝2 ［ωννω-］∧H ψ＝E ψ，ψ＝［b a ］，令E ＝2λ，则［λωννλω---］［b a ］＝0，︱λωννλω---︱＝2λ-2ω-2ν＝0 λ＝±22νω+，E 1＝-2 22νω+，E 2＝222νω+ 当ω?ν，22νω+＝ω（1+22ων）1/2≈ω（1+2 22ων）＝ω+ων22E 1≈-2 ［ω+ων22］，E 2 ＝2［ω+ων22］（b ）∧H ＝ω∧z S +ν∧x S ＝∧H 0+∧H’，∧H 0＝ω∧z S ，∧H ’＝ν∧x S∧H 0本征值为ω 21±，取E 1（0）＝-ω 21，E 2（0）＝ω 21相当本征函数（S z 表象）为ψ1(0)＝[10]，ψ2(0)＝[01 ]则∧H ’之矩阵元（S z 表象）为'11H =0，'22H =0，'12H ＝'21H ＝ν 21E 1＝E 1（0）+'11H +)0(2)0(12'21E E H-＝-ω 21+0-ων2241＝-ω21-ων241 E 2＝E2（0）+'22H +)0(1)0(22'12E E H -＝ω 21+ων2414、E 1＝2222ma π，)(1x ψ＝0sin 2a xa π a x x a x ≥≤<<,00x ＝dx x a ?021ψ＝2sin 202a dx a x x a a=?π x p ＝-i ?=a dx dx d011ψψ-i ?=aa x d a 020)sin 21(2π x xp ＝-i ??-=aaa x d a x x a i dx dx d x 0011)(sin sin 2ππψψ ＝-a a x xd a i 02)(sin 1π ＝0sin [12a a x x a i π --?adx a x 02]sin π＝0+?=ai dx ih 02122 ψ 四项各5分5、（i ），（ii ）各10分（i ）s ＝0，为玻色子，体系波函数应交换对称。
低轨互联网卫星在轨单粒子翻转分析及防护措施尚 琳1,2，刘晓娜1,2，曹彩霞1,2，李国通1,2，朱 野1,2（1. 上海微小卫星工程中心； 2. 中国科学院 微小卫星创新研究院：上海 201203）摘要：空间单粒子翻转（SEU ）对于在轨卫星寿命和可靠性有着较大的影响，然而，针对低轨互联网卫星1000～1200 km 的典型极地轨道空间SEU ，目前缺少在轨试验验证结果。
文章对某型号的两颗卫星在轨7个月以来的SEU 事件记录数据进行处理和分析，给出互联网卫星1050～1425 km 不同轨道高度上的SEU 事件发生的频度、区域及概率，结合在轨运行情况提出互联网卫星在轨单粒子翻转的软硬件防护设计措施。
数据表明，在当前低轨互联网卫星的典型轨道高度上，对于抗单粒子翻转阈值为0.7 MeV·cm 2/mg 的低阈值SRAM 器件，在轨SEU 事件大部分发生在SAA 区域，发生概率约为7.63×10-7 bit -1·d -1。
结合卫星在轨空间防护设计经验，通过加强元器件选用控制、软硬件冗余设计、关键器件限流等措施，可以有效提高低轨互联网卫星的在轨可靠性。
关键词：单粒子翻转；低轨互联网卫星；在轨防护；冗余设计 中图分类号：V474.2; V520.6文献标志码：A 文章编号：1673-1379(2021)05-0503-05DOI: 10.12126/see.2021.05.002Analysis of in-orbit single event upset of low-Earth-orbitinternet satellite and protection measuresSHANG Lin 1,2, LIU Xiaona 1,2, CAO Caixia 1,2, LI Guotong 1,2, ZHU Ye 1,2(1. Shanghai Engineering Center for Microsatellites;2. Innovation Academy for Microsatellites, Chinses Academy of Sciences: Shanghai 201203, China)Abstract: The single event upset (SEU) of low-Earth-orbit (LEO) satellites has a great impact on the lifetime and the reliability of the in-orbit satellites. But the in-orbit verification results are few for the SEU of the internet satellite at a typical polar orbit altitude in the range of 1000 km to 1200 km. This paper analyzes and processes the SEU record data of the two satellites in-orbit for seven months, and gives the frequency, the area and the orbital heights of SEUs at different orbital altitudes from 1050 km to 1425 km for the internet satellites. Itis shown that the probability of the in-orbit SEU for the onboard SRAM is about 7.63×10-7 bit -1·d -1, and most in-orbit single event upsets occur in the South Atlantic Anomaly (SAA) area. And it is shown that the reliability of the LEO internet in-orbit satellites can be effectively improved by strengthening the control of the component selection, the software and hardware redundancy design and the current limiting of the key components.Keywords: single event upset; LEO internet satellite; in-orbit protection; redundancy design收稿日期：2020-11-30；修回日期：2021-08-10基金项目：上海市科委科技创新行动计划项目（编号：17DZ1100700）引用格式：尚琳, 刘晓娜, 曹彩霞, 等. 低轨互联网卫星在轨单粒子翻转分析及防护措施[J]. 航天器环境工程, 2021, 38(5):503-507SHANG L, LIU X N, CAO C X, et al. Analysis of in-orbit single event upset of low-Earth-orbit internet satellite and protection measures[J]. Spacecraft Environment Engineering, 2021, 38(5): 503-507第 38 卷第 5 期航 天 器 环 境 工 程Vol. 38, No. 52021 年 10 月SPACECRAFT ENVIRONMENT ENGINEERING 503E-mail: ***************Tel: (010)68116407, 68116408, 68116544. All Rights Reserved.0 引言近年来，随着卫星技术、电子技术和新材料技术的迅猛发展，国内外纷纷提出包含几千至数万颗低轨卫星的互联网星座建设计划，如美国的StarLink、OneWeb和我国的“虹云”“鸿雁”等卫星星座系统。
Motion in the field of a current sheetGuiding centre approximation Equation of motion of a charged particle isThe case of E= const and B= 0is trivial:Consider next the caseConsequently, the kinetic energy and the speed remain constantcyclotron frequencygyro frequency Larmor frequency cyclotron period, Larmor time ) is:The centre of the gyro motion is called guiding centre (GS)This describes helical motion that -is constant in the direction of B -circular in the xy The radius of the circle is ;(Larmor radius )An electron in the In a uniform magnetic field In a non-uniform magnetic field parallel and perpendicular velocities changesHannes Alfvén: Guiding centre approximation is validin temporally and spatially varying fields when variationsare small during one gyroperiodClearly: is always opposite to B (r L depends on the sign of diamagnetic medium:Placing a large number of charged particles to external magnetic or, in the vector formThe perpendicular components of the eq. of motion areconstant acceleration parallel/antiparallel to very rapid cancellation of large-scale Substitution leads again to gyro motion but now the GC drifts in the y -direction with speed E x /BSame as making Lorentz transformation to GCS:where (non-relat. = 1In vector form:+ ionelectronAll charged particles drift to thesame direction E and Band transformDIn GCS the last two terms must sum to 0 ( )This requires F/qB<< c. If F> qcB, the GC approximation cannot be used! Inserting F= q E into ( )we get the ExB-driftpolarization driftThe corresponding polarization current isExpand the field as a Taylor series about the GC:is the field at GC Straightforward (but a tedious) calculation yields the forcePerpendicular to B:gradient driftNote: the gradient drift separatesdifferent particle species currentionelectronalso curved. The charged particle R C is the radius of curvature(positive inwards) andIf there are no local currents ( B = 0),and v G and v C can be combined tounit vectorsn These are first order drifts . The same procedure can be continued to higher orders:determine the force due to the 1st order driftcalculate the 2nd order drift speed using the same formula ( ) as aboveApplying the forceas before we get the curvature driftPositive charges drift to the west,Negative charges to the east Next we will learn about the mirror effect& p be canonical variables and the motion be almost periodic Example: Consider a charged particle in Larmor motion.Assume that the B does not change much during within one circle.The canonical coordinate is and the canonical momentumcharge!i.e., the magnetic momentis an adiabatic invariantchanged slowly (as compared toNow the energy per mass unit is not conserved. Lorentz asked Einstein at a conference in 1911:What is the conserved quantity in this case?This was a relevant question while quantum mechanics was being Relation to the conservation of magnetic moment:The system is conservative, i.e., total energy is conserved:along the path of GC Change in parallel energy:multiply LHS by v||and RHS byand thusi.e., is constant in the guding centre approximationThe electric field changes the perpendicular energywhich during one gyro period yields the workAs the field changes slowly, replace the time integral by contour integralStokes FaradayFor small changes of the fieldBecause must again remain constant Note that also the flux enclosed by the gyro orbitis constantmirror point When /2, the forceon the GC turns the charge back(mirror force) and the mirror field B mfor a charge that at hasthe pitch-angleIf B2> B1 ,adiabatic heatingOtherwise it is said to be in the loss-cone and escape at the end of the bottlevv|| trappedparticlesloss coneThere are much more complicatedtrapping configurations, e.g. stellarator: The dipole field of the Earth isa large magnetic bottleIf then is constant (second adabatic invariantIf the mirror points move toward each other, p||and thusalso W increases (c.f. a tennis racket hitting a ball).ThisFermi acceleration. It was an early proposalcosmic rays. Today shock acceleration is understood toEnrico Fermiis conserved.This is the third adiabatic invariantThere is a specific energization mechanism for each invariant: W changed by changing the Larmor radius (i.e., |B|)J :W||changed by stretching or shortening the magnetic bottle :W changed by compressing or expanding the drift surfacecomponents of B :Motion in a dipole filed: zero at equator, increases towards north for the Earth), we use k 0= 0M E /4 (also often called dipole moment)As the dipole field is curl-free, it can be given as , wheremagnitude of B :(azimuthal symmetry)Earth’s dipole MThe field lines are found from The length of the line element is Note the important geometrical factorEvery field line is determined by two parameters:and distance r 0where the line crosses the equator polar coordinatesB on a given field line as a function of latitude:for the Earth:In calculations of particle orbits we need the curvature radius ofGuiding centre approximation is applicable in dipole field, if R L<< R Expressed in terms of the rigidity of the particleCondition ( )reduces toThe width of the loss cone at equator is given byThe particle leaks out if itsAt the Earth most particles are lost atFor pitch-anglesA useful order-of-magnitude estimate:In the Earth’s dipole field 1-keV electrons bounce in seconds,As B = 0In case of the Earth: positive ions drift to the west, electrons to the east westward net current (ring current)Often the useful quantityFor particles staying on equator ( 0= /2)For relativistic particles this isTypical energies:protons: 0.1MeV –40 MeV Outer belt electrons:keV –MeV Typical drift periodskeV particles 100s of hours MeV particles 10s of minutesRecall:m p = 938 MeV/m e = 511 keV/ions are nonrelativistic,the most energeticelectrons are relativisticCurrent sheetsCurrent sheets are important in plasma physics. They separate different plasma domains and they are sites of the most important energy releaseprocess, magnetic reconnection (see e.g., the lectures on Space applications of plasma physics (period II) or Advanced space physics (spring 2012))Harris model is a 2D example:BJ y;B n and B 0are constant and The current is according to Ampère’s lawOne-dimensional and two-dimensional Harris modelsWhen a dipole field is stretched, a current-sheet is introduced. For particles, whoseacross the current sheet, regular motion becomes non-adiabatic, i.e., is no more conserved.An example of transition to chaos。
