Dynamics of Open Bosonic Quantum Systems in Coherent State Representation
量子物理中的非对易分析与超算符
单位代码：１０１８３学号：**********吉林大学博士学位论文李根全分类号：0413密级内部————量子物理中的非对易分析与超算符Hyper-operators and Non-commutative Calculus in Quanteum Physics李根全指导教师姓名：吴兆颜职称：教授单位：吉林大学物理学院理论物理研究中心专业名称：理论物理论文答辩日期：授予学位日期：答辩委员会主席：论文评阅人：2004年4月提要量子力学的数学框架被公认是Hilbert空间上的线性算符理论。
但是在过去的文献中更多地受到重视的是它的代数侧面。
我们讨论的是 Hilbert 空间上的线性算符理论的分析侧面。
在第一章介绍了赋范线性空间，Banach空间，内积空间和Hilbert 空间的概念。
主要是赋予了向量以范数，使我们有可能讨论向量序列的极限。
在第一章中我们还介绍了由内积空间A到内积空间X的线性映射构成的线性空间+∈Λ，定义了两个∈Λ的厄米共轭(;)J K A XΛ，定义了线性映射,(;)A X(;)J X AΛ的结构。
A X∈Λ的内积。
讨论了内积空间(;)J K A X线性映射,(;)第二章把数学家建立的Banach 空间上的分析移植到量子物理中的算符函数上。
由于算符函数各阶导函数值是各类超算符，我们系统地建立了超算符的理论。
利用它我们成功地表达出算符函数的各阶导数。
算符函数的Taylor 展开表明，以复数为系数的自变量X（是线性算符）的幂级数，只是“解析的”算符函数的很小的一部分。
超算符理论不仅对非对易分析是必要的代数准备。
超算符概念本身在量子物理，群论与李代数中也有重要的应用。
我们在第三章用例子证明了这一点。
目 录绪论 (1)第一章 态矢量空间、算符空间作为Banach 空间..........................4 § 1.1 节 线性空间..........................................................................................4 § 1.2 节 赋范线性空间..................................................................................6 § 1.3 节 内积空间、Hilbert 空间..................................................................8 § 1.4 节 线性映射构成的线性空间...............................................................9 § 1.5 节 度量空间、开子集. (14)第二章 算符函数的微分学...................................................................17 § 2.1 节 Banach 空间上的分析...................................................................17 § 2.2 节 量子物理中算符函数的微分学.....................................................20 § 2.3 节 各类超算符构成的线性空间);(M M Λ，));(;(M M M ΛΛ，.. (21)第三章 超算符的应用示例...................................................................28 § 3.1 节 Baker-Hausdorff 定理的引理的证明.............................................28 § 3.2 节 利用超算符概念讨论量子力学一些问题.....................................30 结论...............................................................................................................32 附录...............................................................................................................34 参考文献.....................................................................................................39 致谢...............................................................................................................42 学术论文.. (43)中文摘要 (1)英文摘要 (1)绪 论20世纪初物理学经历了一场伟大的革命。
General criterion for the entanglement of two indistinguishable particles
a rXiv:q uant-ph/04165v214J u l24General criterion for the entanglement of two indistinguishable particles ∗GianCarlo Ghirardi †Department of Theoretical Physics of the University of Trieste,and International Centre for Theoretical Physics “Abdus Salam”,and Istituto Nazionale di Fisica Nucleare,Sezione di Trieste,Trieste,Italy and Luca Marinatto ‡Department of Theoretical Physics of the University of Trieste,and Istituto Nazionale di Fisica Nucleare,Sezione di Trieste,Trieste,Italy Abstract We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles.This implies definite constraints on the mathematical form of the state vector associated with the whole system.We then analyze separately the cases of fermion and boson systems,and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neu-mann entropy of the one-particle reduced density operators can supply us with a consistent criterion for detecting entanglement.In particular,the consideration of the von Neumann entropy is particularly useful in deciding whether the correlations of the considered states are simply due to the indistinguishability of the particles involved or are a genuine manifes-tation of the entanglement.The treatment leads to a full clarification of the subtle aspects of entanglement of two identical constituents which have been a source of embarrassment and of serious misunderstandings in the recent literature.Key words:Entanglement;Identical particles;von Neumann entropy.PACS:03.65.Ta,03.67.-a,89.70.+c1IntroductionQuantum entanglement,considered by Schr¨o dinger “the characteristic trait of Quantum Mechan-ics,the one that enforces its entire departure from classical lines of thoughts”[1],has played a central role in the historical development of Quantum Mechanics and nowadays it constitutes an essential resource for many aspects of quantum information and quantum computation theory.In fact,the possibility of performing reliable teleportation processes,of generating uncondition-ally secure private keys in cryptography,or of devising quantum algorithms allowing us to solvecertain computational problems in a more efficient way than the best known classical methods, is essentially based on the peculiar properties of entangled states.However,in spite of the fact that entangled states involving identical constituents are widely used in the experimental imple-mentations of the above mentioned(and many other)processes,the very notion of entanglement for such ubiquitous physical systems seems to be too often misunderstood,or not understood at all,in the current scientific literature on the subject.The most frequent misinterpretations arise in connection with the symmetrization postulate of quantum mechanics which requires definite symmetry properties for the state vectors associated with systems of identical particles.Their nonfactorized form seems to suggest1,when compared with the well-known case of systems composed of distinguishable particles,the occurrence of an unavoidable form of entanglement when identical particles enter into play.In a recent paper[2](see also[3])we analyzed in great detail the problem of entanglement, dealing with systems of two(or more)both distinguishable and identical particles.In accordance with the position of the founding fathers of quantum mechanics[1,4],we strictly related the nonoccurrence of entanglement to the possibility of attributing complete sets of properties with both constituents of the composite system.In this way we were able to formulate an unam-biguous criterion for deciding whether a given state vector is entangled or not,which works for the cases of both distinguishable and identical constituents.It has to be stressed that,contrary to what has sometimes been stated,non-entangled states involving identical constituents can actually occur.Obviously,in the case of distinguishable particles,our criterion is equivalent(as we showed in Ref.[2])to the commonly used criteria to identify whether a system is entangled or not,which involve consideration of the Schmidt number of the biorthonormal decomposition of the state or,equivalently,the evaluation of the von Neumann entropy of the reduced statistical operators.The situation is radically different in the case of composite systems involving identical con-stituents.In the literature,various authors[5,6,7]have suggested identifying the entangled or non-entangled nature of a system of two identical constituents by resorting to natural general-izations of the above mentioned criteria.In so doing they have met various difficulties which emerge when one compares the results obtained with the analogous ones for the case of dis-tinguishable particles.Moreover,the procedure yields apparently contradictory results for the fermion and boson cases[6,7,8].In this paper,we show that the source of the problems rests in not having appropriately taken into account the fact that,even in the case in which it is physically legitimate and correct to consider a state as non-entangled,so that one knows the properties of the constituents,there is an unavoidable lack of information about the actual situation of the constituents,arising from their identity.Furthermore,we prove that by resorting to the general analysis of Refs.[2,3]one can get a complete clarification of the matter and we present a unified criterion for detecting entanglement in systems of identical particles such that(i)it involves both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the state vectors and the von Neumann entropy of the reduced single-particle statistical operators;(ii)it applies equally well for fermions and bosons;and(iii)it is in complete accordance with our original criterion.2Entanglement and propertiesIn this section we briefly review the arguments of Ref.[2],which show that the correct wayto identify the entanglement is to relate it to the impossibility of attributing precise propertiesto the(identical)constituents of a two-particle system S=S1+S2.As is well known,inthe case of non-entangled distinguishable particles the factorized nature of the state vector|ψ(1,2) =|φ 1⊗|χ 2is a necessary and sufficient condition for being allowed to claim that subsystem S1objectively possesses the(complete set of)properties associated with the statevector|φ 1and subsystem S2those associated with|χ 2.We stress that in the case considered weknow not only the properties that are possessed but also to which system they refer.Obviously,in the case of identical constituents one cannot resort to the factorizability criterion to claimthat the two systems are non-entangled,otherwise one would be led to conclude(mistakenly)that non-entangled states cannot exist(an exception being made for two bosons in the samestate),since the necessary symmetry requirements forbid the occurrence of factorized states.However,this naive and inappropriate conclusion derives from taking a purely formal atti-tude about the problem,without paying due attention to the physically meaningful conditionswhich,when satisfied,allow one to legitimately state that two systems are non-entangled.Suchconditions are,primarily,those of being allowed to claim that one particle possesses a preciseand complete set of properties and the other one exhibits analogous features.Obviously,onemust always keep clearly in mind that it is absurd to pretend to individuate the particles,i.e.,to identify which one possesses one set of properties and which one the other set(in the case inwhich such sets are different).Considerations of this kind have led us to identify the following physically appropriate cri-terion characterizing non-entangled states of two identical particles.Definition2.1The identical constituents S1and S2of a composite quantum system S=S1+S2are non-entangled when both constituents possess a complete set of properties. Obviously,we still have to make fully precise the meaning of the expression“both constituents possess a complete set of properties”.To this end we resort,first of all,to the following definition: Definition2.2Given a composite quantum system S=S1+S2of two identical particles de-scribed by the normalized state vector|ψ(1,2) ,we will say that one of the constituents possesses a complete set of properties if and only if there exists a one-dimensional projection operator P, defined on the single particle Hilbert space H,such that:ψ(1,2)|E P(1,2)|ψ(1,2) =1(2.1) whereE P(1,2)=P(1)⊗[I(2)−P(2)]+[I(1)−P(1)]⊗P(2)+P(1)⊗P(2).(2.2) Condition(2.1)gives the probability offinding at least one of the two identical particles(of course we cannot say which one)in the state associated with the one-dimensional projection operator3P2.Since any state vector is a simultaneous eigenvector of a complete set of commuting observables,condition(2.1)allows us to attribute to at least one of the particles the complete set of properties(eigenvalues)associated with the considered set of observables.At this point we must distinguish two cases.If the two identical particles are fermions, then one can immediately prove(see Ref.[2])that Eq.(2.1)implies that there exists another one-dimensional projection operator Q,which is orthogonal to P,such that the operator E Q, which has the expression(2.2)with Q replacing P,also satisfiesψ(1,2)|E Q(1,2)|ψ(1,2) =1.(2.3) Obviously,in such a case,since it is simultaneously true that“there is at least one particle having the properties associated with P”and“there is at least one particle having the properties associ-ated with Q”and,moreover,such properties are mutually exclusive due to the orthogonality of the projection operators,we can legitimately claim that,in the state|ψ(1,2) ,“one particle(it is meaningless to ask which one)has the complete set of properties associated with P and one the complete set associated with Q”.As we showed in Ref.[2]our request for non-entanglement leads,in the fermion case,to the following theorem:Theorem2.1The identical fermions S1and S2of a composite quantum system S=S1+S2 described by the pure normalized state|ψ(1,2) are non-entangled if and only if|ψ(1,2) is obtained by antisymmetrizing a factorized state.In the boson case the situation is slightly different since the two particles can be in the same state.To be completely general let us begin by assuming that at least one of the constituents possesses a complete set of properties,so that there exists a one-dimensional single-particle projection operator P such that the associated E P satisfies Eq.(2.1).At this point we take into account the operator P(1)⊗P(2)and we consider its expectation value in the state|ψ(1,2) . Three possible cases can occur.•If ψ(1,2)|P(1)⊗P(2)|ψ(1,2) =1,then we can say that“both particles possess the complete set of properties associated with P”and|ψ(1,2) is the tensor product of two identical state vectors.This is the only situation in which the unavoidable ambiguities ensuing from the identity of the constituents disappear.•If ψ(1,2)|P(1)⊗P(2)|ψ(1,2) =0then the condition Eq.(2.1)implies that the state is obtained by symmetrizing the product of two orthogonal states,one of which is the one on which P projects,and,consequently,there is another one-dimensional projection operator Q orthogonal to P such that Eq.(2.3)is satisfied,and the mean value of Q(1)⊗Q(2) also vanishes.The situation is perfectly analogous to the fermion case,and one can conclude that“there is precisely one particle possessing the properties identified by P and one possessing the properties identified by Q”.•If ψ(1,2)|P(1)⊗P(2)|ψ(1,2) ∈(0,1),since condition Eq.(2.1)implies that the state is obtained by symmetrizing the product of two states,one of which is the one on which P projects,the second state cannot be orthogonal to thefirst one.If we denote as Q the one-dimensional projection operator on such a state,then we can immediately verify that Eq.(2.3)holds and that ψ(1,2)|Q(1)⊗Q(2)|ψ(1,2) ∈(0,1).In such a situation,in spite of the fact that the two statements“there is at least one particle with the properties associated with P”and“there is at least one particle with the properties associated with Q”are true,one cannot conclude that“there is precisely one particle possessing the properties identified by P and one possessing the properties identified by Q”.The above considerations,as the reader can easily grasp and as has been proved in Ref.[2],lead to the following theorem:Theorem2.2The identical bosons of a composite quantum system S=S1+S2described by the pure normalized state|ψ(1,2) are non-entangled if and only if either the state is obtained by symmetrizing a factorized product of two orthogonal states or it is the product of the same state for the two particles.Concluding,we have shown that,when one deals with the problem of entanglement using the necessary logical rigour and making appeal to the physical meaning of entanglement itself, then the process of(anti)symmetrizing a state vector does not necessarily lead to an entangled state.In addition to the physical motivations we have presented,there are other compelling reasons which show that our criterion for non-entanglement is the correct one.It is in fact easy to show that,when the conditions of Theorems2.1or2.2are satisfied,it is not possible to take advantage of the form of the state vector to perform teleportation processes or to violate Bell’s inequality. These facts further strengthen our conclusion that the state is non-entangled.To clarify our argument we resort to an extremely simple example.Let us consider the following state of two identical spin-1/2fermions,in which we have denoted as|R and|L two precise orthonormal states of the single-particle configuration space having disjoint supports in two distant and nonoverlapping spatial regions Right and Left,respectively,and with|z↑ and |z↓ the eigenvectors of the spin observableσz:|ψ(1,2) =12[|z↑ 1|R 1⊗|z↓ 2|L 2−|z↓ 1|L 1⊗|z↑ 2|R 2].(2.4)Such a state,which can be obtained by antisymmetrizing a factorized state(that is,|z↑ 1|R 1⊗|z↓ 2|L 2),makes perfectly legitimate a statement of the type3:“there is one particle at the right having spin up along the z-direction and the spatial properties associated with|R and a particle at the left having spin down along the z-direction and the spatial properties associated with|L ”.As a consequence,this state does not imply any(nonlocal)correlation between spin measurements performed in the two spacelike separated regions Right and Left and thus it cannot violate Bell’s inequality.This follows immediately from the fact that,when the state is the one of Eq.(2.4),we have for the mean value E( a, b)of the product of the outcomes of twospin measurements along the directions a and b in the two regions Right and Left,the following expression:E( a, b)= ψ| σ(1)· a P(1)R⊗ σ(2)· bP(2)L+ σ(1)· bP(1)L⊗ σ(2)· a P(2)R |ψ= z↑| σ· a|z↑ z↓| σ· b|z↓ ,(2.5) where we have denoted as P R and P L the projection operators on the closed linear manifolds of the spatial wave functions with compact support in the Right and Left regions,respectively. The occurrence of a factorized product of two mean values implies that no choice of the unit vectors a, b, c,and d can lead to a violation of Bell’s inequality[9]:|E( a, b)−E( a, c)|+|E( b, d)+E( c, d)|≤2.(2.6) On the other hand,let us consider a state of the form:|φ(1,2) =14As is well known,given a statistical operatorρ,its von Neumann entropy is defined as S(ρ)≡−Tr[ρlnρ] where the base of the logarithm function is the number e.However,in the present paper,we will rescale this quantity and follow the information theory convention of using all logarithms in base2.6be a projection operator onto a one-dimensional manifold.Correspondingly,its von Neumann entropy S(ρ(1))≡−Tr(1)[ρ(1)logρ(1)]equals zero.This result is correct since such a quantity measures the lack of information about the single-particle subsystem and there is,in fact,no uncertainty at all concerning the state that must be attributed to it.When passing to the more subtle case of interest,that is,to systems composed of two identical constituents,the relations between entanglement and both the Schmidt number and the von Neumann entropy of the reduced statistical operators become less clear and require a careful analysis.The purpose of this Section is to clarify the matter and to present a criterion for determining whether a state is entangled or not which is(i)based on a consideration of both the Slater-Schmidt number and the von Neumann entropy;(ii)consistent with our original criterion summarized in Definition2.1;(iii)equally applicable to fermion and boson systems; andfinally(iv)able to unify in a consistent way various criteria which have appeared recently in the literature[5,6,7].We limit our considerations to the case of afinite-dimensional single-particle Hilbert space and,in accordance with the above remarks,we deal separately with the fermion and the boson cases,since they exhibit quite different features.3.1The fermion caseThe notion of entanglement for systems composed of two identical fermions has been discussed in Ref.[5]where a“fermionic analog of the Schmidt decomposition”was exhibited.Such a decomposition is based on a nice extension to the set of the antisymmetric complex matrices of a well-known theorem holding for antisymmetric real matrices(see,for example,[10]).The Theorem of[5]states the following:Theorem3.1.1For any antisymmetric(N×N)complex matrix A[that is,A∈M(N,C)and A T=−A]there exists a unitary transformation U such that A=UZU T,with Z a block-diagonal matrix of the sortZ=diag[Z0,Z1,...,Z M],Z0=0,Z i= 0z i−z i0 ,(3.1) where Z0is the(N−2M)×(N−2M)null matrix and z i are complex numbers.Equivalently, Z is the direct sum of the(N−2M)×(N−2M)null matrix and the M(2×2)complex antisymmetric matrices Z i.The fermionic analog of the Schmidt decomposition follows from an application of Theorem3.1.1 to systems composed of identical fermions:Theorem3.1.2Any state vector|ψ(1,2) describing two identical fermions of spin s and, consequently,belonging to the antisymmetric manifold A(C2s+1⊗C2s+1),can be written as:|ψ(1,2) =(2s+1)/2i=1a i·12[|2i−1 1⊗|2i 2−|2i 1⊗|2i−1 2],(3.2)where the states{|2i−1 ,|2i }with i=1,...,(2s+1)/2constitute an orthonormal basis of C2s+1,and the complex coefficients a i(some of which may vanish)satisfy the normalization condition i|a i|2=1.7Following the authors of Ref.[5],the number of nonzero coefficients a i appearing in the decomposition(3.2)is called the Slater number of|ψ(1,2) .The relation of such a number to the notion of entanglement has been made explicit in the papers[6,7],where a state displaying the form of Eq.(3.2)is called entangled if and only if its Slater number is strictly greater than1.It is worth noticing that this condition turns out to be totally equivalent to our Definition2.1.In fact, given an arbitrary two-particle state|ψ(1,2) ,suppose that its fermionic Schmidt decomposition has Slater number equal to1.According to Theorem3.1.2,this means that there exist two orthonormal vectors|1 and|2 belonging to C2s+1such that:|ψ(1,2) =12[|1 1⊗|2 2−|2 1⊗|1 2], 1|2 =0.(3.3) Since the state can be obtained by antisymmetrizing the product state|1 1⊗|2 2,the state must be considered as non-entangled in accordance with our criterion.Vice versa,any state obtained by antisymmetrizing a factorized state has Slater number equal to1.On the contrary,if the Slater number is greater(or equal to)two,the form Eq.(3.2)of the state shows immediately that it cannot be obtained by antisymmetrizing a product of two orthonormal vectors,so that the state must be considered as a genuinely entangled one.So far the criterion of Ref.[5,6,7]and ours agree completely;however,some problems arise when one calculates the von Neumann entropy of the reduced density operator associated with one of the two particles5.In fact from Eq.(3.2)one gets:ρ(1)≡T r(2)[|ψ(1,2) ψ(1,2)|]= i|a i|2=1− i|a i|2log|a i|2.(3.5)2It follows trivially that:S(ρ(1))≥1∀|a i|2∈[0,1] i|a i|2=1.(3.6)In analogy with the case of distinguishable particles,one could be tempted to regard this quantity as a measure of entanglement.But,according to the authors of Ref.[6]and[8],this naturally raises the following two puzzling issues:(i)S(ρ(1))of Eq.(3.5)attains its minimum value S min=1in correspondence with a state with Slater number equal to1(which,in accordance with their position,is assumed to identify a non-entangled state),contrary to what happens for a(non-entangled)state of distinguishable particles with Schmidt number equal to1,for which the value of the entropy still takes its minimum value which,however,equals0;(ii)moreover,in the case of two identical bosons,the minimum of the analogous quantity,as we will show later,is null.This seems to imply that the von Neumann entropy is inappropriate to deal with our problem since it gives different measures of entanglement for boson and fermion states.The problematic aspects of this situation derive entirely from not having taken correctly into account the real meaning of the von Neumann entropy as a measure of the uncertainty about the state of a quantum system.To be more precise,let us consider a state|ψ(1,2) with Slater number equal to1,i.e.,a non-entangled state according to our and the Slater number criterion,like that of Eq.(3.3).As we stressed in Section1,in this situation we can attribute definite quantum states|1 and|2 to the particles but,since they are totally indistinguishable, we cannot know whether,for example,particle1is associated to the state|1 or to the state |2 .As a natural consequence,the reduced density operator of each particle and its associated von Neumann entropy reflect such an unavoidable ignorance6.Actually,they are equal to:1ρ(1or2)=6To avoid any misunderstanding we stress that with the expression“ignorance”we do not intend any lack of knowledge about the system.In fact in the considered case this ignorance derives entirely from the fact that, within Quantum Mechanics,there is no conceivable way to individuate identical particles.7With this expression we mean that a measurement aimed to test whether there is one particle in state|1 and one in state|2 gives with certainty a positive outcome.9Theorem3.1.3A state vector|ψ(1,2) describing two identical fermions is non-entangled if and only if its Slater number is equal to1or equivalently iffthe von Neumann entropy of the one-particle reduced density operator S(ρ(1or2))is equal to1.In full agreement with the previous considerations we can say that the von Neumann entropy is a measure both of the amount of uncertainty deriving from the indistinguishability of the involved particles and of the possible uncertainty related to the entangled nature of the states (that is,the impossibility of attributing two definite state vectors to the pair of particles).This quantity,in the case of fermions,is strictly greater than1and the greater it is,the larger is the amount of entanglement of the state.Before coming to the boson case,a last remark is appropriate.Let us consider a non-entangled state of two fermions which has the form of Eq.(3.3).Suppose we choose two arbitrary orthonormal vectors|¯1 and|¯2 belonging to the two-dimensional manifold spanned by the states |1 and|2 ,such that:|1 =α|¯1 +β|¯2 ,|2 =−β∗|¯1 +α∗|¯2 ,(3.8) with|α|2+|β|2=1.Substituting these expressions in Eq.(3.3)we have:|ψ(1,2) =12[|¯1 1⊗|¯2 2−|¯2 1⊗|¯1 2].(3.9) Therefore,when a state is obtained by antisymmetrizing the product of two orthogonal single-particle states,the same state can also be obtained by antisymmetrizing the product of any pair of orthogonal states belonging to the two-dimensional manifold spanned by the original states. This is not surprising but it raises some questions concerning the problem of the assignment of definite properties to the constituents.In fact,Eq.(3.9)shows that,just as we can claim that in the state of Eq.(3.3)there is one particle with the properties associated with|1 and one with the properties associated with|2 ,we can make an analogous statement with reference to any two arbitrary orthogonal states|¯1 and|¯2 of the same manifold.However,this peculiar feature is only apparently problematic.In fact it seems so because here we are confining our consideration to the spin degrees of freedom.When one takes into account also the space degrees of freedom one realizes that the only physically interesting situations are those associated with states like the one of Eq.(2.4),that is,those in which one wants,for example,to investigate the spin properties of a particle that has a precise location8.As we have seen,in such a state one can make precise claims about the spin state of the particle in the Right or Left,respectively, contrary to what happens for a state like the one of Eq.(2.7)which is genuinely entangled. This point has been exhaustively discussed in Ref.[2],to which we refer the reader for a more detailed analysis.2[|z↑ |R +|z↓ |L ]and one in the state1/√3.2The boson caseLet us now pass to the more delicate case of two identical bosons.In such a case,dealing with theproblem of their entanglement requires great care in order to avoid possible misunderstandings.Let us start,as before,by considering the bosonic Schmidt decomposition for an arbitrarystate vector|ψ(1,2) belonging to the symmetric manifold S(C2s+1⊗C2s+1)describing two identical bosons,and let us recall a well-known theorem of matrix analysis(the so-called Takagifactorization theorem[11],a particular instance of a more general theorem named the SingularValue Decomposition)which is particularly useful for this case:Theorem3.2.1For any symmetric(N×N)complex matrix B(that is,B∈M(N,C)and B T=B)there exists a unitary transformation U such that B=UΣU T,whereΣis a real nonnegative diagonal matrixΣ=diag[b1,...,b N].The columns of U are an orthonormal set of eigenvectors of BB†and the diagonal entries ofΣare the nonnegative square roots of the corresponding eigenvalues.It is worth noticing that the columns of the unitary operator U must be built by choosing aprecise set9of eigenvectors of BB†(to be determined in an appropriate manner,see[11]).The bosonic Schmidt decomposition turns out to be a trivial consequence of the just mentionedTheorem3.2.1:Theorem3.2.2Any state vector describing two identical s-spin boson particles|ψ(1,2) and, consequently,belonging to the symmetric manifold S(C2s+1⊗C2s+1)can be written as:|ψ(1,2) =2s+1i=1b i|i 1⊗|i 2,(3.10)where the states{|i },with i=1,...,2s+1,constitute an orthonormal basis for C2s+1,and the real nonnegative coefficients b i are the diagonal elements of the matrixΣand satisfy the normalization condition i b2i=1.The Schmidt decomposition of Eq.(3.10)is always unique when the eigenvalues of the operator BB†are non-degenerate,as happens for the biorthonormal decomposition of states describing distinguishable particles.In analogy with the case of two distinguishable particles and of two identical fermions,one could naively be inclined to term entangled any bosonic state whose Schmidt number[that is, the number of non-zero coefficients in the decomposition(3.10)]is strictly greater than1(as clearly stated,for example,in Ref.[6]).However,as we will show,this criterion turns out to be inappropriate because the states with Schmidt number equal to1(that is,factorized states in which the two bosons are in the same state)do not exhaust the class of non-entangled pairs of identical bosons.In order to identify a satisfactory and unambiguous criterion for detecting entanglement in systems of identical bosons,we resort to the decomposition of Eq(3.10)and we analyze separately the cases of Schmidt number equal to1,2,and greater than or equal to3.。
量子统计力学中的巨正则系综
量子统计力学中的巨正则系综量子统计力学是研究微观粒子的统计行为的物理学分支，它描述了由大量粒子组成的系统的性质。
在量子统计力学中，我们通常使用系综来描述这些系统。
巨正则系综是量子统计力学中的一种重要的系综，它描述了与外界热库和粒子库交换粒子和能量的系统。
本文将介绍巨正则系综的基本概念和数学表达式，并讨论其在研究物理系统中的应用。
巨正则系综是一种描述粒子数和能量都可以变化的系统的统计系综。
在巨正则系综中，系统与外界热库和粒子库交换粒子和能量，从而使得系统的粒子数和能量可以发生变化。
这种交换使得巨正则系综在描述开放系统中的粒子统计行为时非常有用。
巨正则系综的基本概念可以通过巨正则配分函数来描述。
巨正则配分函数是巨正则系综的核心概念，它描述了系统在给定温度、化学势和体积下的统计行为。
巨正则配分函数可以通过对系统的能量本征态进行求和得到，每个能量本征态的权重由玻尔兹曼因子决定。
巨正则配分函数的数学表达式如下：\[\Xi = \sum_{n=0}^{\infty} \sum_{i} e^{-\beta(E_i-\mu N_i)}\]其中，$\Xi$是巨正则配分函数，$n$是粒子数，$i$是能量本征态的标记，$E_i$是能量本征态的能量，$N_i$是能量本征态的粒子数，$\beta=1/kT$是玻尔兹曼因子，$T$是系统的温度，$\mu$是系统的化学势。
巨正则系综的基本概念和数学表达式为我们研究物理系统提供了一个强大的工具。
通过巨正则系综，我们可以计算系统的各种热力学量，如平均粒子数、平均能量和熵等。
这些热力学量可以通过巨正则配分函数的导数来计算，例如：\[\langle N \rangle = \frac{1}{\beta} \frac{\partial \ln \Xi}{\partial \mu}\]\[\langle E \rangle = -\frac{\partial \ln \Xi}{\partial \beta}\]\[S = k(\ln \Xi - \beta \frac{\partial \ln \Xi}{\partial \beta} - \mu \frac{\partial \ln\Xi}{\partial \mu})\]其中，$\langle N \rangle$是平均粒子数，$\langle E \rangle$是平均能量，$S$是熵，$k$是玻尔兹曼常数。
lindblad方程简要介绍
lindblad方程简要介绍标题：Lindblad方程简要介绍引言：Lindblad方程，也被称为量子主方程，是描述开放量子系统演化的一种重要工具。
它在量子信息科学、量子光学、凝聚态物理和化学等领域中被广泛应用。
本文将对Lindblad方程进行简要介绍，包括其基本原理、数学表达形式以及其在量子系统研究中的应用。
1. Lindblad方程的基本原理Lindblad方程由Gorini、Kossakowski和Sudarshan等人在1976年提出，用于描述开放量子系统与外部环境的相互作用。
开放系统指的是与外界环境交换能量、粒子或信息的量子系统。
相比于封闭系统，开放系统的演化更加复杂，需要考虑其与环境间的耦合效应。
2. 数学表达形式Lindblad方程的数学表达形式如下：iħ(dρ/dt) = [H, ρ] + ∑_(k) γ_k (L_k ρ L_k† - 1/2 {L_k† L_k, ρ})其中，ħ表示约化普朗克常数，ρ是系统的密度矩阵，H是系统的哈密顿量，[ , ]表示量子力学的对易子运算，γ_k是开放系统与环境耦合的弛豫速率，L_k是开放系统的湮灭算符。
3. Lindblad方程的应用Lindblad方程在各个领域有着广泛的应用。
以下是一些应用案例的简要介绍：- 量子信息科学：Lindblad方程用于研究量子比特的退相干、弛豫和非马尔可夫过程，从而有效地研究量子计算和量子通讯的可行性。
- 量子光学：Lindblad方程可用于描述自发辐射、吸收和非线性效应对光子态的影响，对于光子态的制备、操作和探测具有重要意义。
- 凝聚态物理：Lindblad方程应用于研究凝聚态系统中的量子相变、量子输运和开放量子系统的热力学性质，如单个自旋和自旋链等系统的非平衡演化过程。
- 化学：Lindblad方程在量子化学中的应用主要集中在描述与溶剂相互作用的开放量子系统的演化，如溶剂重排、电荷转移和激发态弛豫等过程。
总结：Lindblad方程作为描述开放量子系统演化的重要工具，在多个领域中有着广泛的应用。
Rb85能级图
1 Introduction
3
1 Introductiቤተ መጻሕፍቲ ባይዱn
In this reference we present many of the physical and optical properties of 85 Rb that are relevant to various quantum optics experiments. In particular, we give parameters that are useful in treating the mechanical effects of light on 85 Rb atoms. The measured numbers are given with their original references, and the calculated numbers are presented with an overview of their calculation along with references to more comprehensive discussions of their underlying theory. At present, this document is not a critical review of experimental data, nor is it even guaranteed to be correct; for any numbers critical to your research, you should consult the original references. We also present a detailed discussion of the calculation of fluorescence scattering rates, because this topic is often not treated clearly in the literature. More details and derivations regarding the theoretical formalism here may be found in Ref. [1]. The current version of this document is available at /alkalidata, along with “Cesium D Line Data,” “Sodium D Line Data,” and “Rubidium 87 D Line Data.” This is the only permanent URL for this document at present; please do not link to any others. Please send comments, corrections, and suggestions to dsteck@.
Stat_Mech_01
Thermodynamics a nd S ta1s1cal M echanicsSpring 2012Lecture 1Introduc1on t o T hermodynamicsDimi C ulcer微尺度11－002dimi@Office h ours: b y a ppointmentMy E nglish• e.g. = f or e xample • i.e. = t hat m eans • certain ~ 某 • assignment = h omework • mean = a verage • You c an s peak i n C hinese (slowly) • You c an s end e mail i n C hinese • Assignments a re i n E nglish • You c an a nswer i n C hinese• the U K = B ritain • the U S = A merica • TA = t eaching a ssistant • 帮教 • phenomenon, p henomena• momentum, m omentaSome w ords• fundamental a dj . 基础的,必要的; n . 基本法则• empirical 以观察或实验为依据的• law 定律• principle 原理• to v iolate 违反,违背• equa1on 方程• func1on 函数• variable a dj. 可变的; n . 变量• constant 常数• thermometer 温度计• macroscopic 宏观的• microscopic 微观的• arbitrary 任意的 • aVrac1ve 吸引的 • repulsive 排斥的 • Infinitesimal 极微小的 • concept 概念 • parameter 参数 • maVer 物质 • characterize 表征,描绘 • propor1onal 成比例的 • phenomenological 唯象的• concrete 实体的 • abstract 抽象的 • en1ty 实体 • deriva1on 衍生，派生State o f m aVer a nd p hase• High-‐school p hysics u nderstanding• There a re 3 s tates o f m a)er• Solid, l iquid a nd g as• More g enerally, a s tate i s c alled a p hase• Solid p hase• Liquid p hase• Gas p hase o r g aseous p hase• Phase i s a m ore g eneral, m ore u seful c oncept • For e xample F e b ecomes f erromagne1c b elow T c• We s peak o f a n o rdinary p hase a nd a f erromagne1c p hase • Right n ow t his i s j ust v ocabulary• Later y ou w ill l earn t he p hysical d ifferenceWhat i s t hermodynamics?• Thermodynamics i s a m acroscopic t heory • It d escribes l arge, m any-‐par1cle s ystems• Thermodynamics h as t wo t asks• Define a ppropriate p hysical q uan11es i n o rder t ocharacterize m acroscopic p roper1es o f m aVer• Relate t hese q uan11es b y a s et o f e qua1ons, w hich a reuniversally v alid (i.e. d o n ot d epend o n s pecific s ystem) • Thermodynamics i s p henomenological• It t akes i ts c oncepts d irectly f rom e xperiments• Thermodynamics i s a ll c lassical p hysics• No q uantum m echanics• We d o n ot t alk a bout a toms, m oleculesThermodynamics & S ta1s1cal M echanics• Sta1s1cal m echanics i s a m icroscopic t heory • We c are a bout t he d ynamics o f i ndividual p ar1cles• Sta1s1cal m echanics h as t wo b ranches• Classical s ta1s1cal m echanics• Quantum s ta1s1cal m echanics• We w ill s tudy b oth i n t his c ourse• Sta1s1cal m echanics h elps u s t o l ink t he p hysical laws o f t he m icroscopic w orld w ith t hose o f t he macroscopic w orld• One p ar1cle – m echanics• Many p ar1cles – s ta1s1csApplica1ons o f t hermodynamics • Thermodynamics h as m any a pplica1ons• Chemical r eac1ons• Phase t ransi1ons» e.g. s olid t o l iquid, l iquid t o g as» Can b e m ore c omplicated• Solid s tate» Magne1sm» Superconduc1vity a nd s uperfluidity (e.g. l iquid H e)» Charge a nd h eat t ransport• Stars, e.g. t he f orma1on o f s tars, C handrasekhar l imit• The a tmosphere, w eather, c limate• Biology e.g. l ife• All t hese s ystems o bey c ommon & g eneral l awsOverview o f t hermodynamics • In t he first p art o f t he c ourse w e w ill s tudy • The l aws o f t hermodynamics» 0, 1, 2, 3» These a re v ery g eneral• Phase t ransi1ons» Solid t o l iquid t o g as» Paramagne1c t o f erromagne1c» These a re c alled c ri1cal p henomena• Non-‐equilibrium t hermodynamics» Heat t ransport• Kine1c t heory o f g ases» Explains p, V, T b y c onsidering d ynamics o f p ar1clesImportant c oncepts i n t hermodynamics • Thermodynamic s ystem• Dynamical s ystem w ith m any d egrees o f f reedom• Any m acroscopic s ystem i s a t hermodynamic s ystem • Environment = s urroundings = e verything e lse • Thermodynamic p arameters• Measurable m acroscopic q uan11es a ssociated w ith s ystem• Pressure, v olume, t emperature• Thermodynamic s tate• Specified b y a s et o f a ll v alues o f a ll t he t hermodynamicparameters n ecessary t o d escribe t he s ystemIsolated, c losed, a nd o pen s ystems • Isolated s ystem (idealized)• Does n ot i nteract w ith t he e nvironment i n A NY w ay• Separated b y a w all (par11on) w hich d oes n ot a llowexchange o f h eat o r p ar1cles (maVer)• Total e nergy, v olume a nd p ar1cle n umber a re c onserved • Closed s ystem• Can e xchange e nergy w ith t he e nvironment• Cannot e xchange p ar1cles w ith t he e nvironment• Par1cle n umber i s c onserved• Total e nergy i s n ot c onserved• If c losed s ystem i n e quilibrium w ith e nvironment» Total e nergy h as a verage v alue r elated t o t emperature o fenvironment» Can u se t emperature t o c haracterize s tate o f s ystemIsolated, c losed, a nd o pen s ystems • Open s ystem• Can e xchange e nergy a nd p ar1cles w ith t he e nvironment• Total e nergy n ot c onserved• Par1cle n umber n ot c onserved• If o pen s ystem i n e quilibrium w ith e nvironment» Total e nergy h as a verage v alue r elated t o t emperature o fenvironment» Par1cle n umber h as a verage v alue r elated t o t emperature o fenvironment» Can u se t emperature t o c haracterize s tate o f s ystem» Can a lso u se c hemical p oten1al t o c haracterize t he s ystem» We w ill d efine t he c hemical p oten1al l ater• Right n ow w e a re u sing t he w ord t emperature • Later w e w ill d efine i t p roperlyHomogeneous s ystems• Homogeneous s ystem• The p roper1es o f t he s ystem a re t he s ame f or a ny p art • Heterogeneous s ystem• The p roper1es o f t he s ystem c hange d iscon1nuously a tcertain s urfaces• A h eterogeneous s ystem h as h omogeneous p arts • Different p arts o f t he s ystem a re i n d ifferent p hases• Phase b oundary• One s ystem, d ifferent p hases• e.g. p ot c ontaining w ater, a ir a nd s team• There a re t wo p hases – l iquid p hase a nd g aseous p hase• The b oundary b etween t hem i s c alled p hase b oundaryThermodynamic e quilibrium• Think o f a n i solated s ystem• The s ystem i s l ej s tanding f or a l ong 1me• It c omes t o a final s tate w hich d oes n ot c hange • Thermodynamic p arameters a re c onstant• The s ystem i s i n t hermodynamic e quilibrium i f i ts thermodynamic s tate d oes n ot c hange w ith 1me • A t hermodynamic s tate i s a n e quilibrium s tate • Thermodynamic e quilibrium• Also c alled t hermal e quilibrium• Ojen j ust e quilibriumThermal e quilibrium o f t wo s ystems • Generally c onsider t wo i solated s ystems A a nd B • Bring A a nd B i nto c ontact w ith e ach o ther• Ajer a l ong 1me t he t otal s ystem A+B r eaches e quilibrium• Then A a nd B a re i n e quilibrium w ith e ach o ther• A a nd B a re a lso (separately) i n t hermal e quilibrium • Think o f a c losed s ystem• A c losed s ystem e xchanges h eat w ith t he e nvironment• We c an t hink o f t he e nvironment a s a h eat b ath• e.g. t ea o n t able• Ajer a l ong 1me t emperature o f t ea = r oom t emperature• Tea i s i n t hermodynamic e quilibrium w ith r oom• Macroscopic v ariables w hich h ave a d efinite value f or e ach e ach s tate o f t hermal e quilibrium • State v ariables a re d efined o nly i n e quilibrium • Can b e m easured o nly i n e quilibrium• Usually w e o nly n eed a f ew (e.g. 3-‐4) s tate variables t o s pecify t he s tate o f a s ystem• Microscopic q uan11es a re N OT s tate v ariables • posi1on• velocity• momentum• Heat a nd w ork a re a lso N OT s tate v ariables• State v ariables c an b e • Pressure p• Volume V• Temperature T• Energy E (or U)• Entropy S• Par1cle n umber N• Chemical p oten1al μ• Total c harge Q• Total d ipole m oment P• Magne1za1on M• Refrac1ve i ndex ε• Viscosity o f a fluid• Chemical c omposi1onIntensive a nd e xtensive v ariables • Extensive q uan1ty• Scales w ith s ystem s ize• Propor1onal t o t he a mount o f m aVer i n t he s ystem• They a re a ddi1ve• Volume, e nergy, p ar1cle n umber, e ntropy• Intensive q uan1ty• Independent o f t he s ize o f t he s ystem• Independent o f t he a mount o f m aVer i n t he s ystem• Pressure, t emperature, d ensity• Also r efrac1ve i ndex a nd o thers• Can b e d efined l ocally (i.e. f unc1on o f r)• But m ost o f t he 1me w e a ssume t hem t o b e u niformEqua1on o f s tate• Func1onal r ela1onship b etween s tate v ariables • It d escribes a s ystem i n e quilibrium• For e xample i f p arameters a re p, V a nd T• There i s s ome f unc1on f(p, V, T) s uch t hatf(p,V,T)=0• Reduces n umber o f i ndependent p arameters f rom 3 t o 2• f i s g iven w hen t he s ystem i s s pecified = s tate f unc1on• State o f s ystem i s a p oint i n p, V, T s pace• Equa1on o f s tate d efines a s urface i n t his s pace• Any p oint o n t he s urface i s a s tate i n e quilibrium• Other p arameters – o ther s paces, p oints, s urface• Experimentally• All g ases b ehave i n a u niversal w ay w hen t hey a resufficiently d ilute = w hen t he d ensity i s l ow e nough • Ideal g as• Idealiza1on o f t his d ilute l imit• Par1cles a re p oint-‐like (i.e. h ave z ero s ize)• No i nterac1ons b etween p ar1cles• An i deal g as i s a n i dealized s ystem• Parameters f or i deal g as• p, V, T, N• Boyle’s l aw (1664) – a t c onstant t emperaturepV= constant = pVpV =nRT • Ideal g as e qua1on o f s tate• Defines i deal g as t emperature s cale• Like h igh-‐school p hysics• Ideal g as – H e a t v ery l ow d ensity• Measure p V/Nk o f i deal g as a t T a t w hich w ater b oils • Also m easure a t T a t w hich w ater f reezes• Draw a s traight l ine c onnec1ng t hem• Divide i nto 100 u nits – t his i s t he K elvin s cale• How t o u se• Bring o bject i n c ontact w ith i deal g as• Measure p V o f i deal g as• Read o ff t he t emperatureMore a bout t hermometers• Remember• Measuring t emperature i s r elated t o t he e qua1on o f s tate • To d efine a s cale• Choose T0 = 273.15K i n h onour o f K elvin• p0, V0 a re g iven i n y our t extbook• Celsius T C• Water f reezes = 0• Water b oils = 100• Fahrenheit T F• T F = 1.8 T C + 32• T F = T C at -‐40 d egrees• Degrees C elsius a nd d egrees F ahrenheit• But K elvin, n ot d egrees K elvinZeroth l aw o f t hermodynamics • If• System A i s i n e quilibrium w ith s ystem B• System B i s i n e quilibrium w ith s ystem C• Then• System A i s i n e quilibrium w ith s ystem C• Two s ystems A a nd B i n e quilibrium• Func1ons o f s tate f A a nd f B• See b lackboard f or d eriva1on• Systems i n t hermal e quilibrium• They h ave a c ommon i ntensive q uan1ty – t emperature• Systems n ot i n e quilibrium h ave d ifferent t emperatures • See b lackboard f or d eriva1on• First t ake t he i deal g as e qua1on o f s tate• Real g as – p hysical c onsidera1ons• First, p roper v olume• V i s n ot a c onstant b ecause V → 0 a s T → 0• Change V t o V – N b, w here b i s s ome p arameter• Second, i nterac1ons b etween p ar1cles• Interac1on i s m ainly a Vrac1ve• Consider a g lobe w ith a d ensity N/V• Inside t he g lobe t he a verage f orce b etween p ar1cles i s 0• But n ear t he w alls t his i s n ot t rue• Par1cles o n s urface f eel n et f orce t owards i nside• So p f or r eal g as i s s maller t han f or i deal g as• p ideal = p real + p0Exact a nd p ar1al d ifferen1als• State f unc1on = f unc1on o f m any v ariables • We m ake u se o f e xact a nd p ar1al d ifferen1als• For e xample i n e quilibriumf(p,V,T)=0• This m eans w e c an r egard p=p(V,T), V=V(p,T) a nd T=T(p,V) • Differen1al a lso c alled d eriva1ve• Total d eriva1ve, t otal d ifferen1al• See b lackboard f or d eriva1onTypes o f p rocesses• Infinitesimal p rocess• Difference b etween i ni1al a nd final s tate i s i nfinitesimal • Quasi-‐sta1c p rocess (ideal p rocess)• System a nd s urroundings m aintain t hermal e quilibrium• Change m ust b e s low e nough – S LOW p rocess• Quasi-‐sta1c p rocesses a re r eversible – a p rocess i sreversible i f i t r etraces i ts h istory i n 1me w hen t he e xternalcondi1ons r etrace t heir h istory i n 1me• Isothermal• Constant t emperature• Adiaba1c• No t hermal c ontact w ith t he s urroundings• However w ork i s d one o n t he s urroundings o r b y t hesurroundingsThermocouple 热电偶• Seebeck effect – t hermoelectric effect• Temperature d ifference g enerates v oltage i n c onductor• Join s econd c onductor t o m easure t he v oltage• Second c onductor h as o pposite effect• But i f m aterial i s d ifferent n et c urrent flows ∝ T• Used a s a t emperature s ensor• Useful b ecause• Ac1ve o ver w ide r ange o f t emperatures – 000s o f d egrees• Powers i tself• However• Not v ery a ccurate – n ot s ensi1ve b y m ore t han 1 d egree• So y ou c an u se i t w hen y ou d o n ot n eed m uch p recision• e.g. g as t urbine, d iesel e ngine, k iln (=oven), i ndustrySummary• Isolated, o pen a nd c losed s ystems• Thermodynamic e quilibrium• Intensive a nd e xtensive v ariables• State v ariables a nd e qua1on o f s tate • Ideal g as – n o i nterac1ons• Van d e W aals g as – w eak i nterac1ons • Temperature – t hermodynamic d efini1on • Thermometer• Assignments w ill b e g iven n ext w eek。
量子3(薛定谔方程)
（一）海森堡的矩阵力学和
测不准关系
（1）海森堡简介：
德国青年物理学家海森堡（1901~1976）早年就
读于慕尼黑大学和哥廷根大学，1923年获博士学
位后，于1924~1927年在哥本哈根物理研究所深
造，在玻尔的领导下从事量子论的研究工作，在
德布罗意之后创立了矩阵力学，提出了测不准关
朗之万把德布洛意的文章寄给爱因斯坦，
爱因斯坦称赞说：
x 2r
“揭开了自然界巨大帷幕的一角”
“瞧瞧吧，看来疯狂，可真是站得住脚呢”
Ping Qian
例 1:Why atoms do not collapse?为什么原
子不崩塌
解：
电子的最小动量是
关系。
目前测不准关系已被认为是微观粒子的客
观特性，是表明微观粒子基本性质的一个重要
原理，决不仅仅是当初所想的测量手段问题了。
不确定性关系
经典力学中：速度、动量可以同时精确确定
量子概念下：由于概率性，粒子不再具有确定的位置
从而也就不具有确定的动量
电子单缝衍射中的位置与动量：
缝宽a越小，中央明纹分布越宽
(1) 不确定性原理表明不能用经典理论来描述微观
粒子
Ping Qian
(2) the uncertainty principle gives an limitation
using classical model.不确定性原理给出了使用经典
模型的限制
Implications of the position-momentum
系。1932年因提出原子核是由质子和中子组成的
理论，获得了诺贝尔物理学奖，晚年至力于统一
场论的研究。
量子力学英语词汇
1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-partic le system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equation 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical partic les塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。
量子力学索引英汉对照
21-centimeter line, 21厘米线AAbsorption, 吸收Addition of angular momenta, 角动量叠加Adiabatic approximation, 绝热近似Adiabatic process, 绝热过程Adjoint, 自伴的Agnostic position, 不可知论立场Aharonov-Bohm effect, 阿哈罗诺夫—玻姆效应Airy equation, 艾里方程；Airy function, 艾里函数Allowed energy, 允许能量Allowed transition, 允许跃迁Alpha decay, α衰变；Alpha particle, α粒子Angular equation, 角向方程Angular momentum, 角动量Anomalous magnetic moment, 反常磁矩Antibonding, 反键Anti-hermitian operator, 反厄米算符Associated Laguerre polynomial, 连带拉盖尔多项式Associated Legendre function, 连带勒让德多项式Atoms, 原子Average value, 平均值Azimuthal angle, 方位角Azimuthal quantum number, 角量子数BBalmer series, 巴尔末线系Band structure, 能带结构Baryon, 重子Berry's phase, 贝利相位Bessel functions, 贝塞尔函数Binding energy, 束缚能Binomial coefficient, 二项式系数Biot-Savart law, 毕奥—沙法尔定律Blackbody spectrum, 黑体谱Bloch's theorem, 布洛赫定理Bohr energies, 玻尔能量；Bohr magneton, 玻尔磁子；Bohr radius, 玻尔半径Boltzmann constant, 玻尔兹曼常数Bond, 化学键Born approximation, 玻恩近似Born's statistical interpretation, 玻恩统计诠释Bose condensation, 玻色凝聚Bose-Einstein distribution, 玻色—爱因斯坦分布Boson, 玻色子Bound state, 束缚态Boundary conditions, 边界条件Bra, 左矢Bulk modulus, 体积模量CCanonical commutation relations, 正则对易关系Canonical momentum, 正则动量Cauchy's integral formula, 柯西积分公式Centrifugal term, 离心项Chandrasekhar limit, 钱德拉赛卡极限Chemical potential, 化学势Classical electron radius, 经典电子半径Clebsch-Gordan coefficients, 克—高系数Coherent States, 相干态Collapse of wave function, 波函数塌缩Commutator, 对易子Compatible observables, 对易的可观测量Complete inner product space, 完备内积空间Completeness, 完备性Conductor, 导体Configuration, 位形Connection formulas, 连接公式Conservation, 守恒Conservative systems, 保守系Continuity equation, 连续性方程Continuous spectrum, 连续谱Continuous variables, 连续变量Contour integral, 围道积分Copenhagen interpretation, 哥本哈根诠释Coulomb barrier, 库仑势垒Coulomb potential, 库仑势Covalent bond, 共价键Critical temperature, 临界温度Cross-section, 截面Crystal, 晶体Cubic symmetry, 立方对称性Cyclotron motion, 螺旋运动DDarwin term, 达尔文项de Broglie formula, 德布罗意公式de Broglie wavelength, 德布罗意波长Decay mode, 衰变模式Degeneracy, 简并度Degeneracy pressure, 简并压Degenerate perturbation theory, 简并微扰论Degenerate states, 简并态Degrees of freedom, 自由度Delta-function barrier, δ势垒Delta-function well, δ势阱Derivative operator, 求导算符Determinant, 行列式Determinate state, 确定的态Deuterium, 氘Deuteron, 氘核Diagonal matrix, 对角矩阵Diagonalizable matrix, 对角化Differential cross-section, 微分截面Dipole moment, 偶极矩Dirac delta function, 狄拉克δ函数Dirac equation, 狄拉克方程Dirac notation, 狄拉克记号Dirac orthonormality, 狄拉克正交归一性Direct integral, 直接积分Discrete spectrum, 分立谱Discrete variable, 离散变量Dispersion relation, 色散关系Displacement operator, 位移算符Distinguishable particles, 可分辨粒子Distribution, 分布Doping, 掺杂Double well, 双势阱Dual space, 对偶空间Dynamic phase, 动力学相位EEffective nuclear charge, 有效核电荷Effective potential, 有效势Ehrenfest's theorem, 厄伦费斯特定理Eigenfunction, 本征函数Eigenvalue, 本征值Eigenvector, 本征矢Einstein's A and B coefficients, 爱因斯坦A,B系数；Einstein's mass-energy formula, 爱因斯坦质能公式Electric dipole, 电偶极Electric dipole moment, 电偶极矩Electric dipole radiation, 电偶极辐射Electric dipole transition, 电偶极跃迁Electric quadrupole transition, 电四极跃迁Electric field, 电场Electromagnetic wave, 电磁波Electron, 电子Emission, 发射Energy, 能量Energy-time uncertainty principle, 能量—时间不确定性关系Ensemble, 系综Equilibrium, 平衡Equipartition theorem, 配分函数Euler's formula, 欧拉公式Even function, 偶函数Exchange force, 交换力Exchange integral, 交换积分Exchange operator, 交换算符Excited state, 激发态Exclusion principle, 不相容原理Expectation value, 期待值FFermi-Dirac distribution, 费米—狄拉克分布Fermi energy, 费米能Fermi surface, 费米面Fermi temperature, 费米温度Fermi's golden rule, 费米黄金规则Fermion, 费米子Feynman diagram, 费曼图Feynman-Hellman theorem, 费曼—海尔曼定理Fine structure, 精细结构Fine structure constant, 精细结构常数Finite square well, 有限深方势阱First-order correction, 一级修正Flux quantization, 磁通量子化Forbidden transition, 禁戒跃迁Foucault pendulum, 傅科摆Fourier series, 傅里叶级数Fourier transform, 傅里叶变换Free electron, 自由电子Free electron density, 自由电子密度Free electron gas, 自由电子气Free particle, 自由粒子Function space, 函数空间Fusion, 聚变Gg-factor, g—因子Gamma function, Γ函数Gap, 能隙Gauge invariance, 规范不变性Gauge transformation, 规范变换Gaussian wave packet, 高斯波包Generalized function, 广义函数Generating function, 生成函数Generator, 生成元Geometric phase, 几何相位Geometric series, 几何级数Golden rule, 黄金规则"Good" quantum number, “好”量子数"Good" states, “好”的态Gradient, 梯度Gram-Schmidt orthogonalization, 格莱姆—施密特正交化法Graphical solution, 图解法Green's function, 格林函数Ground state, 基态Group theory, 群论Group velocity, 群速Gyromagnetic railo, 回转磁比值HHalf-integer angular momentum, 半整数角动量Half-life, 半衰期Hamiltonian, 哈密顿量Hankel functions, 汉克尔函数Hannay's angle, 哈内角Hard-sphere scattering, 硬球散射Harmonic oscillator, 谐振子Heisenberg picture, 海森堡绘景Heisenberg uncertainty principle, 海森堡不确定性关系Helium, 氦Helmholtz equation, 亥姆霍兹方程Hermite polynomials, 厄米多项式Hermitian conjugate, 厄米共轭Hermitian matrix, 厄米矩阵Hidden variables, 隐变量Hilbert space, 希尔伯特空间Hole, 空穴Hooke's law, 胡克定律Hund's rules, 洪特规则Hydrogen atom, 氢原子Hydrogen ion, 氢离子Hydrogen molecule, 氢分子Hydrogen molecule ion, 氢分子离子Hydrogenic atom, 类氢原子Hyperfine splitting, 超精细分裂IIdea gas, 理想气体Idempotent operaror, 幂等算符Identical particles, 全同粒子Identity operator, 恒等算符Impact parameter, 碰撞参数Impulse approximation, 脉冲近似Incident wave, 入射波Incoherent perturbation, 非相干微扰Incompatible observables, 不对易的可观测量Incompleteness, 不完备性Indeterminacy, 非确定性Indistinguishable particles, 不可分辨粒子Infinite spherical well, 无限深球势阱Infinite square well, 无限深方势阱Inner product, 内积Insulator, 绝缘体Integration by parts, 分部积分Intrinsic angular momentum, 内禀角动量Inverse beta decay, 逆β衰变Inverse Fourier transform, 傅里叶逆变换KKet, 右矢Kinetic energy, 动能Kramers' relation, 克莱默斯关系Kronecker delta, 克劳尼克δLLCAO technique, 原子轨道线性组合法Ladder operators, 阶梯算符Lagrange multiplier, 拉格朗日乘子Laguerre polynomial, 拉盖尔多项式Lamb shift, 兰姆移动Lande g-factor, 朗德g—因子Laplacian, 拉普拉斯的Larmor formula, 拉摩公式Larmor frequency, 拉摩频率Larmor precession, 拉摩进动Laser, 激光Legendre polynomial, 勒让德多项式Levi-Civita symbol, 列维—西维塔符号Lifetime, 寿命Linear algebra, 线性代数Linear combination, 线性组合Linear combination of atomic orbitals, 原子轨道的线性组合Linear operator, 线性算符Linear transformation, 线性变换Lorentz force law, 洛伦兹力定律Lowering operator, 下降算符Luminoscity, 照度Lyman series, 赖曼线系MMagnetic dipole, 磁偶极Magnetic dipole moment, 磁偶极矩Magnetic dipole transition, 磁偶极跃迁Magnetic field, 磁场Magnetic flux, 磁通量Magnetic quantum number, 磁量子数Magnetic resonance, 磁共振Many worlds interpretation, 多世界诠释Matrix, 矩阵；Matrix element, 矩阵元Maxwell-Boltzmann distribution, 麦克斯韦—玻尔兹曼分布Maxwell’s equations, 麦克斯韦方程Mean value, 平均值Measurement, 测量Median value, 中位值Meson, 介子Metastable state, 亚稳态Minimum-uncertainty wave packet, 最小不确定度波包Molecule, 分子Momentum, 动量Momentum operator, 动量算符Momentum space wave function, 动量空间波函数Momentum transfer, 动量转移Most probable value, 最可几值Muon, μ子Muon-catalysed fusion, μ子催化的聚变Muonic hydrogen, μ原子Muonium, μ子素NNeumann function, 纽曼函数Neutrino oscillations, 中微子振荡Neutron star, 中子星Node, 节点Nomenclature, 术语Nondegenerate perturbationtheory, 非简并微扰论Non-normalizable function, 不可归一化的函数Normalization, 归一化Nuclear lifetime, 核寿命Nuclear magnetic resonance, 核磁共振Null vector, 零矢量OObservable, 可观测量Observer, 观测者Occupation number, 占有数Odd function, 奇函数Operator, 算符Optical theorem, 光学定理Orbital, 轨道的Orbital angular momentum, 轨道角动量Orthodox position, 正统立场Orthogonality, 正交性Orthogonalization, 正交化Orthohelium, 正氦Orthonormality, 正交归一性Orthorhombic symmetry, 斜方对称Overlap integral, 交叠积分PParahelium, 仲氦Partial wave amplitude, 分波幅Partial wave analysis, 分波法Paschen series, 帕邢线系Pauli exclusion principle, 泡利不相容原理Pauli spin matrices, 泡利自旋矩阵Periodic table, 周期表Perturbation theory, 微扰论Phase, 相位Phase shift, 相移Phase velocity, 相速Photon, 光子Planck's blackbody formula, 普朗克黑体辐射公式Planck's constant, 普朗克常数Polar angle, 极角Polarization, 极化Population inversion, 粒子数反转Position, 位置；Position operator, 位置算符Position-momentum uncertainty principles, 位置—动量不确定性关系Position space wave function, 坐标空间波函数Positronium, 电子偶素Potential energy, 势能Potential well, 势阱Power law potential, 幂律势Power series expansion, 幂级数展开Principal quantum number, 主量子数Probability, 几率Probability current, 几率流Probability density, 几率密度Projection operator, 投影算符Propagator, 传播子Proton, 质子QQuantum dynamics, 量子动力学Quantum electrodynamics, 量子电动力学Quantum number, 量子数Quantum statics, 量子统计Quantum statistical mechanics, 量子统计力学Quark, 夸克RRabi flopping frequency, 拉比翻转频率Radial equation, 径向方程Radial wave function, 径向波函数Radiation, 辐射Radius, 半径Raising operator, 上升算符Rayleigh's formula, 瑞利公式Realist position, 实在论立场Recursion formula, 递推公式Reduced mass, 约化质量Reflected wave, 反射波Reflection coefficient, 反射系数Relativistic correction, 相对论修正Rigid rotor, 刚性转子Rodrigues formula, 罗德里格斯公式Rotating wave approximation, 旋转波近似Rutherford scattering, 卢瑟福散射Rydberg constant, 里德堡常数Rydberg formula, 里德堡公式SScalar potential, 标势Scattering, 散射Scattering amplitude, 散射幅Scattering angle, 散射角Scattering matrix, 散射矩阵Scattering state, 散射态Schrodinger equation, 薛定谔方程Schrodinger picture, 薛定谔绘景Schwarz inequality, 施瓦兹不等式Screening, 屏蔽Second-order correction, 二级修正Selection rules, 选择定则Semiconductor, 半导体Separable solutions, 分离变量解Separation of variables, 变量分离Shell, 壳Simple harmonic oscillator, 简谐振子Simultaneous diagonalization, 同时对角化Singlet state, 单态Slater determinant, 斯拉特行列式Soft-sphere scattering, 软球散射Solenoid, 螺线管Solids, 固体Spectral decomposition, 谱分解Spectrum, 谱Spherical Bessel functions, 球贝塞尔函数Spherical coordinates, 球坐标Spherical Hankel functions, 球汉克尔函数Spherical harmonics, 球谐函数Spherical Neumann functions, 球纽曼函数Spin, 自旋Spin matrices, 自旋矩阵Spin-orbit coupling, 自旋—轨道耦合Spin-orbit interaction, 自旋—轨道相互作用Spinor, 旋量Spin-spin coupling, 自旋—自旋耦合Spontaneous emission, 自发辐射Square-integrable function, 平方可积函数Square well, 方势阱Standard deviation, 标准偏差Stark effect, 斯塔克效应Stationary state, 定态Statistical interpretation, 统计诠释Statistical mechanics, 统计力学Stefan-Boltzmann law, 斯特番—玻尔兹曼定律Step function, 阶跃函数Stem-Gerlach experiment, 斯特恩—盖拉赫实验Stimulated emission, 受激辐射Stirling's approximation, 斯特林近似Superconductor, 超导体Symmetrization, 对称化Symmetry, 对称TTaylor series, 泰勒级数Temperature, 温度Tetragonal symmetry, 正方对称Thermal equilibrium, 热平衡Thomas precession, 托马斯进动Time-dependent perturbation theory, 含时微扰论Time-dependent Schrodinger equation, 含时薛定谔方程Time-independent perturbation theory, 定态微扰论Time-independent Schrodinger equation, 定态薛定谔方程Total cross-section, 总截面Transfer matrix, 转移矩阵Transformation, 变换Transition, 跃迁；Transition probability, 跃迁几率Transition rate, 跃迁速率Translation,平移Transmission coefficient, 透射系数Transmitted wave, 透射波Trial wave function, 试探波函数Triplet state, 三重态Tunneling, 隧穿Turning points, 回转点Two-fold degeneracy , 二重简并Two-level systems, 二能级体系UUncertainty principle, 不确定性关系Unstable particles, 不稳定粒子VValence electron, 价电子Van der Waals interaction, 范德瓦尔斯相互作用Variables, 变量Variance, 方差Variational principle, 变分原理Vector, 矢量Vector potential, 矢势Velocity, 速度Vertex factor, 顶角因子Virial theorem, 维里定理WWave function, 波函数Wavelength, 波长Wave number, 波数Wave packet, 波包Wave vector, 波矢White dwarf, 白矮星Wien's displacement law, 维恩位移定律YYukawa potential, 汤川势ZZeeman effect, 塞曼效应。
量子光学第二讲,相干态与压缩态
相干态物理性质
于是得到坐标与动量的测不准量为
( ∆pቤተ መጻሕፍቲ ባይዱ)2 = p 2 − p ( ∆q )2 = q 2 − q
2
= ω n+ = ω
(
1 2
)
2
(
n+
1 2
)
测不准关系为
∆p∆q = n +
(
1 2
)
可见最小测不准态是基态 φ0 ( q ) ，最小值为 / 2 。如何通过简单的谐振运动但 保持这个最小测不准波包形状不变？假设在 t = 0 时刻，波函数 ψ (q, 0 ) 具有最 小测不准波包形式，只是在 +q 方向上有一个位移量 q 0 ，于是有
∆q ∆p =
/2
上式说明相干态是最小测不准量子态，因而也是量子理论所容许的最接近 经典极限的量子态。
2004 © Dr. Shutian Liu, Department of Physics, Harbin Institute of Technology 13
相干态物理性质
相干态的能量起伏
利用不等式 A2 + B 2 ≥ 2AB ，我们有
9
相干态物理性质
几率密度随时间的变化为
ω ψ ( q, t ) = π
2
( )
1
2
ω exp ⎡⎢ − ( q − q 0 cos ωt )2 ⎤⎥ ⎣ ⎦
该波包在谐振子势场中来回做简谐振动而形状不变，因此是相干的。该波包 具有最小测不准关系，是最接近经典单模场的量子力学表述形式。相干的最 小测不准波包可以写成
α = exp −
(
1 2 α 2
)
1 2 α exp ( −α*a ) exp ( αa + ) 2 1 = exp − α 2 exp ( αa + ) exp ( −α*a ) 2 ↓ 1 exp ( αa + ) exp ( −α*a ) 0 = exp − α 2 exp ( αa + ) 0 2