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Decoherence Basic Concepts and Their Interpretation

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2Basic Concepts and their Interpretation ?H.D.Zeh (www.zeh-hd.de)2.1The Phenomenon of Decoherence 2.1.1Superpositions The superposition principle forms the most fundamental kinematical con-cept of quantum theory.Its universality seems to have ?rst been postulated by Dirac as part of the de?nition of his “ket-vectors”,which he proposed as a complete 1and general concept to characterize quantum states regardless of any basis of representation.They were later recognized by von Neumann as forming an abstract Hilbert space.The inner product (also needed to de-?ne a Hilbert space,and formally indicated by the distinction between “bra”and “ket”vectors)is not part of the kinematics proper,but required for the probability interpretation,which may be regarded as dynamics (as will be discussed).The third Hilbert space axiom (closure with respect to Cauchy series)is merely mathematically convenient,since one can never decide em-pirically whether the number of linearly independent physical states is in?nite in reality,or just very large.According to this kinematical superposition principle,any two physical states,|1 and |2 ,whatever their meaning,can be superposed in the form c 1|1 +c 2|2 ,with complex numbers c 1and c 2,to form a new physical state (to be be distinguished from a state of information ).By induction,the principle can be applied to more than two,and even an in?nite number,of states,and appropriately generalized to apply to a continuum of states.After postulat-ing the linear Schr¨o dinger equation in a general form,one may furthermore

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conclude that the superposition of two(or more)of its solutions forms again a solution.This is the dynamical version of the superposition principle.

Let me emphasize that this superposition pinciple is in drastic contrast to the concept of the“quantum”that gave the theory its name.Superposi-tions obeying the Schr¨o dinger equation describe a deterministically evolving

continuum rather than discrete quanta and stochastic quantum jumps.Ac-cording to the theory of decoherence,these e?ective concepts“emerge”as a

consequence of the superposition principle when universally and consistently applied.

A dynamical superposition principle(though in general with respect to

real numbers only)is also known from classical waves which obey a linear wave equation.Its validity is then restricted to cases where these equations apply,while the quantum superposition principle is meant to be universal and

exact.However,while the physical meaning of classical superpositions is usu-ally obvious,that of quantum mechanical superpositions has to be somehow determined.For example,the interpretation of a superposition dq e ipq|q as representing a state of momentum p can be derived from“quantization rules”, valid for systems whose classical counterparts are known in their Hamiltonian

form(see Sect.2.2).In other cases,an interpretation may be derived from the dynamics or has to be based on experiments.

Dirac emphasized another(in his opinion even more important)di?er-

ence:all non-vanishing components of(or projections from)a superposition are“in some sense contained”in it.This formulation seems to refer to an en-semble of physical states,which would imply that their description by formal “quantum states”is not complete.Another interpretation asserts that it is the(Schr¨o dinger)dynamics rather than the concept of quantum states which is incomplete.States found in measurements would then have to arise from an initial state by means of an indeterministic“collapse of the wave function”. Both interpretations meet serious di?culties when consistently applied(see Sect.2.3).

In the third edition of his textbook,Dirac(1947)starts to explain the su-perposition principle by discussing one-particle states,which can be described by Schr¨o dinger waves in three-dimensional space.This is an important appli-cation,although its similarity with classical waves may also be misleading. Wave functions derived from the quantization rules are de?ned on their clas-sical con?guration space,which happens to coincide with normal space only for a single mass point.Except for this limitation,the two-slit interference experiment,for example,(e?ectively a two-state superposition)is known to be very instructive.Dirac’s second example,the superposition of two basic photon polarizations,no longer corresponds to a spatial wave.These two basic states“contain”all possible photon polarizations.The electron spin, another two-state system,exhausts the group SU(2)by a two-valued repre-sentation of spatial rotations,and it can be studied(with atoms or neutrons) by means of many variations of the Stern–Gerlach experiment.In his lecture

2Basic Concepts and their Interpretation3 notes(Feynman,Leighton,and Sands1965),Feynman describes the maser mode of the ammonia molecule as another(very di?erent)two-state system.

All these examples make essential use of superpositions of the kind|α = c1|1 +c2|2 ,where the states|1 ,|2 ,and(all)|α can be observed as phys-ically di?erent states,and distinguished from one another in an appropriate setting.In the two-slit experiment,the states|1 and|2 represent the par-tial Schr¨o dinger waves that pass through one or the other slit.Schr¨o dinger’s wave function can itself be understood as a consequence of the superposi-tion principle by being viewed as the amplitudesψα(q)in the superposition of“classical”con?gurations q(now represented by corresponding quantum states|q or their narrow wave packets).In this case of a system with a known classical counterpart,the superpositions|α = dqψα(q)|q are assumed to de?ne all quantum states.They may represent new observable properties (such as energy or angular momentum),which are not simply functions of the con?guration,f(q),only as a nonlocal whole,but not as an integral over corresponding local densities(neither on space nor on con?guration space).

Since Schr¨o dinger’s wave function is thus de?ned on(in general high-dimensional)con?guration space,increasing its amplitude does not describe an increase of intensity or energy density,as it would for classical waves in three-dimensional space.Superpositions of the intuitive product states of composite quantum systems may not only describe particle exchange sym-metries(for bosons and fermions);in the general case they lead to the fun-damental concept of quantum nonlocality.The latter has to be distinguished from a mere extension in space(characterizing extended classical objects).For example,molecules in energy eigenstates are incompatible with their atoms being in de?nite quantum states themselves.Although the importance of this “entanglement”for many observable quantities(such as the binding energy of the helium atom,or total angular momentum)had been well known,its consequence of violating Bell’s inequalities(Bell1964)seems to have sur-prised many physicists,since this result strictly excluded all local theories conceivably underlying quantum theory.However,quantum nonlocality ap-pears paradoxical only when one attempts to interpret the wave function in terms of an ensemble of local properties,such as“particles”.If reality were de?ned to be local(“in space and time”),then it would indeed con?ict with the empirical actuality of a general superposition.Within the quantum formalism,entanglement also leads to decoherence,and in this way it ex-plains the classical appearance of the observed world in quantum mechanical terms.The application of this program is the main subject of this book(see also Zurek1991,Mensky2000,Tegmark and Wheeler2001,Zurek2001,or www.decoherence.de).

The predictive power of the superposition principle became particularly evident when it was applied in an ingenious step to postulate the existence of superpositions of states with di?erent particle numbers(Jordan and Klein 1927).Their meaning is illustrated,for example,by“coherent states”of dif-

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ferent photon numbers,which may represent quasi-classical states of the elec-tromagnetic?eld(cf.Glauber1963).Such dynamically arising(and in many cases experimentally con?rmed)superpositions are often misinterpreted as representing“virtual”states,or mere probability amplitudes for the occur-rence of“real”states that are assumed to possess de?nite particle number. This would be as mistaken as replacing a hydrogen wave function by the probability distribution p(r)=|ψ(r)|2,or an entangled state by an ensem-ble of product states(or a two-point function).A superposition is in general observably di?erent from an ensemble consisting of its components with cor-responding probabilities.

Another spectacular success of the superposition principle was the pre-diction of new particles formed as superpositions of K-mesons and their an-tiparticles(Gell-Mann and Pais1955,Lee and Yang1956).A similar model describes the recently con?rmed“neutrino oscillations”(Wolfenstein1978), which are superpositions of energy eigenstates.

The superposition principle can also be successfully applied to states that may be generated by means of symmetry transformations from asymmet-ric ones.In classical mechanics,a symmetric Hamiltonian means that each asymmetric solution(such as an elliptical Kepler orbit)implies other solu-tions,obtained by applying the symmetry transformations(e.g.rotations). Quantum theory requires in addition that all their superpositions also form solutions(cf.Wigner1964,or Gross1995;see also Sect.9.6).A complete set of energy eigenstates can then be constructed by means of irreducible linear representations of the dynamical symmetry group.Among them are usually symmetric ones(such as s-waves for scalar particles)that need not have a counterpart in classical mechanics.

A great number of novel applications of the superpositon principle have been studied experimentally or theoretically during recent years.For exam-ple,superpositions of di?erent“classical”states of laser modes(“mesoscopic Schr¨o dinger cats”)have been prepared(Monroe et al.1996),the entangle-ment of photon pairs has been con?rmed to persist over tens of kilometers (Tittel et al.1998),and interference experiments with fullerene molecules were successfully performed(Arndt et al.1999).Even superpositions of a macroscopic current running in opposite directions have been shown to ex-ist,and con?rmed to be di?erent from a state with two(cancelling)currents (Mooij et al.1999,Friedman et al.2000).Quantum computers,now under intense investigation,would have to perform“parallel”(but not spatially sep-arated)calculations,while forming one superposition that may later have a coherent e?ect.So-called quantum teleportation requires the advanced prepa-ration of an entangled state of distant systems(cf.Busch et al.2001for a consistent description in quantum mechanical terms).One of its components may then later be selected by a local measurement in order to determine the state of the other(distant)system.

2Basic Concepts and their Interpretation5 Whenever an experiment was technically feasible,all components of a superposition have been shown to act coherently,thus proving that they exist simultaneously.It is surprising that many physicists still seem to regard superpositions as representing some state of ignorance(merely characterizing unpredictable“events”).After the fullerene experiments there remains but a minor step to discuss conceivable(though hardly realizable)interference experiments with a conscious observer.Would he have one or many“minds”(when being aware of his path through the slits)?

The most general quantum states seem to be superpositions of di?er-ent classical?elds on three-or higher-dimensional space.2In a perturbation expansion in terms of free“particles”(wave modes)this leads to terms cor-responding to Feynman diagrams,as shown long ago by Dyson(1949).The path integral describes a superposition of paths,that is,the propagation of wave functionals according to a generalized Schr¨o dinger equation,while the individual paths under the integral have no physical meaning by themselves.

(A similar method could be used to describe the propagation of classical waves.)Wave functions will here always be understood in the generalized sense of wave functionals if required.

One has to keep in mind this universality of the superposition princi-ple and its consequences for individually observable physical properties in order to appreciate the meaning of the program of decoherence.Since quan-tum coherence is far more than the appearance of spatial interference fringes observed statistically in series of“events”,decoherence must not simply be understood in a classical sense as their washing out under?uctuating envi-ronmental conditions.

2.1.2Superselection Rules

In spite of this success of the superposition principle it soon became evi-dent that not all conceivable superpositions are found in Nature.This led some physicists to postulate“superselection rules”,which restrict this prin-ciple by axiomatically excluding certain superpositions(Wick,Wightman,

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and Wigner1970,Streater and Wightman1964).There are also attempts to derive some of these superselection rules from other principles,which can be postulated in quantum?eld theory(see Chaps.6and7).In general,these principles merely exclude“unwanted”consequences of a general superposi-tion principle by hand.

Most disturbing in this sense seem to be superpositions of states with integer and half-integer spin(bosons and fermions).They violate invariance under2π-rotations(see Sect.6.2),but such a non-invariance has been exper-imentally con?rmed in a di?erent way(Rauch et al.1975).The theory of supersymmetry(Wess and Zumino1971)postulates superpositions of bosons and fermions.Another supposedly“fundamental”superselection rule forbids superpositions of di?erent charge.For example,superpositions of a proton and a neutron have never been directly observed,although they occur in the isotopic spin formalism.This(dynamically broken)symmetry was later successfully generalized to SU(3)and other groups in order to characterize further intrinsic degrees of freedom.However,superpositions of a proton and a neutron may“exist”within nuclei,where isospin-dependent self-consistent potentials may arise from an intrinsic symmetry breaking.Similarly,superpo-sitions of di?erent charge are used to form BCS states(Bardeen,Cooper,and Schrie?er1957),which describe the intrinsic properties of superconductors. In these cases,de?nite charge values have to be projected out(see Sect.9.6) in order to describe the observed physical objects,which do obey the charge superselection rule.

Other limitations of the superposition principle are less clearly de?ned. While elementary particles are described by means of wave functions(that is,superpositions of di?erent positions or other properties),the moon seems always to be at a de?nite place,and a cat is either dead or alive.A general superposition principle would even allow superpositions of a cat and a dog(as suggested by Joos).They would have to de?ne a”new animal”–analogous to a K long,which is a superposition of a K-meson and its antiparticle.In the Copenhagen interpretation,this di?erence is attributed to a strict conceptual separation between the microscopic and the macroscopic world.However, where is the border line that distinguishes an n-particle state of quantum mechanics from an N-particle state that is classical?Where,precisely,does the superposition principle break down?

Chemists do indeed know that a border line seems to exist deep in the microscopic world(Primas1981,Woolley1986).For example,most molecules (save the smallest ones)are found with their nuclei in de?nite(usually ro-tating and/or vibrating)classical“con?gurations”,but hardly ever in super-positions thereof,as it would be required for energy or angular momentum eigenstates.The latter are observed for hydrogen and other small molecules. Even chiral states of a sugar molecule appear“classical”,in contrast to its parity and energy eigenstates,which correctly describe the otherwise analo-gous maser mode states of the ammonia molecule(see Sect.3.2.4for details).

2Basic Concepts and their Interpretation7 Does this di?erence mean that quantum mechanics breaks down already for very small particle number?

Certainly not in general,since there are well established superpositions of many-particle states:phonons in solids,super?uids,SQUIDs,white dwarf stars and many more!All properties of macroscopic bodies which can be cal-culated quantitatively are consistent with quantum mechanics,but not with any microscopic classical description.As will be demonstrated throughout the book,the theory of decoherence is able to explain the apparent di?er-ences between the quantum and the classical world under the assumption of a universally valid quantum theory.

The attempt to derive the absence of certain superpositions from(exact or approximate)conservation laws,which forbid or suppress transitions between their corresponding components,would be insu?cient.This“traditional”ex-planation(which seems to be the origin of the name“superselection rule”) was used,for example,by Hund(1927)in his arguments in favor of the chiral states of molecules.However,small or vanishing transition rates require in addition that superpositions were absent initially for all these molecules(or their constituents from which they formed).Similarly,charge conservation does not explain the charge superselection rule!Negligible wave packet dis-persion(valid for large mass)may prevent initially presumed wave packets from growing wider,but this initial condition is quantitatively insu?cient to explain the quasi-classical appearance of mesoscopic objects,such as small dust grains or large molecules(see Sect.3.2.1),or even that of celestial bodies in chaotic motion(Zurek and Paz1994).Even initial conditions for conserved quantities would in general allow one only to exclude global superpositions, but not local ones(Giulini,Kiefer and Zeh1995).

So how can superselection rules be explained within quantum theory? 2.1.3Decoherence by“Measurements”

Other experiments with quantum objects have taught us that interference,for example between partial waves,disappears when the property characterizing these partial waves is measured.Such partial waves may describe the passage through di?erent slits of an interference device,or the two beams of a Stern–Gerlach device(“Welcher Weg experiments”).This loss of coherence is indeed required by mere logic once measurements are assumed to lead to de?nite re-sults.In this case,the frequencies of events on the detection screen measured in coincidence with a certain passage can be counted separately,and thus have to be added to de?ne the total probabilities.3It is therefore a plausible

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experience that the interference disappears also when the passage is“mea-sured”without registration of a de?nite result.The latter may be assumed to have become a“classical fact”as soon the measurement has irreversibly“oc-curred”.A quantum phenomenon may thus“become a phenomenon”without being observed(in contrast to this early formulation of Bohr’s,which is in accordance with Heisenberg’s idealistic statement about a trajectory coming into being by its observation–while Bohr later spoke of objective irreversible events occurring in the counter).However,what presicely is an irreversible quantum event?According to Bohr,it can not be dynamically analyzed.

Analysis within the quantum mechanical formalism demonstrates nonethe-less that the essential condition for this“decoherence”is that complete infor-mation about the passage is carried away in some physical form(Zeh1970, 1973,Mensky1979,Zurek1981,Caldeira and Leggett1983,Joos and Zeh 1985).Possessing“information”here means that the physical state of the environment is now uniquely quantum correlated(entangled)with the rele-vant property of the system(such as a passage through a speci?c slit).This need not happen in a controllable form(as in a measurement):the“informa-tion”may as well be created in the form of noise.However,in contrast to statistical correlations,quantum correlations de?ne pure(completly de?ned) nonlocal states,and thus individual physical properties,such as the total spin of spatially separated objects.Therefore,one cannot explain entanglement in terms of the concept of information(cf.Brukner and Zeilinger2000).This terminology would mislead to the popular misunderstanding of the collapse as a“mere increase of information”(which would require an initial ensem-ble describing ignorance).Since environmental decoherence a?ects individual physical states,it can neither be the consequence of phase averaging in an ensemble,nor one of phases?uctuating uncontrollably in time(as claimed in some textbooks).For example,nonlocal entanglement exists in the static quantum state of a relativstic physical vacuum(even though it is then often visualized in terms of particles as“vacuum?uctuations”).

When is unambiguous“information”carried away?If a macroscopic ob-ject had the opportunity of passing through two slits,we would always be able to convince ourselves of its choice of a path by simply opening our eyes in order to“look”.This means that in this case there is plenty of light that contains information about the path(even in a controllable manner that al-lows“looking”).Interference between di?erent paths never occurs,since the

2Basic Concepts and their Interpretation9 path is evidently“continuously measured”by light.The common textbook argument that the interference pattern of macroscopic objects be too?ne to be observable is entirely irrelevant.However,would it then not be su?cient to dim the light in order to reproduce(in principle)a quantum mechanical interference pattern for macroscopic objects?

This could be investigated by means of more sophisticated experiments with mesoscopic objects(see Brune et al.1996).However,in order to precisely determine the subtle limit where measurement by the environment becomes negligible,it is more economic?rst to apply the established theory which is known to describe such experiments.Thereby we have to take into account the quantum nature of the environment,as discussed long ago by Brillouin (1962)for an information medium in general.This can usually be done easily, since the quantum theory of interacting systems,such as the quantum the-ory of particle scattering,is well understood.Its application to decoherence requires that one averages over all unobserved degrees of freedom.In tech-nical terms,one has to“trace out the environment”after it has interacted with the considered system.This procedure leads to a quantitative theory of decoherence(cf.Joos and Zeh1985).Taking the trace is based on the prob-ability interpretation applied to the environment(averaging over all possible outcomes of measurements),even though this environment is not measured. (The precise physical meaning of these formal concepts will be discussed in Sect.2.4.)

Is it possible to explain all superselection rules in this way as an e?ect induced by the environment4–including the existence and position of the border line between microscopic and macroscopic behaviour in the realm of molecules?This would mean that the universality of the superposition principle could be maintained–as is indeed the basic idea of the program of decoherence(Zeh1970,Zurek1982;see also Chap.4of Zeh2001).If physical states are thus exclusively described by wave functions rather than by points in con?guration space–as originally intended by Schr¨o dinger in space by means of narrow wave packets instead of particles–then no uncertainty relations are available for states in order to explain the probabilistic aspects of quantum theory:the Fourier theorem applies to a given wave function(al).

As another example,consider two states of di?erent charge.They inter-act very di?erently with the electromagnetic?eld even in the absence of radiation:their Coulomb?elds carry complete“information”about the total charge at any distance.The quantum state of this?eld would thus decohere a superposition of di?erent charges if considered as a quantum system in a bounded region of space(Giulini,Kiefer,and Zeh1995).This instantaneous action of decoherence at an arbitrary distance by means of the Coulomb?eld gives it the appearance of a kinematical e?ect,although it is based on the

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dynamical law of charge conservation,compatible with a retarded?eld that would“measure”the charge(see Sect.6.4).

There are many other cases where the unavoidable e?ect of decoherence can easily be imagined without any calculation.For example,superpositions of macroscopically di?erent electromagnetic?elds,f(r),may be described by a?eld functionalΨ[f(r)].However,any charged particle in a su?ciently narrow wave packet would then evolve into di?erent packets,depending on the?eld f,and thus become entangled with the state of the quantum?eld (K¨u bler and Zeh1973,Kiefer1992,Zurek,Habib,and Paz1993;see also Sect.4.1.2).The particle can be said to“measure”the quantum state of the ?eld.Since charged particles are in general abundant in the environment,no superpositions of macroscopically di?erent electromagnetic?elds(or di?erent “mean?elds”in other cases)are observed under normal conditions.This result is related to the di?culty of preparing and maintaining“squeezed states”of light(Yuen1976)–see Sect.3.3.3.1.Therefore,the?eld appears to be in one of its classical states(Sect.4.1.2).

In all these cases,this conclusion requires that the quasi-classical states (or“pointer states”in measurements)are robust(dynamically stable)under natural decoherence,as pointed out already in the?rst paper on decoherence (Zeh1970;see also Di′o si and Kiefer2000).

A particularly important example of a quasiclassical?eld is the metric of general relativity(with classical states described by spatial geometries on space-like hypersurfaces–see Sect.4.2).Decoherence caused by all kinds of matter can therefore explain the absence of superpositions of macroscop-ically distinct spatial curvatures(Joos1986,Zeh1986,1988,Kiefer1987), while microscopic superpositions would describe those hardly ever observ-able gravitons.

Superselection rules thus arise as a straightforward consequence of quan-tum theory under realistic assumptions.They have nonetheless been dis-cussed mainly in mathematical physics–apparently under the in?uence of von Neumann’s and Wigner’s“orthodox”interpretation of quantum mechan-ics(see Wightman1995for a review).Decoherence by“continuous measure-ment”seems to form the most fundamental irreversible process in Nature.It applies even where thermodynamical concepts do not(such as for individual molecules–see Sect.3.2.4),or when any exchange of heat is entirely negligi-ble.Its time arrow of“microscopic causality”requires a Sommerfeld radiation condition for microscopic scattering(similar to Boltzmann’s chaos),viz.,the absence of any dynamically relevant initial correlations,which would de?ne a“conspiracy”in common terminology(Joos and Zeh1985,Zeh2001).

2Basic Concepts and their Interpretation11 2.2Observables as a Derived Concept

Measurements are usually described by means of“observables”,formally rep-resented by Hermitian operators,and introduced in addition to the concepts of quantum states and their dynamics as a fundamental and independent ingredient of quantum theory.However,even though often forming the start-ing point of a formal quantization procedure,this ingredient should not be separately required if physical states are well described by these formal quan-tum states.This understanding,to be further explained below,complies with John Bell’s quest for the replacement of observables with“beables”(see Bell 1987).It was for this reason that his preference shifted from Bohm’s theory to collapse models(where wave functions are assumed to completely describe reality)during his last years.

Let|α be an arbitrary quantum state,de?ned operationally(up to a complex numerical factor)by a“complete preparation”procedure.The phe-nomenological probability for?nding the system during an appropriate mea-surement in another quantum state|n ,say,is given by means of their inner product as p n=| n|α |2(where both states are assumed to be normalized). The state|n is here de?ned by the speci?c measurement.(In a position mea-surement,for example,the number n has to be replaced with the continuous coordinates x,y,z,leading to the“improper”Hilbert states|r .)For measure-ments of the“?rst kind”(to which all others can be approximately reduced –see Sect.2.3),the system will again be found in the state|n with certainty if the measurement is immediately repeated.Preparations can be regarded as such measurements which select a certain subset of outcomes for further mea-surements.n-preparations are therefore also called n-?lters,since all“not-n”results are thereby excluded from the subsequent experiment proper.The above probabilities can also be written in the form p n= α|P n|α ,with an“observable”P n:=|n n|,which is thus derived from the kinematical concept of quantum states.

Instead of these special“n or not-n measurements”(with?xed n),one can also perform more general“n1or n2or...measurements”,with all n i’s mutually exclusive( n i|n j =δij).If the states forming such a set{|n }are pure and exhaustive(that is,complete, P n=1l),they represent a basis of the corresponding Hilbert space.By introducing an arbitrary“measurement scale”a n,one may construct general observables A= |n a n n|,which permit the de?nition of“expectation values” α|A|α = p n a n.In the

,and expectation special case of a yes-no measurement,one has a n=δnn

values become probabilities.Finding the state|n during a measurement is then also expressed as“?nding the value a n of an observable”.A change of scale,b n=f(a n),describes the same physical measurement;for position measurements of a particle it would simply represent a coordinate transfor-mation.Even a measurement of the particle’s potential energy is equivalent to a position measurement(up to degeneracy)if the function V(r)is given.

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According to this de?nition,quantum expectation values must not be

understood as mean values in an ensemble that represents ignorance of the precise state.Rather,they have to be interpreted as probabilities for poten-

tially arising quantum states|n –regardless of the latters’interpretation.

If the set{|n }of such potential states forms a basis,any state|α can be represented as a superposition|α = c n|n .In general,it neither forms an n0-state nor any not-n0state.Its dependence on the complex coe?cients c n requires that states which di?er from one another by a numerical factor must

be di?erent“in reality”.This is true even though they represent the same

“ray”in Hilbert space and cannot,according to the measurement postulate, be distinguished operationally.The states|n1 +|n2 and|n1 ?|n2 could not be physically di?erent from another if|n2 and?|n2 were the same state. (Only a global numerical factor would be“redundant”.)For this reason,pro-jection operators|n n|are insu?cient to characterize quantum states(cf. also Mirman1970).

The expansion coe?cients c n,relating physically meaningful states–for example those describing di?erent spin directions or di?erent versions of the K-meson–must in principle be determined(relative to one another)by ap-propriate experiments.However,they can often be derived from a previously known(or conjectured)classical theory by means of“quantization rules”. In this case,the classical con?gurations q(such as particle positions or?eld variables)are postulated to parametrize a basis in Hilbert space,{|q },while the canonical momenta p parametrize another one,{|p }.Their correspond-ing observables,Q= dq|q q q|and P= dp|p p p|,are required to obey commutation relations in analogy to the classical Poisson brackets.In this way,they form an important tool for constructing and interpreting the spe-ci?c Hilbert space of quantum states.These commutators essentially deter-mine the unitary transformation p|q (e.g.as a Fourier transform e i pq)–thus more than what could be de?ned by means of the projection operators |q q|and|p p|.This algebraic procedure is mathematically very elegant and appealing,since the Poisson brackets and commutators may represent generalized symmetry transformations.However,the concept of observables (which form the algebra)can be derived from the more fundamental one of state vectors and their inner products,as described above.

Physical states are assumed to vary in time in accordance with a dynam-

ical law–in quantum mechanics of the form i?t|α =H|α .In contrast, a measurement device is usually de?ned regardless of time.This must then also hold for the observable representing it,or for its eigenbasis{|n }.The probabilities p n(t)=| n|α(t) |2will therefore vary with time according to the time-dependence of the physical states|α .It is well known that this (Schr¨o dinger)time dependence is formally equivalent to the(inverse)time dependence of observables(or the reference states|n ).Since observables “correspond”to classical variables,this time dependence appeared sugges-tive in the Heisenberg–Born–Jordan algebraic approach to quantum theory.

2Basic Concepts and their Interpretation13 However,the absence of dynamical states|α(t) from this Heisenberg picture, a consequence of insisting on classical kinematical concepts,leads to para-doxes and conceptual inconsistencies(complementarity,dualism,quantum logic,quantum information,and all that).

An environment-induced superselection rule means that certain superpo-sitions are highly unstable with respect to decoherence.It is then impossible in practice to construct measurement devices for them.This empirical situa-tion has led some physicists to deny the existence of these superpositions and their corresponding observables–either by postulate or by formal manipu-lations of dubious interpretation,often including in?nities.In an attempt to circumvent the measurement problem(that will be discussed in the follow-ing section),they often simply regard such superpositions as“mixtures”once they have formed according to the Schr¨o dinger equation(cf.Primas1990).

While any basis{|n }in Hilbert space de?nes formal probabilities,p n= | n|α |2,only a basis consisting of states that are not immediately destroyed by decoherence de?nes a practically“realizable observable”.Since realizable observables usually form a genuine subset of all formal observables(diagonal-izable operators),they must contain a nontrivial“center”in algebraic terms. It consists of those of them which commute with all the rest.Observables forming the center may be regarded as“classical”,since they can be mea-sured simultaneously with all realizable ones.In the algebraic approach to quantum theory,this center appears as part of its axiomatic structure(Jauch 1968).However,since the condition of decoherence has to be considered quan-titatively(and may even vary to some extent with the speci?c nature of the environment),this algebraic classi?cation remains an approximate and dy-namically emerging scheme.

These“classical”observables thus de?ne the subspaces into which super-positions decohere.Hence,even if the superposition of a right-handed and a left-handed chiral molecule,say,could be prepared by means of an appropri-ate(very fast)measurement of the?rst kind,it would be destroyed before the measurement may be repeated for a test.In contrast,the chiral states of all individual molecules in a bag of sugar are“robust”in a normal envi-ronment,and thus retain this property individually over time intervals which by far exceed thermal relaxation times.This stability may even be increased by the quantum Zeno e?ect(Sect.3.3.1).Therefore,chirality appears not only classical,but also as an approximate constant of the motion that has to be taken into account in the de?nition of thermodynamical ensembles(see Sect.2.3).

The above-used description of measurements of the?rst kind by means of probabilities for transitions|α →|n (or,for that matter,by correspond-ing observables)is phenomenological.However,measurements should be de-scribed dynamically as interactions between the measured system and the measurement device.The observable(that is,the measurement basis)should thus be derived from the corresponding interaction Hamiltonian and the ini-

14H.D.Zeh(www.zeh-hd.de)

tial state of the device.As discussed by von Neumann(1932),this interaction must be diagonal with respect to the measurement basis(see also Zurek1981).

Its diagonal matrix elements are operators which act on the quantum state of the device in such a way that the“pointer”moves into a position appropriate for being read,|n |Φ0 →|n |Φn .Here,the?rst ket refers to the system, the second one to the device.The states|Φn ,representing di?erent pointer positions,must approximately be mutually orthogonal,and“classical”in the

explained sense.

Because of the dynamical superposition principle,an initial superposition c n|n does not lead to de?nite pointer positions(with their empirically ob-served frequencies).If decoherence is neglected,one obtains their entangled superposition c n|n |Φn ,that is,a state that is di?erent from all poten-tial measurement outcomes|n |Φn .This dilemma represents the“quantum measurement problem”to be discussed in Sect.2.3.Von Neumann’s inter-action is nonetheless regarded as the?rst step of a measurement(a“pre-measurement”).Yet,a collapse seems still to be required–now in the mea-surement device rather than in the microscopic system.Because of the en-tanglement between system and apparatus,it then a?ects the total system.5 If,in a certain measurement,a whole subset of states|n leads to the same pointer position|Φn

0 ,these states are not distinguished in this mea-surement.The pointer state|Φn

0 now becomes dynamically correlated with the whole projection of the initial state, c n|n ,on the subspace spanned by this subset.A corresponding collapse was indeed postulated by L¨u ders(1951) in his generalization of von Neumann’s“?rst intervention”(Sect.2.3).

In this dynamical sense,the interaction with an appropriate measuring device de?nes an observable up to arbitrary monotoneous scale transforma-tions.The time dependence of observables according to the Heisenberg pic-ture would thus describe an imaginary time dependence of the states of this device(its pointer states),paradoxically controlled by the intrinsic Hamilto-nian of the system.

The question of whether a formal observable(that is,a diagonalizable operator)can be physically realized can only be answered by taking into ac-count the unavoidable environment of the system(while the measurement device is always asssumed to decohere into its macroscopic pointer states). However,environment-induced decoherence by itself does not solve the mea-surement problem,since the“pointer states”|Φn may be assumed to include the total environment(the“rest of the world”).Identifying the thus arising

2Basic Concepts and their Interpretation15 global superposition with an ensemble of states,represented by a statistical operatorρ,that merely leads to the same expectation values A =tr(Aρ)

for a limited set of observables{A}would obviously beg the question.This argument is nonetheless found wide-spread in the literature(cf.Haag1992, who used the subset of all local observables).

In Sect.2.4,statistical operatorsρwill be derived from the concept of quantum states as a tool for calculating expectation values,while the latter

are de?ned,as described above,by means of probabilities for the occurrence of new states in measurements.In the Heisenberg picture,ρis often regarded as in some sense representing the ensemble of potential“values”for all ob-

servables that are here postulated to formally replace the classical variables. This interpretation is suggestive because of the(incomplete)formal analogy ofρto a classical phase space distribution.However,the prospective“values”

would be physically meaningful only if they characterized di?erent physical states(such as pointer states).Note that Heisenberg’s uncertainty relations refer to potential outcomes which may arise in di?erent(mutually exclusive)

measurements.

2.3The Measurement Problem

The superposition of di?erent measurement outcomes,resulting according to the Schr¨o dinger equation(as discussed above),demonstrates that a“naive

ensemble interpretation”of quantum mechanics in terms of incomplete knowl-edge is ruled out.It would mean that a quantum state(such as c n|n |Φn ) represents an ensemble of some as yet unspeci?ed fundamental states,of

which a subensemble(for example represented by the quantum state|n |Φn ) may be“picked out by a mere increase of information”.If this were true,then the subensemble resulting from this measurement could in principle be traced back in time by means of the Schr¨o dinger equation in order to determine also the initial state more completely(to“postselect”it–see Aharonov and Vaid-man1991for an inappropriate attempt).In the above case this would lead to the initial quantum state|n |Φ0 that is physically di?erent from–and thus inconsistent with–the superposition( c n|n )|Φ0 that had been prepared (whatever it means).

In spite of this simple argument,which demonstrates that an ensemble

interpretation would require a complicated and miraculous nonlocal“back-ground mechanism”in order to work consistently(cf.Footnote3regarding Bohm’s theory),the ensemble interpretation of the wave function seems to remain the most popular one because of its pragmatic(though limited)value.

A general and rigorous critical discussion of problems arising in an ensemble interpretation may be found in d’Espagnat’s books(1976and1995).

A way out of this dilemma in terms of the wave function itself requires one of the following two possibilities:(1)a modi?cation of the Schr¨o dinger equation that explicitly describes a collapse(also called“spontaneous local-

16H.D.Zeh(www.zeh-hd.de)

ization”–see Chap.8),or(2)an Everett type interpretation,in which all measurement outcomes are assumed to coexist in one formal superposition, but to be perceived separately as a consequence of their dynamical decoupling under decoherence.While this latter suggestion may appear“extravagant”(as it requires myriads of coexisting parallel quasi-classical“worlds”),it is similar in principle to the conventional(though nontrivial)assumption,made tacitly in all classical descriptions of observations,that consciousness is local-ized in certain(semi-stable and su?ently complex)spatial subsystems of the world(such as human brains or parts thereof).For a dispute about which of the above-mentioned two possibilities should be preferred,the fact that en-vironmental decoherence readily describes precisely the apparently occurring “quantum jumps”or“collapse events”(as will be discussed in great detail throughout this book)appears most essential.

The dynamical rules which are(explicitly or tacitly)used to describe the e?ective time dependence of quantum states thus represent a“dynamical dualism”.This was?rst clearly formulated by von Neumann(1932),who distinguished between the unitary evolution according to the Schr¨o dinger equation(remarkably his“zweiter Eingri?”or“second intervention”),

iˉh

6Thus also Bohr(1928)in a subsection entitled“Quantum postulate and causal-ity”about“the quantum theory”:“...its essence may be expressed in the so-called quantum postulate,which attributes to any atomic process an essential discontinuity,or rather individuality,completely foreign to classical theories and symbolized by Planck’s quantum of action”(my italics).The later revision of these early interpretations of quantum theory(required by the important role of entangled quantum states for much larger systems)seems to have gone unnoticed by many physicists.

2Basic Concepts and their Interpretation17 In scattering theory,one usually probes only part of quantum mechanics by restricting consideration to asymptotic states and their probabilities(dis-

regarding their superpositions).All quantum correlations between them then appear statistical(“classical”).Occasionally even the unitary scattering am-plitudes m out|n in = m|S|n are confused with the probability amplitudes φm|ψn which describe measurements to?nd a state|φm in an initial|ψn . In his general S-matrix theory,Heisenberg temporarily speculated about de-

riving the latter from the former.Since macroscopic systems never become asymptotic because of their dynamical entanglment with the environment, they can not be described by an S-matrix at all.

The Born/von Neumann dynamical dualism was evidently the major mo-tivation for an ignorance interpretation of the wave function,which attempts to explain the collapse not as a dynamical process in the system,but as an increase of information about it(the reduction of an ensemble of pos-sible states).However,even though the dynamics of ensembles in classical description uses a formally similar dualism,an analogous interpretation in quantum theory leads to the severe(and apparently fatal)di?culties indi-cated above.They are often circumvented by the invention of“new rules of logic and statistics”,which are not based on any ensemble interpretation or incomplete information.

If the state of a classical system is incompletely known,and the cor-

responding point p,q in phase space therefore replaced by an ensemble(a probability distribution)ρ(p,q),this ensemble can be“reduced”by a new observation that leads to increased information.For this purpose,the system must interact in a controllable manner with the“observer”who holds the information(cf.Szilard1929).His physical state of memory must thereby change in dependence on the property-to-be-measured of the observed sys-tem,leaving the latter unchanged in the ideal case(no“recoil”).Accord-ing to deterministic dynamical laws,the ensemble entropy of the combined system,which initially contains the entropy corresponding to the unknown microscopic quantity,would remain constant if it were de?ned to include the entropy characterizing the?nal ensemble of di?erent outcomes.Since the ob-server is assumed to“know”(to be aware of)his own state,this ensemble is reduced correspondingly,and the ensemble entropy de?ned with respect to his state of information is lowered.

This is depicted by the?rst step of Fig.2.1,where ensembles of states are represented by areas.In contrast to many descriptions of Maxwell’s de-mon,the observer(regarded as a device)is here subsumed into the ensemble description.Physical entropy,unlike ensemble entropy,is usually understood as a local(additive)concept,which neglects long range correlations for being “irrelevant”,and thus approximately de?nes an entropy density.Physical and ensemble entropy are equal in the absence of correlations.The information I,given in the?gure,measures the reduction of entropy according to the increased knowledge of the observer.

18H.D.Zeh (www.zeh-hd.de)

S ensemble = S

S physical = S o + kln2

I = 0

S ensemble =: S o

S physical = S o

I = 0measurement system observer environment

S ensemble o S physical = S o - kln2

I = kln2

Fig.2.1.Entropy relative to the state of information in an ideal classical mea-surement.Areas represent sets of microscopic states of the subsystems (while those of uncorrelated combined systems would be represented by their direct products).During the ?rst step of the ?gure,the memory state of the observer changes de-terministically from 0to A or B ,depending on the state a or b of the system to be measured.The second step depicts a subsequent reset,required if the measure-ment is to be repeated with the same device (Bennett 1973).A ′and B ′are e?ects which must thereby arise in the thermal environment in order to preserve the to-tal ensemble entropy in accordance with presumed microscopic determinism.The “physical entropy”(de?ned to add for subsystems)measures the phase space of all microscopic degrees of freedom,including the property to be measured,while depending on given macroscopic variables.Because of its presumed additivity,this physical entropy neglects all remaining statistical correlations (dashed lines,which indicate sums of products of sets)for being “irrelevant”in the future –hence S physical ≥S ensemble .I is the amount of information held by the observer.The min-imum initial entropy,S 0,is k ln 2in this simple case of two equally probable values a and b .

This description does not necessarily require a conscious observer (al-though it may ultimately rely upon him).It applies to any macroscopic mea-surement device,since physical entropy is not only de?ned to be local,but also relative to “given”macroscopic properties (as a function of them).The dynamical part of the measurement transforms “physical”entropy (here the ensemble entropy of the microscopic variables)deterministically into entropy of lacking information about controllable macroscopic properties.Before the observation is taken into account (that is,before the “or”is applied),both parts of the ensemble after the ?rst step add up to give the ensemble entropy.When it is taken into account (as done by the numbers given in the ?gure),

2Basic Concepts and their Interpretation19 the ensemble entropy is reduced according to the information gained by the observer.

Any registration of information by the observer must use up his memory capacity(“blank paper”),which represents non-maximal entropy.If the same measurement is to be repeated,for example in a cyclic process that could be used to transform heat into mechanical energy(Szilard,l.c.),this capacity would either be exhausted at some time,or an equivalent amount of entropy must be absorbed by the environment(for example in the form of heat)in order to reset the measurement or registration device(second step of Fig.2.1). The reason is that two di?erent states cannot deterministically evolve into the same?nal state(Bennett1973).7This argument is based on an arrow of time of“causality”,which requires that all correlations possess local causes in their past(no“conspiracy”).The irreversible formation of“irrelevant”correlations then explains the increase of physical(local)entropy,while the ensemble entropy is conserved.

The unsurmountable problems encountered in an ensemble interpretation of the wave function(or of any other superposition,such as|a +|b )are re-?ected by the fact that there is no ensemble entropy that would represent the unknown property-to-be-measured(see the?rst step of Fig.2.2or2.3–cf. also Zurek1984).The“ensemble entropy”is now de?ned by the“correspond-ing”expression S ensemble=?k tr{ρlnρ}(but see Sect.2.4for the meaning of the density matrixρ).If the entropy of observer plus environment is the same as in the classical case of Fig.2.1,the total initial ensemble entropy is now lower;in the case of equal initial probabilities for a and b it is S0?k ln2. It would even vanish for pure statesφandχof observer and environment, respectively:(|a +|b )|φ0 |χ0 .The Schr¨o dinger evolution(depicted in Fig.

2.3)would then be described by three dynamical steps,

(|a +|b )|φ0 |χ0 →(|a |φA +|b |φB )|χ0

→|a |φA |χA′′ +|b |φB |χB′′

→(|a |χA′A′′ +|b |χB′B′′ )|φ0 ,(2.3) with an“irrelevant”(inaccessible)?nal quantum correlation between system and environment as a relic from the initial superposition.In this unitary evo-lution,the two“branches”recombine to form a nonlocal superposition.It “exists,but it is not there”.Its local unobservability characterizes an“ap-parent collapse”(as will be discussed).For a genuine collapse(Fig.2.2), the?nal correlation would be statistical,and the ensemble entropy would increase,too.

As mentioned in Sect.2.2,the general interaction dynamics that is re-quired to describe“ideal”measurements according to the Schr¨o dinger equa-tion(2.1)is derived from the special case where the measured system is

20H.D.Zeh

(www.zeh-hd.de)

or (by reduction of

S ensemble = S o

S physical = S o

+ kln2

I = 0

(forget)

S ensemble = S o - kln2(

S physical = S o + kln2)

I = ?O S ensemble = S o - kln2

S physical = S o - kln2

I = 0

system observer environment

'measurement'

interaction S ensemble = S o - kln2 (each)

S physical = S o - kln2

I = kln2Fig.2.2.Quantum measurement of a superposition |a +|b by means of a collapse process,here assumed to be triggered by the macroscopic pointer position.The initial entropy is smaller by one bit than in Fig.2.1(and may in principle vanish),since there is no initial ensemble a/b for the property to be measured.Dashed lines before the collapse now represent quantum entanglement.(Compare the ensemble entropies with those of Fig.2.1!)Increase of physical entropy in the ?rst step is appropriate only if the arising entanglement is regarded as irrelevant .The collapse itself is often divided into two steps:?rst increasing the ensemble entropy by re-placing the superposition with an ensemble,and then lowering it by reducing the ensemble (applying the “or”–for macroscopic pointers only).The increase of en-semble entropy,observed in the ?nal state of the Figure,is a consequence of this ?rst step of the collapse.It brings the entropy up to its classical initial value of Fig.2.1

prepared in an eigenstate |n before measurement (von Neumann 1932),

|n |Φ0 →|n |Φn .(2.4)

Here,|n corresponds to |a or |b in the ?gures,the pointer positions |Φn to the states |φA and |φB .(During non-ideal measurements,the state |n would change,too.)However,applied to an initial superposition, c n |n ,the interaction according to (2.1)leads to an entangled superposition, c n |n |Φ0 → c n |n |Φn .(2.5)As explained in Sect.2.1.1,the resulting superposition represents an individ-ual physical state that is di?erent from all components appearing in this sum.While decoherence arguments teach us (see Chap.3)that neglecting the envi-ronment of (2.5)is absolutely unrealistic if |Φn describes the pointer state of

VB程序设计课后习题答案(科学出版社)

同步练习1 二、选择题 01——05 CADAB 06——10 ACDAB 11——15 CBDBB 同步练习2 二、选择题 01——05 ABDCA 06——10 CACBC 11——15 DADAD 16——20 BDBBB 三、填空题 1．可视 2．LEFT、TOP、WIDTH、HEIGHT 3．按字母顺序 4．查看代码 5．工具、编辑器 6．FORM窗体、FONT 7．MULTILINE 8．在运行时设计是无效的 9．工程、工程属性、通用、FORM1.SHOW 10．TABINDEX、0 同步练习3 二、选择题 01——05 BCADB 06——10 ADBBC 11——15 DBCBA 16——20 BAABB 三、填空题 1．整型、长整型、单精度型、双精度型 2．SIN(30*3.14/180)+SQR(X+EXP(3))/ABS(X-Y)-LOG(3*X) 3．164、今天是：3-19 4．FALSE 5．-4、3、-3、3、-4、4 6．CDEF 7．(X MOD 10)*10+X\10 8．(35\20)*20=20 ( 35 \ 20 )* 20 = 20 9．X MOD 3=0 OR X MOD 5=0 10．27.6、8.2、8、1、100、397、TRUE、FALSE 同步练习4 一、选择题 01——05 DBCAD 06——10 CBBAB

11——15 D25BAC 16——20 CBACB 21——25 DAABC 二、填空题 1．正确性、有穷性、可行性、有0个或多个输入、有1个或多个输出2．1 2 3 3．X>=7 4．X

vb课后习题答案

VB 课后练习题参考答案第一章一、 1、C 2、C 3、B 4、B 5、D 6、B 7、B 8、D 二、 1、学习版、专业版、企业版 2、alt+Q 或 alt+F4 3、.vbp 、 .frm 4、固定、浮动 5、"abcd"、"VB Programing" 6、属性窗口、运行 7、对象框、事件框 8、窗体模块、标准模块、类模块第二章一、 1、B 2、B 3、B 4、B 5、D 6、D 二、 1、((x+y)+z)*80-5*(C+D) 2、cos(x)*sin(sin(x)+1 3、2*a*(7+b) 4、8*EXP(3)*LOG(2) 5、good morning 、 good morning 6、2001/8/25 8 2001 7 第三章一、 1、C 2、B 3、D 4、A 5、D 、 3 6、C 7、B 8、C 9、C 10、D 11、B 12、C 13、B 14、B 15、A 16、B 17、D 18、C 19、C 二、 1、AutoSize 2、text1.setfocus 3、0 、 0 4、 picture1.picture=loadpic ture("yy.gif") 5、stretch 6、interval 7、enable 8、下拉式组合框、简单组合框、下拉式列表框、style 9、下拉式列表框 10、条目1 、条目3 11、欢迎您到中国来、 welcome to china!! 第四章一、 1、B 2、C 3、C 4、B 5、C 6、B 7、C 8、B 9、D 10、A 11、B 12、A 13、B 14、D 15、A 16、B 17、A 18、C 19、B 二、 1、2542=57 2、beijing 3、002.45、2.449、 24.49e-01、-2.449 4、9 10 11 5、9 6、1 2 3 7、 iif(x<=0,y=0,iif(x<=10, y=5+2*x,iif(x<=15,y=x- 5,y=0))) 8、x=7 或 x>6 或 x>5 9、x>=0 、x

VB第一章课后习题答案讲课教案

习题一、单项选择题 1. 在设计阶段，当双击窗体上的某个控件时，所打开的窗体是_____。 A. 工程资源管路器窗口 B. 工具箱窗体 C. 代码窗体 D. 属性窗体 2. VB中对象的含义是_____。 A. 封装了数据和方法的实体 B. 封装的程序 C. 具有某些特性的具体事物的抽象 D. 创建对象实例的模板 3. 窗体Form1的Name属性是MyForm，它的单击事件过程名是_____。 A. MyForm_Click B. Form_Click C. Form1_Click D. Frm1_Click 4. 如果要改变窗体的标题，需要设置窗体对象的_____属性。 A. BackColor B. Name C. Caption D. Font 5. 若要取消窗体的最大化功能，可将其_____属性设置为False来实现。 A. Enabled B.ControlBox C. MinButton D. MaxButton 6. 若要以代码方式设置窗体中显示文本的字体大小，可通过设置窗体对象_____属性来实现。 A. Font B.FontName C.FontSize D. FontBold 7. 确定一个控件在窗体上位置的属性是_____。 A. Width或Height B. Width和Height C. Top或Left D. Top和Left 8. 以下属性中，不属于标签的属性是_____。 A. Enabled B. Default C. Font D. Caption 9. 若要设置标签控件中文本的对齐方式，可通过_____属性实现。 A.Align B. AutoSize C. Alignment D. BackStyle 10. 若要使标签控件的大小自动与所显示文本的大小相适宜，可将其_____属性设置为True来实现。 A.Align B. AutoSize C. Alignment D. Visible 11. 若要设置或返回文本框中的文本，可通过设置其_____属性来实现。 A.Caption B. Name C. Text D. （名称） 12. 若要设置文本框最大可接受的字符数，可通过设置其_____属性来实现。 A.MultiLine B. Max C. Length D. MaxLength

机械工程及自动化专业本科专业人才培养目标体系-上海交通大学

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vb课后练习答案习题解答 (5)

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