当前位置：文档之家› Connes' trace formula and Dirac realization of Maxwell and Yang-Mills action

Connes' trace formula and Dirac realization of Maxwell and Yang-Mills action

a r

t h

a y

CONNES’TRACE FORMULA AND DIRAC REALIZATION OF

MAXWELL AND YANG-MILLS ACTION PETER M.ALBERTI AND RAINER MATTHES Abstract.The paper covers known facts about the Dixmier trace (with some generalities about traces),the Wodzicki residue,and Connes’trace theorem,including two variants of proof of the latter.Action formulas are treated very sketchy,because they were considered in other lectures of the workshop.Contents 1.The Dixmier-Traces 21.1.Generalities on traces on C ?-and W ?-algebras 21.1.1.Basic topological notions,notations 21.1.2.Traces on C ?-and W ?-algebras 31.2.Examples of traces 41.2.1.Traces on compact linear operators 41.2.2.Traces on B (H )81.3.Examples of singular traces on B (H )91.3.1.Step one:Some ideal of compact operators 91.3.2.Step two:Scaling invariant states 121.3.3.Constructing the Dixmier-traces 131.3.4.The Dixmier-trace as a singular trace 141.4.Calculating the Dixmier-trace 151.4.1.Simple criteria of measurability 151.4.2.A residue-formula for the Dixmier-trace 172.The Connes’trace theorem and its application 182.1.Preliminaries

182.1.1.Basic facts about pseudodi?erential operators

182.1.2.De?nition of the Wodzicki residue

202.2.Connes’trace theorem

212.2.1.Formulation of Connes’trace theorem

212.2.2.On the proof of Connes’trace theorem

222.2.3.An alternative proof of the Connes’trace formula

252.3.Classical Yang-Mills actions

282.3.1.The classical Dirac operator and integration on manifolds

282.3.2.Classical gauge ?eld actions in terms of Dixmier-trace

29References

2P.M.ALBERTI,R.MATTHES

1.The Dixmier-Traces

The essential ingredients of the famous trace formula[3,4]of A.Connes are operator algebraic constructs over the W?-algebra B(H)of all bounded linear oper-

ators over some in?nite dimensional separable Hilbert space H which are known as

Dixmier-traces.The constructions will be explained in this section.For a general operator-algebraic background the reader is referred e.g.to[9,31,36,22,23].

1.1.Generalities on traces on C?-and W?-algebras.

1.1.1.Basic topological notions,notations.A C?-algebra M is a Banach?-algebra,

with?-operation x?→x?and norm · obeying x?x = x 2,for each x∈M (?-quadratic property).Let M+={x?x:x∈M}be the cone of positive elements

of M.For x∈M let x≥0be synonymous with x∈M+.The Banach space

of all continuous linear forms on M(dual)will be denoted by M?,the dual norm (functional norm)be · 1.As usual,a linear form f over M is termed positive,

f≥0,if f(x?x)≥0,for each x∈M.Remind that positive linear forms are

automatically continuous and are generating for M?.Thus M?+={f∈M?:f≥0} is the set of all these forms.There is a fundamental result of operator theory

saying that if M possesses a unit1(unital C?-algebra),then f≥0if,and only if,f(1)= f 1.It is common use to refer to positive linear forms of norm one as

states.Thus,in a unital C?-algebra the set of all states on M,S(M),is easily seen

to be a convex set which according to the Alaoglu-Bourbaki theorem isσ(M?,M)-compact.Here,theσ(M?,M)-topology(also w?-topology in the special context

at hand)is the weakest locally convex topology generated by the seminormsρx,

x∈M,withρx(f)=|f(x)|,for each f∈M?.The generalization of the notion of positive linear form on M(which refers to positive linear maps into the special

C?-algebra of complex numbers C)is the notion of the positive linear map.Say that a linear mapping T:M?→N which acts from one C?-algebra M into another

one N is positive if T(x?x)≥0within N for each x∈M.In the unital case(for

M)one then knows that T(1) = T holds( T refers to the operator norm of T as a linear operator acting from the one Banach space M into the other N).On

the other hand,each linear map T:M?→N which obeys this relation is known

to be positive.If both M and N have a unit,a linear map T:M?→N is said to be unital if T(1)=1is ful?lled(the units being the respective units).Thus,

and in particular,each unital linear map T:M?→N of norm one beween unital C?-algebras has to be a positive map.

Remind that a C?-algebra M is a W?-algebra,that is,is?-isomorphic to some

vN-algebra on some Hilbert space H,if and only if,there exists a(unique)Banach subspace M?(the predual space)of M?such that M is the(continuous)dual of

M?,M=(M?)?.Note that a non-zero W?-algebra is always unital.Suppose now

that M is a W?-algebra.The forms of M?will be referred to as normal linear forms.One then knows that the normal positive linear forms M?+=M?∩M?+ (resp.the normal states S0(M)=M?∩S(M))are generating for M?in the sense that each normal linear form may be represented as a complex linear combination

of at most four normal states.As a consequence of this,for each ascendingly

directed(in the sense of≤)bounded net{xα}?M+there exists a lowest upper bound l.u.b.xαwithin M+.On the other hand,a positive linear formω∈M?+is normal if,and only if,for each ascendingly directed bounded net{xα}?M+the relationω(l.u.b.xα)=l.u.b.ω(xα)=limαω(xα)is valid.Note that in the latter

CONNES’TRACE FORMULA3 characterization it su?ces if the mentioned continuity be ful?lled for ascendingly directed nets of orthoprojections of M.

In contrast to the previous,ν∈M?+is called singular if to each orthoprojection p∈M withν(p)>0there is another orthoprojection q∈M with0

The simplest example of a W?-algebra where singular positive linear forms can exist is the commutative W?-algebra?∞=?∞(N)of all bounded sequences x= (x n)=(x1,x2,...)of complex numbers,with norm x ∞=sup n∈N|x n|,and algebra multiplication x·y=(x n y n)and?-operation x?=(ˉx n)de?ned com-ponentwise.Recall that in this case the predual space?∞?is the Banach space of all absolutely summable sequences?1(N),with norm ω 1= n∈N|ωn|for ω=(ωn)∈?1(N)(=?1).Thereby,each suchωcan be identi?ed with an element in the dual Banach space(?∞)?via the identi?cation with the linear functionalω(·) given asω(x)= n∈Nωn x n,for each x∈?∞.For simplicity,also this functional ω(·)will be referred to asω,ω=ω(·).

In generalizing from the setting of a normal positive linear form,call a positive linear map T:M?→N from one W?-algebra M into another W?-algebra N normal if T(l.u.b.xα)=l.u.b.T(xα)holds for each ascendingly directed bounded net{xα}?M+.Note that in a W?-algebra the Alaoglu-Bourbaki theorem may be applied on M,and then yields that the(closed)unit ball M1of M isσ(M,M?)-compact.From this it follows that the unit ball within the bounded linear operators which map the W?-algebra M into itself is compact with respect to the topology determined by the system of seminormsρx,f,labelled by x∈M and f∈M?, and which are de?ned at T byρx,f(T)=|f?T(x)|.Refer to this topology as theσ(M,M?)-weak operator topology on the Banach algebra of bounded linear operators B(M)over the Banach space M.Thereby,by convention for T,S∈B(M) let the product T S∈B(M)be de?ned through successive application of maps to the elements of M in accordance with the rule‘apply right factor?rst’,that is T S(x)=(T?S)(x)=T(S(x)).

1.1.

2.Traces on C?-and W?-algebras.We recall the very basic facts on traces as found e.g.in[9,6.1.].A functionτ:M+?→

4P.M.ALBERTI,R.MATTHES

(which according to the previous is Mτ+)onto[Mτ+]is unique,by tacit under-standing the notationτ(x)will be also used at non-positive x of the de?ning ideal ofτif the evaluation?τ(x)of the linear functional?τat x is meant.

The traceτis termed?nite trace if Mτ+=M+,and semi?nite trace ifτ(x)= sup{τ(y):y≤x,y∈Mτ+},for each x∈M+.If M is a W?-algebra,with group of unitary elements U(M),the above condition on invariance usually is replaced with a seemingly weaker requirement upon unitary invariance,that isτ(u?xu)=τ(x)be ful?lled,for each x∈M+and u∈U(M).However,both conditions are equivalent there(and are so even on unital C?-algebras).Also,in the W?-case the traceτis said to be normal provided for each ascendingly directed bounded net{xα}?M+ the relationτ(l.u.b.xα)=l.u.b.τ(xα)=limατ(xα)is ful?lled.

Now,suppose I?M is a proper two-sided ideal of the W?-algebra M.Then,I is also a?-subalgebra of M,with generating positive cone I+=I∩M+,that is,I= [I+]is ful?lled(these facts are consequences of the polar decomposition theorem, essentially).Under these premises we have the following extension principle: Lemma1.1.Suppose I+is a hereditary subcone of M+.Then,each additive, positive homogeneous and invariant mapτ0:I+?→

CONNES’TRACE FORMULA5 And secondly,each non-zero eigenvalue of x has only?nite multiplicity,that is, m(μ)=#{k:μk(x)=μ}obeys m(μ)<∞,for eachμ∈R+\{0}.

Recall that C B(H)is also a closed?-ideal of B(H).Hence,according to polar

√decomposition,x∈B(H)is compact if,and only if,the module|x|=

√

x?n,

6P.M.ALBERTI,R.MATTHES

?From(1.1b)–(1.1d)one now concludes some useful relations and estimates.The ?rst is a rather trivial consequence of the de?nition ofσk(x)and says that

?x∈C B(H),λ∈C,k∈N:σk(λx)=|λ|σk(x).(1.2a) It is stated here only for completeness.In the special case of positive compact operators from(1.1c)we get the following often used estimates:

?x,y∈C B(H)+,k∈N:y≤x=?σk(y)≤σk(x).(1.2b) The next estimate is due to[25]and at once gets obvious from(1.1b),and tells us that the following holds:

?x,y∈C B(H),k∈N:σk(x+y)≤σk(x)+σk(y).(1.2c) The third estimate deals with an upper bound ofσk(x)+σk(y)in case of posi-tive operators x,y∈C B(H)+and arises from(1.1d).In line with the latter,let orthoprojections p,q of rank k be given such thatσk(x)= ∞n=1 xp?n,?n and σk(y)= ∞n=1 yq?n,?n are ful?lled.Then,the least orthoprojection p∨q ma-jorizing both p and q has rank2k at most.Thus there is an orthoprojection Q of rank2k and obeying p∨q≤Q.Hence,in view of the choice of p,q and with the help of(??)one infers that ∞n=1 xQ?n,?n =σk(x)+ ∞n=1 x(Q?p)?n,?n ≥σk(x) and ∞n=1 yQ?n,?n =σk(y)+ ∞n=1 x(Q?q)?n,?n ≥σk(y).In view of(1.1d) from this thenσk(x)+σk(y)≤σ2k(x+y)follows.For positive compact operators the previous together with(1.2c)may be summarized into the following one:?x,y∈C B(H)+,k∈N:σk(x+y)≤σk(x)+σk(y)≤σ2k(x+y).(1.2d) Note that,since C B(H)is a two-sided ideal,from(1.1a)for each y∈C B(H)and a,b∈B(H)the estimate

?k∈N:μk(ayb)≤ a b μk(y)(1.2e) can be obtained.Thus,under these conditions one has

?a,b∈B(H),y∈C B(H),k∈N:σk(ayb)≤ a b σk(y).(1.2f) Especially,if x=u|x|is the polar decomposition of x∈C B(H)within B(H),then with the partial isometry u∈B(H)one has both,xx?=ux?xu?and x?x=u?xx?u. In the special cases of(1.2f)with y=xx?,a=u?,b=u and y=x?x,a=u,b= u?we arrive at estimates which?t together into the following assertion:

?x∈C B(H),k∈N:σk(x?x)=σk(xx?).(1.2g) In the following,a traceτis said to be non-trivial if there is at least one x≥0with 0<τ(x)<∞.The relations given in eqs.(1.2)are the key facts that the theory of traces on both algebras C B(H)and B(H)can be based on.

Lemma1.2.Let tr:C B(H)+?→

CONNES’TRACE FORMULA7 is in?nite dimensional,tr x=∞will occur for some positive compact operators. To see that tr is semi?nite requires to prove that for x∈C B(H)+with tr x=∞there existed a sequence{x n}?C B(H)+with x n≤x and tr x n<∞such that

lim n→∞tr x n=∞.Note that by de?nition of tr,tr x=∞implies that x cannot be of?nite rank.Hence,x can be written as x= ∞k=1μk(x)p k,with in?nitely many mutually orthogonal one-dimensional orthoprojections p k and allμk(x)=0. Clearly,for each n∈N the operators x n= n k=1μk(x)p k are of?nite rank and obey0≤x1≤x2≤x3≤...≤x.Also,owing toσk(x n)=σn(x)for k≥n,one has tr x n=σn(x),and therefore lim n→∞tr x n=∞follows.Thus tr is semi?nite. Supposeτis a non-trivial trace.Thus0<τ(y)<∞,for some positive compact y.Supposeτ(x)>0for some x≥0of?nite rank.According to additivity and homogeneity ofτthere has to exist a one-dimensional subprojection p of a spectral orthoprojection of x withτ(p)>0.The same arguments for y ensure that τ(q)<∞,for some one-dimensional subprojection q of some spectral projection of y.But since q=vv?and p=v?v,with v∈C B(H),by invariance ofτone hasτ(q)=τ(p).Hence∞>τ(p)>0,andτ(q)=τ(p)for each one-dimensional orthoprojection q.Putλ=τ(p)?1.Thenλ·τ(q)=tr q,and thusλ·τ(x)=tr x for each positive operator x of?nite rank.Finally,if x∈C B(H)+is not of?nite rank,let0≤x1≤x2≤x3≤...≤x be the above approximating sequence of x by ?nite rank operators x n.Also in such case lim n→∞tr x n=lim n→∞σn(x)=tr x follows.Hence,in view of the above relation over the operators of?nite rank,and sinceτ(x n)≤τ(x)holds,tr x=lim n→∞tr x n=λlim n→∞τ(x n)≤λ·τ(x).

For completeness,we give yet the most famous formula relating tr and which makes that this trace is so extremely useful.

Corollary1.1.For each maximal orthonormal system{ψn}?H and x∈C B(H)+ one has tr x= ∞n=1 xψn,ψn .

Proof.Let{?j}be an o.n.s.with x?k=μk(x)?k,for all k∈N,and be p n the orthoprojection with p n H=[?1,...,?n].Then,by positivity of x one has xψn,ψn = √xψn ≥ p k√xψn ,and therefore and in view of(??) one gets ∞n=1 xψn,ψn ≥ ∞n=1 p k√xψn = k j=1 x?j,?j =σk(x).Ac-cording to Lemma1.2then ∞n=1 xψn,ψn ≥tr x follows.On the other hand,if q k is the orthoprojection onto[ψ1,...,ψk],according to(1.1d)for each k∈N cer-tainly k n=1 xψn,ψn = ∞n=1 xq kψn,ψn ≤σk(x).From this in view of Lemma 1.2once more again ∞n=1 xψn,ψn ≤tr x is seen.Taking together this with the above estimate provides that equality has to occur.

A non-zero traceτon C B(H)will be said to be singular ifτ(x)=0for each x≥0 of?nite rank.Relating this and non-trivial traces there is the following result. Corollary1.2.Letτbe a non-trivial trace on C B(H).Then,eitherτ=λ·tr holds,for a uniqueλ∈R+,or there exist a singular traceτs and a uniqueα∈R+ such thatτ=τs+α·tr.

Proof.Ifτ=λ·tr is ful?lled,thenτ(p)=λtr p,for each one-dimensional ortho-projection p.Owing to tr p=1(see Corollary1.1)thenλ=τ(p)follows.

Supposeτ∈R+tr.Then,τ=0,and if a decompositionτ=τs+α·tr with singularτs exists,thenτ(p)=α,for some(and thus any)one-dimensional orthoprojection p,and the following two alternatives have to be dealt with:?rstly, ifτis vanishing on all positive operators of?nite rank,τis singular,andτ=τs

8P.M.ALBERTI,R.MATTHES

andα=0have to be chosen(see above).Secondly,ifτdoes not vanish on all positive operators of?nite rank,according to Lemma1.2there exists uniqueλ>0 withλ·τ(x)≥tr x,for each x∈C B(H)+,with equality occuring on any operator of?nite rank.Hence,in de?ningτs(x)=τ(x)?λ?1tr x,for each x with tr x<∞, andτs(x)=∞else,we get a positive mapτs which does not vanish identically on the positive compact operators,but which is vanishing on all positive operators of ?nite rank.From the previous and since bothτand tr are traces,also additivity, positive homogeneity and invariance ofτs at once follow.Hence,τs is a singular trace,which is easily seen to obeyτ=τs+α·tr,withα=λ?1.

1.2.2.Traces on B(H).Remind in short the theory of traces on M=B(H),with separable in?nite dimensional Hilbert space H.Let F B(H)be the two-sided ideal of all operators of?nit rank in B(H).In the following an ideal I will be termed non-trivial if I={0}and I=B(H).Both F B(H)and C B(H)are non-trivial two-sided ideals.Thereby,the compact operators form a closed ideal,with F B(H) being dense within C B(H).Start with a useful criterion on non-compactness for a positive operator.

Lemma1.3.A positive operator x≥0is non-compact if,and only if,there exist realλ>0and in?nite dimensional orthoprojection p obeyingλp≤x.

Proof.Note that,in contrast to the spectral characterization of positive compact operators,the spectral theorem in case of a non-compact x≥0with#spec(x)<∞provides thatλp≤x has to be ful?lled,for some non-zeroλand orthopro-jection p with dim p H=∞(for oneλ∈spec(x)\{0}at least the correspond-ing spectral eigenprojection p has to meet the requirement).But then,due to normclosedness of the compact operators,and since for each positive x one has x∈

CONNES’TRACE FORMULA 9

upon de?ning τ(x )=τ0(x )for x ∈C B (H )+,and τ(x )=∞for x ∈B (H )+\C B (H )+,a non-zero trace τon B (H )is given.Thus,traces (resp.non-trivial traces)on all bounded linear operators are in one-to-one correspondence with traces (resp.non-trivial traces)on the compact operators.

For the unique extension of the trace tr of Lemma 1.2from compact operators onto B (H )the same notation tr will be used.Note that in view of Lemma 1.3with the help of 1.2.1(×)and (??)easily follows that for non-compact x ≥0and each m.o.n.s.{?n }one has ∞n =1 x?n ,?n =∞.Hence,the formula given in Corollary 1.1extends on all x ∈B (H )+.From this formula it is plain to see that tr is a non-trivial normal trace on B (H ).Up to a positive multiple,tr is also unique on B (H )as non-trivial trace with this property :

Corollary 1.4.A non-trivial normal trace τhas the form τ=α·tr ,with α>0.Proof.Let p 1

√x =x .Note that x n =√x ∈F B (H )+holds.Since also τ|C B (H )+is a non-trivial trace,by Corollary 1.2there is unique α>0with τ(x n )=α·tr x n ,for each n ∈N .Hence,by normality of τand since tr is normal,τ(x )=α·tr x follows,for each x ≥0.

Note that in view of the mentioned one-to-one correspondence with traces on the compact operators Corollary 1.2extends to non-trivial traces on B (H )accordingly.In line with this and Corollary 1.4the theory of traces on B (H )with separable in?nite dimensional H essentially is the theory of the one normal trace tr and myriads of singular traces.

1.3.Examples of singular traces on B (H ).Examples of singular traces have been invented by J.Dixmier in [8].Nowadays this class is referred to as Dixmier-traces.In the following,only the singular traces of this class will be constructed and considered.Thereby,in constructing these traces we will proceed in two steps.

In a ?rst step we are going to de?ne some non-trivial two-sided ideal in B (H ),with hereditary positive cone,which later will prove to belong to the de?ning ideal of each of the singular traces to be constructed.As has been already noticed in context of Theorem 1.1,each such ideal then is an ideal of compact operators.For such ideals one knows that these can be completely described in terms of the classes (Schatten-classes)of the characteristic sequences coming along with the operators of the ideal,see [32,Theorem 12].In these sequences,which are in ?∞(N )+,the full information on the ideal is encoded.

In a second step,a class of states on ?∞(N )is constructed which,in restriction to the mentioned sequences from the ideal,yields a map which vanishes on those sequences which correspond to operators of ?nite rank.If taken as functions on the positive operators of the ideal these maps will be shown to be additive,positive homogeneous and invariant.Hence,the extension via the extension principle of Lemma 1.1on all of B (H )+?nally will provide us with a class of singular traces.

1.3.1.Step one :Some ideal of compact operators.For compact x with the help of the characteristic sequence {μn (x )}de?ne

?k ∈N \{1}:γk (x )=1

log k .(1.3a)

10P.M.ALBERTI,R.MATTHES

Then,{γn(x):n>1}is a sequence of non-negative reals which may be bounded or not.The bounded situation deserves our special interest.Let a subset L1,∞(H)?C B(H)be de?ned as follows:

L1,∞(H)= x∈C B(H):sup n≥2γn(x)<∞ .(1.3b)

It is plain to see that by L1,∞(H)an ideal is given in B(H),for some corresponding terminology see[28,3,4],and e.g.[14].

Proposition1.1.L1,∞(H)is a non-trivial two-sided ideal in B(H),and thus is an ideal of compact operators,with hereditary cone L1,∞(H)+of positive elements. Proof.In view of the de?nitions(1.3)and since C B(H)is a two-sided ideal,the validity of the?rst assertion follows as an immediate consequence of(1.2a),(1.2c) and(1.2f)together with the fact that for each operator x of?nite rank{γn(x): n>1}is a null-sequence and thus is bounded.Finally,owing to Corollary1.3for x∈L1,∞(H)+and y∈B(H)with0≤y≤x one infers y∈C B(H)+,and then y≤x according to(1.2b)implies also y∈L1,∞(H)+.

For completeness yet another characterization of L1,∞(H)will be noted(without proof,see e.g.in[4,IV.2.β]),and a class of L1,∞-elements,which can be charac-terized through the asymptotic behavior of the singular values,will be given.

Let L1(H)be the ideal of all operators of trace-class,that is,the de?ning ideal which corresponds to the normal trace tr,cf.Lemma1.2and Corollary 1.1.From(1.3b)and Lemma1.2then especially follows that the inclusion rela-tion L1(H)?L1,∞(H)takes place amongst L1,∞(H)and the ideal of trace-class operators.Moreover,if in line with[28]another Banach space L∞,1(H)(=Sωin [28])is de?ned through

L∞,1(H)= y∈C B(H):∞ n=1n?1μn(y)<∞ ,(1.4)

then it is essentially due to(1.2c),(1.2e)and by monotonicity of the sequences of the σn(y)’s and1

CONNES’TRACE FORMULA11 (the ordering ofμn(x)’s is of importance in this context).Since also the sequence ofμn(y)’s is in decreasing order,it is not hard to see that by successively ex-ploiting the just mentioned conditions on r,for n≤N with N∈N,the validity of k≤Nμk(x)(μk(y)?r k)≥0can be derived,for each N∈N,and any given r which is subject to the above conditions.From this the left-hand side estimate of(1.5a) gets evident.Now,for any two given compact linear operators x,y in view of the polar decomposition theorem and owing to compactness of both operators partial isometries u,w can be chosen such that u|x|w|y|≥0holds,with the singular val-ues of the compact operator u|x|w|y|obeyingμn(u|x|w|y|)=μn(x)μn(y),for each n∈N.In accordance with Lemma1.2one then has|tr u|x|v|y||=tr u|x|w|y|= lim n→∞σn(u|x|w|y|)= μn(u|x|w|y|)= μn(x)μn(y).Hence,since in our par-ticular situation of x,y we have u|x|w|y|∈L1(H)and the above proved left-hand side estimate of(1.5a)has been shown to hold,(1.5a)is completely seen.

Especially,in view of Proposition1.2the estimate(1.5a)can be applied with x∈L1,∞(H)and I=L∞,1(H).Relating asymptotic properties of singular values of x∈L1,∞(H)we thus get the following information:

x∈L1,∞(H)=??y∈L∞,1(H):

∞

n=1μn(x)μn(y)<∞.(1.5b)

Viewing(1.4)and(1.5b)together suggests compact x with asymptotic behavior of singular values likeμn(x)=O(n?1)as good candidates for elements of L1,∞(H).1 In fact,such asymptotic behavior implies that,with some C>0,for all n≥2σn(x)= 1≤k≤nμk(x)≤C 1+ 2≤k≤n k?1 ≤C 1+ n1t?1dt =C(1+log n) is ful?lled.In view of(1.3)we therefore arrive at the following result:

Corollary1.5.x∈C B(H),μn(x)=O(n?1)=?x∈L1,∞(H).

Remark1.1.(1)It is easy to see that for compact x with bounded multiplicity function,m(λ)≤N<∞for allλ,the condition imposed by(1.5b)upon x amounts toμn(x)=O(n?1).Unfortunately,in case of unbounded m this can fail to hold.2That this can even occur for x within L1,∞(H)can be seen by the following counterexample:3

(2)Let x be positive and compact withμ1(x)=1,and with singular values

which for k≥2with(m?1)!

Since the function f(t)=log(1+t/m!)?t{log(m+1)/(m+1)!}is non-negative for0≤t≤m·m!,from the previous alsoσk(x)≤1+log k can be followed whenever m!

12P.M.ALBERTI,R.MATTHES

1.3.

2.Step two:Scaling invariant states.Let us come back now to the construction of the Dixmier-traces.The construction will be based on considering a certain class

of states on the commutative W?-algebra M=?∞,see1.1.1for basic notations. Relating special further notations,for each k∈N let e k∈?∞be the k-th atom in?∞,with j-th component obeying(e k)j=δkj(Kronecker symbol),and let E k

be the special orthoprojection of rank k given as E k= j≤k e j.The ascendingly directed sequence{E n}obeys l.u.b.E n=1and the following equivalence is valid: x∈?∞: · ∞?lim n→∞E n x=x??lim n x n=0.(1.6) Also,for x∈?∞+,{E n x}??∞+is ascendingly directed,with l.u.b.E n x=x.

For the following,let a mapping s:?∞?→?∞(scaling)be de?ned on x∈?∞through s(x)j=x2j,for all j∈N.It is obvious that s is a normal?-homomorphism onto?∞.Hence,s is a unital normal positive linear map onto itself,and?∞s={x∈?∞:s(x)=x}is a W?-subalgebra of?∞(the?xpoint algebra of s). Lemma1.4.There exists a conditional expectation E:?∞?→?∞s projecting onto the?xpoint algebra?∞s such that the following properties hold:

(1)E?s=E;

(2)E(x)=(lim n→∞x n)·1,for each x∈?∞with lim n→∞x n existing. Proof.Let us consider the sequence{s n }of partial averages s n =1

n hold,for each n∈N,

and since owing to normality of s for eachω∈?1alsoω?s∈?1is ful?lled,by argueing with the mentioned subnet one infers that E?s=s?E=E.From this

{x∈?∞:E(x)=x}??∞s and s n ?E=E follow,for each n.Thus in view of the above E2=E?E=E follows.Hence,E is a projection of norm one(conditional

expectation)projecting onto the?xpoint algebra of s and which satis?es(1).

To see(2),note?rst that owing to s(e k)=0for k odd,and s(e k)=e k/2for k

even,one certainly has s n(E k)=0,for each n>log k/log2.Hence,the action of the n-th average s n to the orthoprojection E k can be estimated as s n (E k) ∞≤[log k/log2]/n(here[·]means the integer part),and thus for all k∈N one has · ∞?lim n→∞s n (E k)=0.?From this and E=σ(?∞,?1)?weak limλs nλ then especiallyω(E(E k))=0follows,for eachω∈?1.Hence E(E k)=0,for each k.Since for each y∈?∞with0≤y≤1one has0≤E k y≤E k,from the previous together with positivity of E also E(E k y)=0follows.By linearity of E and since ?∞is the linear span of?∞+∩(?∞)1this remains true for each y∈?∞.But then, for x∈?∞withα=lim n→∞x n by continuity of E and in view of(1.6)one infers E(x?α·1)= · ∞?lim k→∞E(E k(x?α·1))=0,which is equivalent with(2).

Corollary1.6.There is a stateω∈S(?∞)satisfying the following properties:

(1)ω?s=ω;

(2)ω(x)=lim n→∞x n,provided lim n→∞x n exists.

CONNES’TRACE FORMULA13 The setΓs(?∞)of all such states is a w?-compact convex subset of singular states.4 Proof.Let E be constructed as in Lemma1.4.By positivity and unitality of E,for eachν∈S(?∞)alsoω=ν?E is a state.In view of(1)–(2)this state then obviously satis?es(1)–(2).5ThatΓs(?∞)is w?-compact and convex is evident from the linear nature of the conditions(1)–(2).Finally,in accordance with(2)one hasω(E k)=0, for eachω∈Γs(?∞)and all k∈N.Now,let p∈?∞be any orthoprojection with ω(p)>0.Then,p=0,and owing to l.u.b.E n p=p there has to exist k∈N with q=E k p=0.Thus0

1.3.3.Constructing the Dixmier-traces.For given x∈L1,∞(H),let a sequenceγ(x) be given throughγ(x)=(γ2(x),γ3(x),...),withγn(x)in accordance with(1.3a). Then,by de?nition(1.3b)one hasγ(x)∈?∞+.Hence,if for each?xed scaling invariant stateω∈Γs(?∞),see Corollary1.6,following[8]we de?ne

?x∈L1,∞(H)+:Trω(x)=ω(γ(x)),(1.7) then according to Proposition1.1and sinceωis a positive linear form,we are given a positive map Trω:L1,∞(H)+?x?→Trω(x)∈R+de?ned on the positive cone of the ideal L1,∞(H).The key idea of[8]is that additivity of Trωcan be shown. Lemma1.5.Trωis an additive,positive homogeneous and invariant map from L1,∞(H)+into R+.

Proof.Since L1,∞(H)+is the positive cone of a two-sided ideal of compact op-erators,for x,y∈L1,∞(H)+andλ∈R+we have that x+y,λx,x?x,xx?∈L1,∞(H)+,and these are compact operators again.Hence,in view of(1.3a)from (1.2a)and(1.2g)bothλ·γ(x)=γ(λx)andγ(x?x)=γ(xx?)follow,which in line with(1.7)means that Trωis positive homogeneous and invariant.It remains to be shown that Trωis additive.First note that according to the left-hand side estimate of(1.2d)within?∞+one hasγ(x+y)≤γ(x)+γ(y).Hence,by positivity and linearity ofω,(1.7)yields

Trω(x+y)≤Trω(x)+Trω(y).(?) Now,to each compact operator z letγ0(z)=(γ3(z),γ4(z),...),that is,γ0(z)arises fromγ(x)by application of the one-step left-shift.Also,on?∞let a linear map m be de?ned by m(β)n=log2

≤log2log k γ(z) ∞.

log k?1

14P.M.ALBERTI,R.MATTHES On the other hand,we also have

γk(z)?γk+1(z)≥σk(z)?σk+1(z)

log(k+1)≥?

R+).Since each stateω∈Γs(?∞)obeys Corollary1.6(2),in view of the previous and(1.7)also(2)follows.Finally,for each x∈F B(H)+the sequenceγ(x)is a null-sequence,and therefore especially x∈L1,∞(H)+,and as a special case of(2)then Trω(x)=0follows.Hence,Trωis a singular trace.

Note the remarkable feature of the Dixmier-traces coming along with Theorem 1.2(2)and saying that provided certain circumstances are ful?lled for x,e.g.if the sequence{γn(x)}has a limit,then independent of the state-parameterωall these Dixmier-traces may yield the same common value at this x.It is such case of independence one usually is tacitely addressing to when speaking simply of the Dixmier-trace of x,whereas the operator itself then is referred to as measurable

CONNES’TRACE FORMULA15 operator,cf.[4,IV.2,De?nition7].Some criteria of measurability,which however all reduce upon showing that the above mentioned special case of existence of lim n→∞γn(x)would happen,subsequently will be discussed in more detail.

1.4.Calculating the Dixmier-trace.

1.4.1.Simple criteria of measurability.We start with discussing conditions which read in terms of spectral theory and which ensure that–for a given operator x∈L1,∞(H)which is not simply of?nite rank–the above-mentioned special case of measurability occurs,that is,the limit lim n→∞γn(x)exists.As a?rst result of that kind one has the following one:6

Lemma1.6.Suppose x∈C B(H),withμn(x)～L·n?1.Then lim n→∞γn(x)=L.7 Proof.For compact operator x suppose lim n→∞nμn(x)=L to be ful?lled.Then, in case of L>0,forδwith L>δ>0,let M(δ)∈N be chosen such that

?n≥M=M(δ):(L?δ)n?1≤μn(x)≤(L+δ)n?1.

?From this for each n≥M we get

(L?δ) M

Since0

M+1

dt t?1≤ M

which holds for n>M the above estimate(×)implies

M+1

dt(L?δ)t?1≤σn(x)?σM(x)≤ n M dt(L+δ)t?1.

?From this for all n>M=M(δ)

(L?δ){1?log(M+1)/log n}≤γn(x)?σM(x)/log n≤(L+δ){1?log M/log n} is obtained.Considering these estimates for n→∞then yields

(L?δ)≤lim inf

n→∞γn(x)≤lim sup

n→∞

γn(x)≤L+δ.

Note that in case of L=0by positivity of allγn(x)instead of the previous one ?nds0≤lim inf n→∞γn(x)≤lim sup n→∞γn(x)≤δ,for anyδ>0.Thus,since δ>0can be chosen arbitrarily small,in either case lim n→∞γn(x)=L follows.

Now,let us suppose x∈C B(H),withμn(x)=O(1

6We are grateful to C.Portenier,Marburg,for suggesting some details around this and related subjects[30].

7For f:N→R+and g:N→R+\{0}the notation f(n)～L·g(n)stands for lim n→∞f(n)/g(n)=L(and accordingly de?ned with R+instead of N).

16P.M.ALBERTI,R.MATTHES Lemma1.7.Let x∈C B(H),withμn(x)=O(1

CONNES’TRACE FORMULA17 Remark1.2.(1)Relating Lemma1.6remark that there are examples of opera-tors where lim n→∞γn(x)=L exists butμn(x)～L·n?1,see[42,Beispiel

A.27]or[15,Lemma7.35].

(2)On the one hand,the conditions imposed onμn(x)andζx in Lemma1.7

simply reproduce the usual conditions for the standard results of Tauberian type8to become applicable.On the other hand,that the behavior of the extension ofζx at the whole line?z=1(and not only at z=1)has to be of relevance can be seen also by example:there is x∈C B(H)with μn(x)=O(1

n )one has x∈L1,∞(H)+,

and then byζx(z)=tr x z a holomorphic function in the half-plane?z>1is given. Supposeζx extends onto the half-plane?z≥1and is continuous there except for a simple pole at z=1,at worst.Then the Dixmier-trace of x is obtained as

Trω(x)=lim

n γn(x)=lim

s→1+

(s?1)tr x s.(1.10a)

Especially,whenζx extends to a meromorphic function on the whole complex plane, with a simple pole at z=1at worst,this formula turns into

Trω(x)=Res|z=1(ζx),(1.10b) with the residue Res(ζx)of the extended complex function,taken at z=1.

For completeness remark that by our Corollary1.7,which is su?cient to cope with our later needs around Connes’trace theorem,in the special cases at hand the implication(1)?(2)of[4,IV,Proposition4]is reproduced.

Clearly,from both the theoretical and practical point of view,in context of the previous those situations deserve the main interest where formula(1.10b)could be applied.According to the results in[17,Theorem7.1,7.2]this happens e.g.if the context of the classical pseudodi?erential operators of order?n acting on the sectionsΓ(E)of a complex vector bundle E→M of a n-dimensional compact Riemannian manifold M is considered.

In fact,in[17]one proves that as a consequence of the good function-theoretic properties ofζx for each such operator the Weyl’s formula of the asymptotic dis-tribution of the spectral values[43]can be seen to hold.Thus,in particular the conditionμk(x)=O(1/k)is then ful?lled automatically and does not appear as an independent condition any longer.

18P.M.ALBERTI,R.MATTHES

But then,upon combining formula(1.10b)with a method[17,Theorem7.4,7.5] (or see[44])of expressing the residue in terms of the principal symbol of the classi-cal pseudodi?erential operator in question,one?nally will arrive at Connes’trace theorem.

2.The Connes’trace theorem and its application

In the following we are going to comment on the way along to Connes’trace theorem in a more detailed manner and will give some indications on applications of this formula as to classical Yang-Mills theory.

2.1.Preliminaries.

2.1.1.Basic facts about pseudodi?erential operators.Let?be an open set in R n, and let C∞0(?)be the space of smooth functions with compact support inside?. De?nition2.1.Let p∈C∞(?×R).p is called a symbol of order(at most)m∈R, if it satis?es the estimates

|?αξ?βx p(x,ξ)|≤CαβK(1+ ξ )m?|α|,x∈K,ξ∈R n,(2.1) for any choice of multiindicesα,βand compact K??.The space of the symbols of order m is denoted by S m(?×R n)or simply S m.

Note that our de?nition corresponds to the special case with?=1andδ=0of a more general class of symbols as considered e.g.in[34,De?nition1.1.],to which and to[10,11]the reader might refer also for other details on pseudodi?erential operators.10It is obvious that S m?S k for m≤k.For p∈S m,let p(x,D)denote the operator

(p(x,D)u)(x)=(2π)?n/2 p(x,ξ)e i x,ξ ?u(ξ)dξ.(2.2)

?u(ξ)=(2π)?n/2 e?i x,ξ u(x)dx(2.3) is the Fourier transform of u.Note that di?erent p,p′∈S m may lead to the same operator,p(x,D)=p′(x,D).

De?nition2.2.A pseudodi?erential operator(ψDO)of order(at most)m is an operator of the form

P=p(x,D),(2.4) where p∈S m.The class ofψDO’s of order m is denoted by L m.

The mapping S m?→L m,p→p(x,D),is surjective,but in general it will not be injective.Its kernel is contained in S?∞= m∈R S m.TheψDO’s corresponding to S?∞form the space L?∞of smoothing operators11.The principal symbolσm(P) of aψDO P of order m with symbol p∈S m is the class of p in S m/S m?1.

CONNES’TRACE FORMULA19 De?nition2.3.p∈S m is called classical,if it has an“asymptotic expansion”

p～

∞

j=0p m?j,(2.5)

i.e.p m?j∈S m?j and

p?N?1

j=0p m?j∈S m?N,?N,(2.6)

and if p m?j is positive homogeneous inξ“away from0”,i.e.

p m?j(x,tξ)=t m?j p m?j(x,ξ), ξ ≥1,t≥1.(2.7) AψDO is said to be classical if its symbol is classical.The spaces of classical symbols andψDOs are denoted by S m cl and L m cl respectively.

Let p0m?j(x,ξ)be homogeneous functions inξon?×(R n\{0})coinciding with p m?j for ξ ≥1.These functions are uniquely determined,and one writes also

p～

∞

j=0p0m?j(2.8)

instead of(2.5).The principal symbol of a classicalψDO can be identi?ed with the leading term p0m in the asymptotic expansion(2.8).

Theorem2.1.Let F:?′?→?be a di?eomorphism of domains in R n.

Then to everyψDO P on?with symbol p∈S m(?×R n)corresponds aψDO P′on?′with symbol p′∈S m(?′×R n)such that:

F?(P u)=P′(F?(u)),u∈C∞0(?),F??pull-back,(2.9)

p′(x,ξ)?p(F(x),(t F′(x))?1ξ)∈S m?1(?′×R n).(2.10) If P is a classicalψDO then so is P′.

The theorem makes it possible to de?neψDO’s on manifolds.Let M be a paracompact smooth manifold,and consider an operator A:C∞0(M)?→C∞(M). If?is some coordinate neighborhood of M,there are a natural extension map i?:C∞0(?)?→C∞0(M)and a natural restriction map p?:C∞(M)?→C∞(?).

A is calledψDO of order m if all the local restrictions A?:=p??A?i?:C∞0(?)?→C∞(?)areψDO of order m.By Theorem2.1,this is a good de?nition,and also classicalψDO can be de?ned in this manner.Moreover,equation(2.10)says that the principal symbol has an invariant meaning as a function on the cotangent bundle T?M.

On the other hand,ψDO on a manifold can be constructed by gluing:Let j?j= M be a locally?nite covering of M by coordinate neighbourhoods,and let A j be ψDO’s of order m on?j.Furthermore,let jψj=1be a partition of unity subordinate to the given covering,and letφj∈C∞0(?j)withφj|suppψj=1.Then A:= jφj?A j?ψj(φj,ψj considered as multiplication operators)is aψDO of order m on M whose restrictions A?

coincide with A j.

ψDO’s acting on sections of vector bundles are de?ned with appropriate mod-i?cations:They are glued from localψDO’s which are de?ned using matrices of

20P.M.ALBERTI,R.MATTHES

symbols.The principal symbol is then a function on T?M with values in the endo-

morphisms of E,i.e.a section of the bundleπ?(End(E)),whereπ:T?M?→M is the projection of the cotangent bundle,and End(E)is the bundle of endomorphisms

of E.

ψDO’s are operators from C∞0(M)to C∞(M).ψDO’s of order m can be extended to bounded linear operators H s(M)?→H s?m(M),s∈R(Sobolev

spaces).Notice that,by the Sobolev embedding theorems,everyψDO of order ≤0,H s?→H s?m,can be considered as an operator H s?→H s.In particular, taking the case s=0,everyψDO of order≤0may be considered as an operator

L2?→L2.For the case of manifolds,a Riemannian metric is used in the de?nition of the L2scalar products,for vector bundles in addition a?bre metric.L2(M,E) denotes the corresponding space of L2sections.We will need the following list of facts(for some terminology and the corresponding generalities see[34,De?nition 3.1.,24.3]and[7,23.26.12.]e.g.):

1.The product(which exists,if at least one of the factors is“properly supported”) of twoψDO’s of orders m,m′is aψDO of order m+m′.

2.The principal symbol of the product of twoψDO’s is the product of the principal

symbols of the factors.

3.AψDO of order≤0is bounded.For order<0it is compact.

4.AψDO of order less than?n on a manifold of dimension n is trace class.

5.If A is aψDO on a manifold,and ifφj andψj are as above,then A may be written

A= jψj Aφj+A′

with A′∈L?∞(smoothing operator).

Remark2.1.Note that the classicalψDO’s form an algebra which is an example of a more abstract object which usually is referred to as Weyl algebra.According to[17], it is a Weyl algebra corresponding to the symplectic cone Y=T?M\{0}({0}the zero section),with its standard symplectic formωand R+-actionρt(x,ξ)=(x,tξ). That is,Y is an R+-principal bundle such thatρ?tω=tω.The properties listed above,however,are only part of the conditions assumed in[17,2.,A.1.-E.].

2.1.2.De?nition of the Wodzicki residue.There are at least two equivalent de?ni-

tions of the Wodzicki residue:As a residue of a certainζ-function and as an integral

of a certain local density[44],[24].We take as starting point the second de?nition which can be used most directly for writing classical gauge?eld Lagrangians.The ?rst de?nition will show up in the second proof of Connes theorem.

De?nition2.4.Let M be an n-dimensional compact Riemannian manifold.Let T be a classical pseudodi?erential operator of order?n acting on sections of a complex vector bundle E?→M.The Wodzicki residue of T is de?ned by

Res W(T)=

狄拉克符号(Dirac)

狄拉克符号（Dirac ） 1狄拉克符号量子体系状态的描述，前述波动力学和矩阵力学两种方法，其共同特点是：与体系有关的所有信息都有波函数给出；极为重要的是波函数可以写成各类力学量的本征函数的线性组合，而展开系数模平方具有力学量概率的含义。问题：能否不从单一角度描述体系，而用统一的方式全面概括体系的所有性质及概念？狄拉克从数学理论方面，构造了一个抽象的、一般矢量--态矢，并引进了一套“狄拉克符号”，简洁、灵活地描述量子力学体系的状态。 1.1狄拉克符号的引入 1.1.1 态空间任何力学量完全集的本征函数系{})(x u n 作为基矢构成希尔伯特空间（以离散谱为例），微观体系的状态波函数ψ作为该空间的一个态矢，有 ∑=n n n u a ψ （1） n a 即为态矢ψ在基矢n u 上的分量，态矢ψ在所有基矢{}n u 上的分量{}n a 构成了态矢在{}n u 这个表象中的表示（矩阵） ????? ?? ? ??= n a a a 21ψ () ,,,,* *2*1n a a a =+ψ （2）微观体系所有可以实现的状态都与此空间中某个态矢相对应，故称该空间为态空间注意：（1）式中的n u 只是表示某力学量的本征态，而抛开其具体表象；（2）式的右方是ψ的{}n u 表象 1.1.2 态空间中内积（标积）的定义设态空间中两个任意态矢A ψ与B ψ在同一表象{}n u 中的分量表示各为{}n a 与{}n b ，则两态矢内积的定义为 () ∑=????? ?? ? ??=+n n n n n B A b a b b b a a a *21**2*1,,,, ψψ （3）注意：A B B A ψψψψ+ +≠ 1.1.3狄拉克符号的引入态空间中的ψ与+ψ在形式上具有明显的不对称性，狄拉克认为它们应该分属于两个不同的空间?伴随空间

Tracepro入门与进阶1-40

Tracepro 入门与进阶
CYQ DESIGN STUDIO
1

Tracepro 入门与进阶
CYQ DESIGN STUDIO
内容简介
本书以美国 Lambda Research Corporation 的最新 3.24 版本为蓝本进行编写，内容涵盖了 tracepro3.24 光学仿真设计的概念、tracepro 软件的配置和用户定制、光学元件模型的创建、描光、分析等内容。本书章节的安排次序采用由浅入深，前后呼应的教学原则，在内容安排上，为方便读者更快、更深入地理解 tracepro 软件中的一些相关概念、命令和功能，并对运用该软件进行光学仿真设计的过程有一个全局的了解，本书中介绍了单片 LCD 投影机的仿真设计全过程，同时在本书的最后一章详细介绍了背光源等光学仿真设计过程，增强了本书的可读性和实用性，摆脱单个概念、命令、功能的枯燥讲解和介绍。本书可作为光学专业人员的自学教程和参考书籍，也可作为大专院校光学、光电专业的学生学习 tracepro 的使用教材。
2

Tracepro 入门与进阶
CYQ DESIGN STUDIO
前
言
Tracepro 是一套可以做照明光学系统分析、传统光学分析，辐射度以及光度分析的软件，它也是第一套由符合工业标准的 ACIS 立体模型绘图软件发展出来的光机软件。功能强大的 Tracepro 减轻了光学设计人员的劳动强度，节约了大量的人力资源，缩短了设计周期，还可以开发出更多质量更高的光学产品。但目前 Tracepro 学习教程甚少，不少初学者苦于无参考学习资料而举步为艰。本人根据从事光学设计的经验与运用 Tracepro 的体会，汇集成书，目的是使 Tracepro 的初学人员能快速入门，快速见效，使已入门者能进一步提高 Tracepro 的应用水平和操作能力，从而在工作中发挥更大的效益，为中国的光学事业作出贡献！本书乃仓促而成，虽然几经校对，但错误之处在所难免，恳请广大读者朋友予以指正，不甚感谢！电子邮箱： cyqdesign@https://www.doczj.com/doc/00698620.html,
陈涌泉 2004 年 12 月 4 日
3

量子力学考试大纲

876 量子力学考试大纲一、考试性质与范围本《量子力学》考试大纲用于北京科技大学物理学相关各专业硕士研究生的入学考试。本科目考试的重点是要求熟练掌握波函数的物理解释，薛定谔方程的建立、基本性质和精确的以及一些重要的近似求解方法，理解这些解的物理意义，熟悉其实际的应用。掌握量子力学中一些特殊的现象和问题的处理方法，包括力学量的算符表示、对易关系、不确定性关系、态和力学量的表象、电子的自旋、粒子的全同性、泡利不相容原理、量子跃迁及光的发射与吸收的半经典处理方法等，并具有综合运用所学知识分析问题和解决问题的能力。二、考试基本要求（一）波函数和薛定谔方程 1．了解波粒二象性的物理意义及其主要实验事实。 2．熟练掌握波函数的标准化条件：有限性、连续性和单值性。深入理解波函数的概率解释。 3．理解态叠加原理及其物理意义。 4．熟练掌握薛定谔方程的建立过程。深入了解定态薛定谔方程，定态与非定态波函数的意义及相互关系。了解连续性方程的推导及其物理意义。（二）一维势场中的粒子 1．熟练掌握一维无限深方势阱的求解方法及其物理讨论，掌握一维有限深方势阱束缚态问题的求解方法。 2．熟练掌握势垒贯穿的求解方法及隧道效应的解释。掌握一维有限深方势阱的反射、透射的处理方法。 3．熟练掌握一维谐振子的能谱及其定态波函数的一般特点及其应用。 4．了解 --函数势的处理方法。（三）力学量的算符表示 1. 掌握算符的本征值和本征方程的基本概念。 2．熟练掌握厄米算符的基本性质及相关的定理。 3．熟练掌握坐标算符、动量算符以及角动量算符，包括定义式、相关的对易关系及本征值和本征函数。 4．熟练掌握力学量取值的概率及平均值的计算方法，理解两个力学量同时具有确定值的条件和共同本征函数。 5．熟练掌握不确定性关系的形式、物理意义及其一些简单的应用。 6．理解力学量平均值随时间变化的规律。掌握如何根据哈密顿算符来判断该体系的守

LED(Tracepro官方LED建模光学仿真设计教程)

Requirements Models: None Properties: None Editions: TracePro LC, Standard and Expert Introduction In this example you will build a source model for a Siemens LWT676 surface mount LED based on the manufacturer’s data sheet. The dimensions will be used to build a solid model and the source output will be defined to match the LED photometric curve. Copyright ? 2013 Lambda Research Corporation.

Create a Thin Sheet First analyze the package to determine the best method of constructing the geometry in TracePro. The symmetry of the package suggests starting from a Thin Sheet and extruding the top and bottom halves with a small draft angle. Construct Thin Sheet in the XY plane. 1. Start TracePro 2. Select View|Profiles|XY or click the View XY button on the toolbar, and switch to silhouette mode, View|Silhouette. 3. Select Insert|Primitive Solid and select the Thin Sheet tab. 4. Enter the four corners of the Thin Sheet in mm in the dialog box, as shown below, and click Insert. 5. Click the Zoom All button or select View|Zoom|All to see the new object.

用宏表函数与公式

用宏表函数与公式 1. 首先：点CTRL+F3打开定义名称，再在上面输入“纵当页”，在下面引用位置处输入： =IF(ISNA(MATCH(ROW(),GET.DOCUMENT(64))),1,MATCH(ROW(),GET.DOCUMENT(64))+1) 2.然后再继续添加第二个名称：“横当页”，在下面引用位置处输入： =IF(ISNA(MATCH(column(),GET.DOCUMENT(65))),1,MATCH(column(),GET.DOCUMENT(65))+1) 3.再输入“总页”；引用位置处输入：（在MSoffice2007不管有多少页，都只显示共有1页，不知为什么） =GET.DOCUMENT(50)+RAND()*0 4.最后再定义“页眉”，引用位置： ="第"&IF(横当页=1,纵当页,横当页+纵当页)&"页/共"&总页&"页" 5.在函数栏使用应用即可得到需要的页码。另外一般情况下，一般的表册都要求每页25行数据，同时每页还需要设置相同的表头，虽然上面的方法可以在任意单元格内计算所在页面的页码，但是如果公式太多的话，计算特别慢。如果每页行数是固定的（比如25行）话，就可以采用下面的笨方法。 1、设置顶端标题行，“页面设置”→“工作表”→“顶端标题行”中输入“$1:$4”（第1行到第4行） 2、在工作表中数据输入完毕后，设置好各种格式，除表头外，保证每页是25行数据。 3、在需要设置该行所在页面的页码的单元格内输入如下公式： =INT((ROW()-ROWS(Print_Titles)-1)/25)+1 （公式里面的Print_Titles就是前面第1步所设置的顶端标题行区域。） 4、通过拖动或者复制的方法复制上面的公式，即可得到页码。

Excel常用函数公式大全(实用)

Excel常用函数公式大全 1、查找重复内容公式：=IF(COUNTIF(A:A,A2)>1,"重复","")。 2、用出生年月来计算年龄公式：=TRUNC((DAYS360(H6,"2009/8/30",FALSE))/360,0)。 3、从输入的18位身份证号的出生年月计算公式： =CONCATENATE(MID(E2,7,4),"/",MID(E2,11,2),"/",MID(E2,13,2))。 4、从输入的身份证号码内让系统自动提取性别，可以输入以下公式： =IF(LEN(C2)=15,IF(MOD(MID(C2,15,1),2)=1,"男","女"),IF(MOD(MID(C2,17,1),2)=1,"男","女"))公式内的“C2”代表的是输入身份证号码的单元格。 1、求和：=SUM(K2:K56) ——对K2到K56这一区域进行求和； 2、平均数：=AVERAGE(K2:K56) ——对K2 K56这一区域求平均数； 3、排名：=RANK(K2，K$2:K$56) ——对55名学生的成绩进行排名； 4、等级：=IF(K2>=85,"优",IF(K2>=74,"良",IF(K2>=60,"及格","不及格"))) 5、学期总评：=K2*0.3+M2*0.3+N2*0.4 ——假设K列、M列和N列分别存放着学生的“平时总评”、“期中”、“期末”三项成绩； 6、最高分：=MAX(K2:K56) ——求K2到K56区域（55名学生）的最高分； 7、最低分：=MIN(K2:K56) ——求K2到K56区域（55名学生）的最低分； 8、分数段人数统计：（1）=COUNTIF(K2:K56,"100") ——求K2到K56区域100分的人数；假设把结果存放于K57单元格；（2）=COUNTIF(K2:K56,">=95")－K57 ——求K2到K56区域95～99.5分的人数；假设把结果存放于K58单元格；（3）=COUNTIF(K2:K56,">=90")－SUM(K57:K58) ——求K2到K56区域90～94.5分的人数；假设把结果存放于K59单元格；（4）=COUNTIF(K2:K56,">=85")－SUM(K57:K59) ——求K2到K56区域85～89.5分的人数；假设把结果存放于K60单元格；

准直TIR透镜Tracepro实例

准直TIR透镜的TracePro模拟过程说明：本例只讲解我用TP的模拟过程，不是TP的使用手册之类，讲解有误或不清楚的地方请见谅。本例不讲解透镜的设计方法，请不要追问如何设计透镜。最后提一个要求：不喜勿喷。作者：虫洞里的猫准直TIR透镜，是指在原点的点光源经过透镜后光线能平行出射的透镜，但由于LED的发光面都是面光源，因此LED经过此透镜后不可能是平行光出射，但其出光角度会是最小值。本实例以已设计好的准直TIR透镜为例，逐步演示TracePro的模拟过程。 1.插入3D文件 TracePro可以打开多种3D格式的文件，最方便的是直接插入零件，但此过程只能使用.SAT格式的文件，如下图的过程。

如果你的3D文件是其它格式，如STEP等，则可以用TracePro直接打开，具体过程为：文件-打开，在打开的对话框的下拉菜单中选择合适的格式。 2.设置光源 2.1 设置档案光源 2.1.1 方法一设置光源可以有很多方式，但最直接也最准确的是使用光源文件，在TracePro中也称为档案光源，TracePro可用的档案光源主要有.DAT或.RAY格式的。此文件可以从LED厂家的官网上下载，本实例使用的LED为CREE公司的XLamp XP-E。如下图，XP-E Cool White Optical Source Model - TracePro (zip) (42 MB)是适合TracePro使用的光源文件，其网站地址为：https://www.doczj.com/doc/00698620.html,/LED-Components-and-Modules/Products/XLamp/Discrete-Directional/XLa mp-XPE。

tracepro实验报告范文

2020 tracepro实验报告范文Contract Template

tracepro实验报告范文前言语料：温馨提醒，报告一般是指适用于下级向上级机关汇报工作，反映情况，答复上级机关的询问。按性质的不同，报告可划分为：综合报告和专题报告；按行文的直接目的不同，可将报告划分为：呈报性报告和呈转性报告。体会指的是接触一件事、一篇文章、或者其他什么东西之后，对你接触的事物产生的一些内心的想法和自己的理解本文内容如下：【下载该文档后使用Word打开】一.实验概况实验时间：实验地点：合肥工业大学仪器学院平房实验室指导老师：郎贤礼实验要求：1.熟练TracePro软件基本功能及实际操作方法； 2.掌握光学器件设计的原理及一般步骤； 3.会对设计好的光学器件进行数据图像分析； 4.能够自己设计简单的光学器件。二.实验内容（一）软件介绍TracePro是一套普遍用于照明系统、光学分析、辐射度分析及光度分析的光线模拟软体。它是第一套以ACISsolidmodelingkernel为基本的光学软体。第一套结合真实固体模型、强大光学分析功能、资料转换能力强及易上手的使用介面的模拟软件。

TracePro可利用在显示器产业上，它能模仿所有类型的显示系统，从背光系统，到前光、光管、光纤、显示面板和LCD投影系统。应用领域包括：照明、导光管、背光模组、薄膜光学、光机设计、投影系统、杂散光、雷射邦浦常建立的模型：照明系统、灯具及固定照明、汽车照明系统（前头灯、尾灯、内部及仪表照明）、望远镜、照相机系统、红外线成像系统、遥感系统、光谱仪、导光管、积光球、投影系统、背光板。TracePro作为下一代偏离光线分析软件，需要对光线进行有效和准确地分析。为了达到这些目标，TracePro具备以下这些功能：处理复杂几何的能力，以定义和跟踪数百万条光线；图形显示、可视化操作以及提供3D实体模型的数据库；导入和导出主流CAD软件和镜头设计软件的数据格式。通过软件设计和仿真功能，可以:得到灯具的出光角度：只需有灯具的3D模块便可通过软件仿真功能预判灯具出光角度，以此判断灯具是否达到设计目标。得到灯具出光光斑图和照度图：可以模拟灯具打在不同距离得到的光斑、照度图分布情况,以此判断灯具出光性能。灯具修改建议功能：如果通过软件判断初步设计灯具性能不符合要求，TracePro光线可视图可以看到形成配光图每段曲线是由罩那段曲线形成，以提供修改建议。准配光图和IES文件：可导出标准配光图和IES文件，用于照明工程设计。实际效益通过软件的仿真功能，可以一次次在软件中完成灯具结构不同状态下时的出光性能，而不需每次灯具修改都需开模或做手板后测试才知道，这就大大缩短了产品开发周期、节省开模成本费用、提高产品设计准确性。

宏表函数

宏表函数贡献者：zuazua日期：2010-11-18 阅读：2484 相关标签：et2010 > 公式 > 函数 > 宏表函数 EVALUATE 对以文字表示的一个公式或表达式求值，并返回结果 INDIRECT函数贡献者：843211日期：2008-07-21 阅读：58024 相关标签：et2007 > 公式 > 函数 > 函数类型 > 查找与引用函数 > INDIRECT 返回由文本字符串指定的引用。此函数立即对引用进行计算，并显示其内容。当需要更改公式中单元格的引用，而不更改公式本身，请使用函数INDIRECT。语法 INDIRECT(ref_text,a1) Ref_text 为对单元格的引用，此单元格可以包含A1-样式的引用、R1C1-样式的引用、定义为引用的名称或对文本字符串单元格的引用。如果ref_text 不是合法的单元格的引用，函数INDIRECT 返回错误值#REF!。 ? 如果ref_text 是对另一个工作簿的引用（外部引用），则那个工作簿必须被打开。如果源工作簿没有打开，函数INDIRECT 返回错误值#REF!。 A1 为一逻辑值，指明包含在单元格ref_text 中的引用的类型。 ? 如果a1 为TRUE 或省略，ref_text 被解释为A1-样式的引用。 ? 如果a1 为FALSE，ref_text 被解释为R1C1-样式的引用。示例如果您将示例复制到空白工作表中，可能会更易于理解该示例。 A B 1数据数据 2B2 1.333 3B345 4George10 5562 公式说明（结果） =INDIRECT($A$2)单元格A2中的引用值(1.333) =INDIRECT($A$3)单元格A3中的引用值(45) =INDIRECT($A$4)如果单元格B4有定义名“George”，则返回定义名的值(10) =INDIRECT("B"&$A$5)单元格A5中的引用值(62)

tracepro实验报告范文

tracepro实验报告范文一.实验概况实验时间：实验地点：合肥工业大学仪器学院平房实验室指导老师：郎贤礼实验要求：1.熟练TracePro软件基本功能及实际操作方法； 2.掌握光学器件设计的原理及一般步骤； 3.会对设计好的光学器件进行数据图像分析； 4.能够自己设计简单的光学器件。二.实验内容（一）软件介绍 TracePro是一套普遍用于照明系统、光学分析、辐射度分析及光度分析的光线模拟软体。它是第一套以ACIS solid modeling kernel为基本的光学软体。第一套结合真实固体模型、强大光学分析功能、资料转换能力强及易上手的使用介面的模拟软件。 TracePro可利用在显示器产业上，它能模仿所有类型的显示系统，从背光系统，到前光、光管、光纤、显示面板和LCD投影系统。应用领域包括：照明、导光管、背光模组、薄膜光学、光机设计、投影系统、杂散光、雷射邦浦常建立的模型：照明系统、灯具及固定照明、汽车照明系统（前头灯、尾灯、内部及仪表照明）、望远镜、照相机系统、红外线成像系统、遥感系统、光谱仪、导光管、积光球、投影系统、背光板。 TracePro作为下一代偏离光线分析软件，需要对光线进行有效和准确地分析。为了达到这些目标，TracePro具备以下这些功能：处理复杂几何的能力，以定义和跟踪数百万条光线；图形显示、可视化操作以及提供3D实体模型的数据库；导入和导出主流CAD软件和镜头设计软件的数据格式。通过软件设计和仿真功能，可以: 得到灯具的出光角度：只需有灯具的 3D模块便可通过软件仿真功能预判灯具出光角度，以此判断灯具是否达到设计目标。得到灯具出光光斑图和照度图：可以模拟灯具打在不同距离得到的光斑、照度图分布情况,以此判断灯具出光性能。灯

excel常用宏

1.拆分单元格赋值 Sub 拆分填充() Dim x As Range For Each x In https://www.doczj.com/doc/00698620.html,edRange.Cells If x.MergeCells Then x.Select x.UnMerge Selection.Value = x.Value End If Next x End Sub 2.E xcel 宏按列拆分多个excel Sub Macro1() Dim wb As Workbook, arr, rng As Range, d As Object, k, t, sh As Worksheet, i& Set rng = Range("A1:f1") Application.ScreenUpdating = False Application.DisplayAlerts = False arr = Range("a1:a" & Range("b" & Cells.Rows.Count).End(xlUp).Row) Set d = CreateObject("scripting.dictionary") For i = 2 To UBound(arr) If Not d.Exists(arr(i, 1)) Then Set d(arr(i, 1)) = Cells(i, 1).Resize(1, 13) Else Set d(arr(i, 1)) = Union(d(arr(i, 1)), Cells(i, 1).Resize(1, 13)) End If Next k = d.Keys t = d.Items For i = 0 To d.Count - 1 Set wb = Workbooks.Add(xlWBATWorksheet) With wb.Sheets(1) rng.Copy .[A1] t(i).Copy .[A2] End With wb.SaveAs Filename:=ThisWorkbook.Path & "\" & k(i) & ".xlsx" wb.Close Next

量子力学之狄拉克符号系统与表象

Dirac 符号系统与表象一、Dirac 符号 1. 引言我们知道任一力学量在不同表象中有不同形式，它们都是取定了某一具体的力学量空间,即某一具体的力学量表象。量子描述除了使用具体表象外，也可以不取定表象，正如几何学和经典力学中也可用矢量形式 A 来表示一个矢量，而不用具体坐标系中的分量(A x , A y , A z )表示一样。量子力学可以不涉及具体表象来讨论粒子的状态和运动规律。这种抽象的描述方法是由 Dirac 首先引用的，本质是一个线性泛函空间，所以该方法所使用的符号称为 Dirac 符号。 2. 态矢量（1）. 右矢空间力学量本征态构成完备系，所以本征函数所对应的右矢空间中的右矢也组成该空间的完备右矢（或基组），即右矢空间中的完备的基本矢量（简称基矢）。右矢空间的任一矢量 |ψ> 可按该空间的某一完备基矢展开。例如： =n n a n ψ∑ （2）. 左矢空间右矢空间中的每一个右矢量在左矢空间都有一个相对应的左矢量，记为 < |。右矢空间和左矢空间称为伴空间或对偶空间，<ψ | 和 |ψ> 称为伴矢量。

的关系 |ψ >按 Q 的左基矢 |Q n > 展开： |ψ > = a 1 |Q 1> + a 2 |Q 2> + ... + a 3 |Q 3 > + ... 展开系数即相当于 Q 表象中的表示： 12 n a a a ψ?? ? ? ?= ? ? ?? ? <ψ| 按 Q 的左基矢和 <φ| 的标积为：*n n n b a ?ψ=∑。显然<φ|ψ>* = <ψ|φ>。对于满足归一化条件的内积有：*1n n n a a ψψ= =∑。这样，本征态的归一化条件可以写为：

excel常用宏集合

65：删除包含固定文本单元的行或列 Sub 删除包含固定文本单元的行或列() Do Cells.Find(what:="哈哈").Activate Selection.EntireRow.Delete '删除行 ' Selection.EntireColumn.Delete '删除列 Loop Until Cells.Find(what:="哈哈") Is Nothing End Sub 72：在指定颜色区域选择单元时添加/取消"√"（工作表代码） Private Sub Worksheet_SelectionChange(ByVal Target As Range) Dim myrg As Range For Each myrg In Target If myrg.Interior.ColorIndex = 37 Then myrg = IIf(myrg <> "√", "√", "") Next End Sub 73：在指定区域选择单元时添加/取消"√"（工作表代码） Private Sub Worksheet_SelectionChange(ByVal Target As Range) Dim Rng As Range If Target.Count <= 15 Then If Not Application.Intersect(Target, Range("D6:D20")) Is Nothing Then For Each Rng In Selection With Rng If .Value = "" Then .Value = "√" Else .Value = "" End If End With Next End If End If End Sub 74：双击指定单元，循环录入文本（工作表代码）

tracepro仿真设计实例

三、实验步骤下面以导光管为例： 1.运行Tracepro并用File|New打开一个新模型； 2.打开Insert|Primative Solid对话框，并选择Cylinder/Cone标签； 3.输入主半径为2，长度为30，并按下Insert按钮； 4.按下Zoom All缩放全部或者View|菜单以便看到新物件 5.用Edit|Select|Surface菜单选择棒的右端面right end；用鼠标“选取”棒的端面。 6.选择Edit|Surface|Revolve Surface Selection旋转填料选项； 7.输入角度90°与弯曲半径25； 8.将轴置于点（0,-25,30），并定义轴的指向为空间中的X轴方向； 9.按下旋转填料Revolve Surface按钮来进行弯曲；

10.通过Edit|Surface|Sweep打开表面拉伸填料选项Sweep Surface Selection对话框； 11.在拉伸长度Distance框中输入15，拉伸角度Draft angle为-2°，表面沿着设定长度被拉伸的同时，以2°的角度逐渐变细： 12.用Edit|Select|Object菜单选择导光管，并用鼠标点击导光管； 13.用Define|Apply Properties(设定材料)，打开Apply Properties应用特性对话框： 14.选择Plastic目录Catalog，选择名称Acrylic并按下Apply按钮；

15.选择菜单Insert|Lens Element，透镜参数为Surface1 Radius : 25，Thickness 3.5mm，Material BK7，Position (0, 0, -40)在导光管前安置一汇聚透镜，以达到较好的光路耦合性能。 16.设定光源，菜单选择,Analysis|Grid Raytrace,网格光线沿圆周排列 Annular,半径 Outer Radius 10mm,网格光线圆周数量 Rings : 10, 光线的起点位置 (0, 0, -48).

宏表函数纺织

身定制使铁马佛山定,制软件企业管理更出,渠道通软件数量无限,制可定制数据安全可,安装在自己服务器上,渠道通对客格订单促,仓库快软件加密安全,变简单卖软件难新的,为软件加密后可同时,访硬件虚拟和试件天,带为大量发佛山定制,软件行业软件十二年,佛山定制软件经验为,企业量身定制使铁马,佛山定制软件企业管,理更出渠道通软件数,量无限制数据安全可,安装在自己服务器上,渠道通对客格订单促,仓库快软件加密安全,变简单卖新的为软件,加密后可同时访硬件,虚拟和试件天带为大,量发年全新计做账软,件下载计做账软件下,载全套程计实务工做,账出纳程抄报税流程,纳税实务流股票软件,下载排行验摇术股票,软件下载排行您买买,卖独大股票池买卖示,仓位媒软件自己在脑,上录制歌曲平时通话,的耳行我录制了一首,歌曲放在脑上放很清,晰但放在扩音器上却,很杂音别的歌试扩音,器也很清晰录制歌曲,的时候到高潮部分的,时候原唱的声音去除,原唱然录制下来的歌,曲高潮部分杂音为卡,拉歌厅录制现场演唱,的歌曲然后压缩出售,给顾客的生咋高解答,下载唱吧里别的录制,的歌曲软件可以将你,自己和歌曲的字幕以,及你自己的声音录制,呢好一点的录制歌曲,的址录制歌曲时选择,分类呢指最动听的歌,那些只需要首左右轻松玩,乐自由靠近一点点周,杰伦夜曲浪漫珊瑚海,枫发雪黑毛衣梁静丝,路可惜你瘦瘦的陈科,妤曹璐猫咪陶晶莹走,路去纽约心事我变了,离开我以前的但可以,听魏雪漫懂了香郑中,基无最近很喜欢听千,华唱了更喜欢绿光森,林题曲三个字无法开,口过大好找哈郑畅版,恶作剧之吻原声带恶,作剧王蓝茵遇到全世,界的都李克勤可以谁,自

然卷乐坐在巷口的,那对小持久至我深过,一次林忆莲杜德伟走,钟一生嫁个我吴宗宪,窗外存张瑶荧光七天,张震岳我别走可你们,听过了但还很好听的,歌夜曲水煮鱼我你解,脱笔记心我的心里只,你他我最摇摆南拳妈,妈破晓陈依依音谷考岂自保雄才一,作才李斯税驾,苦早华亭鹤唳,讵可闻上蔡苍,鹰足道君见吴,中张翰称达生,称一作真秋风,忽忆江东行且,乐生前一杯酒,须身后千载鸟,夜啼云城边乌,栖归飞哑哑枝,上啼中织锦秦,川中织锦一作,闺中织妇川一,作家碧纱烟隔,窗语停梭怅然,忆远怅然忆远,一作向故夫独,宿孤房泪雨飞,龙引二首帝铸,鼎于荆山炼丹,砂丹砂骑龙飞,上太清家云愁,海思令嗟宫中,颜花飘然挥凌,紫霞从风纵鸾,车鸾车侍轩辕,遨游青天中其,乐可言其二鼎,湖流水清且闲,轩辕去时弓剑,古传道留其间,后宫婵娟花颜,乘鸾飞烟亦还,骑龙攀天造天,关造天关闻天,语长云河车载,玉载玉过紫皇,紫皇乃赐白兔,所捣之方后天,而老凋三光下,瑶池见王母蛾,眉萧飒秋霜长,相思长相思在,长安络纬秋啼,井阑微霜凄凄,簟寒微一作凝,孤灯明思绝明,一作寐卷帏望,月空长叹花隔,云端花一作佳,期迢迢上青冥,之高天高一作,长下渌水之波,澜天长路远魂,飞苦梦魂到关,山难长相思摧,心肝上留田行,行至上留田孤,坟峥嵘积此万,古恨春草复生,悲风边来肠断,白杨声借谁家,地埋蒿里茔古,老向余言言上,留田蓬科马鬣,今已平昔之死,兄葬他于此举,铭旌一鸟死百,鸟鸣一兽走百,兽惊桓山之禽,别离苦去翔征,田氏仓卒骨肉,分青天白日摧,紫荆交柯之木,本同形东枝憔,悴西枝荣无心,之物尚此参胡,乃寻天兵孤竹,延陵让扬高风,缅邈颓波清尺,布之谣塞耳听,春日行深

华中师范大学量子力学

一、填空题（共10题，每题2分，共20分） 1. 若某种光的波长为λ，则其光子的能量为，动量大小为。 2. 戴维逊—革末实验主要表现出电子具有。 3. 若(),x t ψ 是归一化的波函数，则()2,x t dx ψ 表示 4. 设力学量算符?F 与?G 不对易，且其对易子为???,F G ik ??=?? ，则它们的不确定性关系为。 5. 厄米算符在自身表象是矩阵。 6. 从量子力学的观点看，氢原子中核外电子的运动不再是圆轨道上的运动，而是电子云的图像，电子云是； 7. 设氢原子处于态()211021112,,()(,)()(,)33 r R r Y R r Y ψθ?θ?θ?-= -，求氢原子的角动量z 分量的平均值 8. 证明电子具有自旋的实验是。 9. 两个角动量121 1, 2 j j ==耦合的总角动量J = 10. 共振跃迁意思是。共振跃迁意思是当周期性微扰的频率不等于两能级间的玻尔频率时，即 mn ωω≠±时，跃迁概率不大；当mn ωω=±时，即吸收过程和辐射过程，其跃迁概率随时间而增大，称为共振跃迁。二、判断题（共10题，每题1分，共10分） 1. 光电效应证实了光的粒子性，康普顿效应进一步证实了光的粒子性。 2. 若12,,,,n ψψψ 是体系的一系列可能的状态，则这些态的线性叠加 1122 n n C C C ψψψψ=++++ （其中12,,,,n C C C 为复常数）也是体系的一个可能状态。 3. 不同定态的线性叠加还是定态。 4. 因为坐标与动量算符均是厄米算符，所以它们的乘积一定是厄米算符。 5. 若两个力学量算符不对易，则它们一般没有共同本征态。 6. 粒子在中心力场中运动，若角动量L z 是守恒量，那么L x 就不是守恒量。 7. 在一维势场中运动的粒子，势能对原点对称：)()(x U x U =-，则粒子的波函数一定具有确定的宇称。 8. 费米子体系的哈密顿算符H ?必须是交换反对称的，波色子体系的哈密顿算符H ?必须是交换对称的。

表格常用函数大全

1、查找重复内容公式：=IF(COUNTIF(A:A,A2)>1,"重复","")。 2、用出生年月来计算年龄公式： =TRUNC((DAYS360(H6,"2009/8/30",FALSE))/360,0)。 3、从输入的18位身份证号的出生年月计算公式： =CONCATENATE(MID(E2,7,4),"/",MID(E2,11,2),"/",MID(E2,13,2))。 4、从输入的身份证号码内让系统自动提取性别，可以输入以下公式： =IF(LEN(C2)=15,IF(MOD(MID(C2,15,1),2)=1,"男","女 "),IF(MOD(MID(C2,17,1),2)=1,"男","女"))公式内的“C2”代表的是输入身份证号码的单元格。 1、求和：=SUM(K2:K56) ——对K2到K56这一区域进行求和； 2、平均数：=AVERAGE(K2:K56) ——对K2 K56这一区域求平均数； 3、排名：=RANK(K2，K$2:K$56) ——对55名学生的成绩进行排名； 4、等级：=IF(K2>=85,"优",IF(K2>=74,"良",IF(K2>=60,"及格","不及格")))

5、学期总评：=K2*0.3+M2*0.3+N2*0.4 ——假设K列、M列和N列分别存放着学生的“平时总评”、“期中”、“期末”三项成绩； 6、最高分：=MAX(K2:K56) ——求K2到K56区域（55名学生）的最高分； 7、最低分：=MIN(K2:K56) ——求K2到K56区域（55名学生）的最低分； 8、分数段人数统计：（1）=COUNTIF(K2:K56,"100") ——求K2到K56区域100分的人数；假设把结果存放于K57单元格；（2）=COUNTIF(K2:K56,">=95")－K57 ——求K2到K56区域95～99.5分的人数；假设把结果存放于K58单元格；（3）=COUNTIF(K2:K56,">=90")－SUM(K57:K58) ——求K2到K56区域90～94.5分的人数；假设把结果存放于K59单元格；（4）=COUNTIF(K2:K56,">=85")－SUM(K57:K59) ——求K2到K56区域85～89.5分的人数；假设把结果存放于K60单元格；

-第1章-量子力学基础详细讲解

第1章、量子力学基础 1.1 量子力学和量子光学发展简史 1900，Planck （普朗克），黑体辐射，能量量子化： h εν= 1905，Einstein （爱因斯坦）, 光电效应，光量子–光子： E h ν=， h p λ= （h h E p c c νλ===） 1913，Bohr （玻尔）, 原子光谱和原子结构，定态、量子跃迁及跃迁频率： ()/mn m n E E h ν=- 1923， de Broglie （德布罗意）, 物质粒子的波动性，物质波： E h ν=，h p λ= 1925， Heisenberg （海森堡），矩阵力学 1926, Schr?dinger （薛定谔）, 波函数(),r t ψ ，波动方程- Schr?dinger 方程，波动力学: ()(),,i r t H r t t ψψ?=? 1926， Born （波恩）, 波函数的统计诠释：()2 ,r t ψ 为概率密度， ()2,1dr r t ψ=? 1926， Dirac （狄拉克），狄拉克符号、态矢量ψ、量子力学的表象理论 1927， Dirac ，电磁场的量子化 1928， Dirac ，相对论性波动方程至此，量子力学的基本架构已建立，起初主要用其处理原子、分子、固体等实物粒子问题。尽管量子力学在处理实际问题中获得了巨大成功，但是关于量子力学的基本解释和适用范围一直存在争论，最著名的有： 1935， Schr?dinger 猫态 1935， EPR 佯谬 1960 前后，量子理论用于电磁场：量子光学 1956， Hanbury Brown 和Twiss ，强度关联实验 1963， Glauber （2005年诺奖得主），光的量子相干性 1963， Jaynes & Cummings, J-C 模型：量子单模电磁场与二能级原子的相互作用 1962-1964，激光理论（Lamb, Haken, Lax 三个主要学派） 1970’s, 光学瞬态、共振荧光、超荧光、超辐射 1980’s ，光学双稳态 1990’s ，光场的非经典性质（反群聚效应、亚泊松分布、压缩态）、

文档之家

Connes' trace formula and Dirac realization of Maxwell and Yang-Mills action

狄拉克符号(Dirac)

Tracepro入门与进阶1-40

量子力学考试大纲

LED(Tracepro官方LED建模光学仿真设计教程)

用宏表函数与公式

Excel常用函数公式大全(实用)

准直TIR透镜Tracepro实例

tracepro实验报告范文

宏表函数

tracepro实验报告范文

excel常用宏

量子力学之狄拉克符号系统与表象

excel常用宏集合

tracepro仿真设计实例

宏表函数纺织

华中师范大学量子力学

表格常用函数大全

-第1章-量子力学基础详细讲解

最新最全excel表格常用函数及用法