# Relativistic Quantum Mechanics

Size: px
Start display at page:

Transcription

1 Chapte Relativistic Quantum Mechanics In this Chapte we will addess the issue that the laws of physics must be fomulated in a fom which is Loentz invaiant, i.e., the desciption should not allow one to diffeentiate between fames of efeence which ae moving elative to each othe with a constant unifom velocity v. The tansfomations beween such fames accoding to the Theoy of Special Relativity ae descibed by Loentz tansfomations. In case that v is oiented along the x axis, i.e., v = v ˆx, these tansfomations ae x = x v t, t = t v c v c x v c, x = x ; x 3 = x 3. which connect space time coodinates x, x, x 3, t in one fame with space time coodinates x, x, x 3, t in anothe fame. Hee c denotes the velocity of light. We will intoduce below Loentz-invaiant diffeential equations which take the place of the Schödinge equation of a paticle of mass m and chage q in an electomagnetic field [c.f. efeq:ham, 8.45] descibed by an electical potential V, t and a vecto potential A, t i [ t ψ, t = m i q ] A, c t + qv, t ψ, t. The eplacement of. by Loentz invaiant equations will have two supising and extemely impotant consequences: some of the equations need to be fomulated in a epesentation fo which the wave functions ψ, t ae vectos of dimension lage one, the components epesenting the spin attibute of paticles and also epesenting togethe with a paticle its anti-paticle. We will find that actually seveal Loentz invaiant equations which eplace. will esult, any of these equations being specific fo cetain classes of paticles, e.g., spin paticles, spin paticles, etc. As mentioned, some of the equations descibe a paticle togethe with its anti-paticle. It is not possible to uncouple the equations to descibe only a single type paticle without affecting negatively the Loentz invaiance of the equations. Futhemoe, the equations need to be intepeted as actually descibing many paticle-systems: the equivalence of mass and enegy in elativistic fomulations of physics allows that enegy convets into paticles such that any paticle descibed will have companions which assume at least a vitual existence. Obviously, it will be necessay to begin this Chapte with an investigation of the goup of Loentz tansfomations and thei epesentation in the space of position and time t. The epesentation 87

2 88 Relativistic Quantum Mechanics in Sect.. will be extended in Sect..4 to cove fields, i.e., wave functions ψ, t and vectos with functions ψ, t as components. This will povide us with a geneal set of Loentz invaiant equations which fo vaious paticles take the place of the Schödinge equation. Befoe intoducing these geneal Loentz invaiant field equations we will povide in Sects..5,.7 a heuistic deivation of the two most widely used and best known Loentz invaiant field equations, namely the Klein Godon Sect..5 and the Diac Sect..7 equation.. Natual Repesentation of the Loentz Goup In this Section we conside the natual epesentation of the Loentz goup L, i.e. the goup of Loentz tansfomations.. Rathe than stating fom., howeve, we will povide a moe basic definition of the tansfomations. We will find that this definition will lead us back to the tansfomation law., but in a setting of epesentation theoy methods as applied in Secti. 5 to the goups SO3 and SU of otation tansfomations of space coodinates and of spin. The elements L L act on 4 dimensional vectos of position and time coodinates. We will denote these vectos as follows x µ def = x, x, x, x 3.3 whee x = ct descibes the time coodinate and x, x, x 3 = descibes the space coodinates. Note that the components of x µ all have the same dimension, namely that of length. We will, hencefoth, assume new units fo time such that the velocity of light c becomes c =. This choice implies dimtime = dimlength. Minkowski Space Histoically, the Loentz tansfomations wee fomulated in a space in which the time component of x µ was chosen as a puely imaginay numbe and the space components eal. This space is called the Minkowski space. The eason fo this choice is that the tansfomations. leave the quantity s = x x x x 3.4 invaiant, i.e., fo the tansfomed space-time codinates x µ = x, x, x, x 3 holds x x x x 3 = x x x x 3..5 One can intepete the quantity s as a distance in a 4 dimensional Euclidean space if one chooses the time component puely imaginay. In such a space Loentz tansfomations coespond to 4-dimensional otations. Rathe than following this avenue we will intoduce Loentz tansfomations within a setting which does not equie eal and imaginay coodinates. The Goup of Loentz Tansfomations L = O3, The Loentz tansfomations L descibe the elationship between space-time coodinates x µ of two efeence fames which move elative to each othe with unifom fixed velocity v and which might be eoiented elative to each othe by a otation aound a common oigin. Denoting by x µ the

3 .: Natual Repesentation of the Loentz Goup 89 coodinates in one efeence fame and by x µ the coodinates in the othe efeence fame, the Loentz tansfomations constitute a linea tansfomation which we denote by 3 x µ = L µ νx ν..6 ν= Hee L µ ν ae the elements of a 4 4 matix epesenting the Loentz tansfomation. The uppe index close to L denotes the fist index of the matix and the lowe index ν futhe away fom L denotes the second index. [ A moe conventional notation would be L µν, howeve, the latte notation will be used fo diffeent quantities futhe below.] The following possibilities exist fo the positioning of the indices µ, ν =,,, 3: 4-vecto: x µ, x µ ; 4 4 tenso: A µ ν, A µ ν, A µν, A µν..7 The eason fo the notation is two-fold. Fist, the notation in.6 allows us to intoduce the so-called summation conventon: any time the same index appeas in an uppe and a lowe position, summation ove that index is assumed without explicitly noting it, i.e., y µ x µ } {{ } new = 3 y µ x µ µ= } {{ } old ; A µ νx ν } {{ } new = 3 A µ νx ν ν= } {{ } old ; A µ νb ν ρ } {{ } new = 3 A µ νb ν ρ ν= } {{ } old..8 The summation convention allows us to wite.6 x µ = L µ νx ν. The second eason is that uppe and lowe positions allow us to accomodate the expession.5 into scala poducts. This will be explained futhe below. The Loentz tansfomations ae non-singula 4 4 matices with eal coefficients, i.e., L GL4, R, the latte set constituting a goup. The Loentz tansfomations fom the subgoup of all matices which leave the expession.5 invaiant. This condition can be witten x µ g µν x ν = x µ gµν x ν.9 whee g µν = Combining condition.9 and.6 yields Since this holds fo any x µ it must be tue = g.. L µ ρg µν L ν σ x ρ x σ = g ρσ x ρ x σ.. L µ ρg µν L ν σ = g ρσ.. This condition specifies the key popety of Loentz tansfomations. We will exploit this popety below to detemine the geneal fom of the Loentz tansfomations. The subset of GL4, R, the

4 9 Relativistic Quantum Mechanics elements of which satisfy this condition, is called O3,. This set is identical with the set of all Loentz tansfomations L. We want to show now L = O3, GL4, R is a goup. To simplify the following poof of the key goup popeties we like to adopt the conventional matix notation fo L µ ν L = L µ ν = L L L L 3 L L L L 3 L L L L 3 3 L 3 L 3 L 3 L 3 3 Using the definition. of g one can ewite the invaiance popety. Fom this one can obtain using gl T gl = and, hence, the invese of L L = g L T g = The coesponding expession fo L T is obviously To demonstate the goup popety of O3,, i.e., of..3 L T gl = g..4 g =.5 L L L L 3 L L L L 3 L L L L 3 L 3 L 3 L 3 L L T = L T = g L g..7 O3, = { L, L GL4, R, L T gl = g },.8 we note fist that the identity matix is an element of O3, since it satisfies.4. We conside then L, L O3,. Fo L 3 = L L holds L T 3 g L 3 = L T L T g L L = L T L T gl L = L T g L = g,.9 i.e., L 3 O3,. One can also show that if L O3,, the associated invese obeys.4, i.e., L O3,. In fact, employing expessions.6,.7 one obtains L T g L = glgggl T g = glgl T g.. Multiplying.4 fom the ight by gl T and using.5 one can deive L T glgl T multiplying this fom the left by by gl T yields = L T and L g L T = g. Using this esult to simplify the.h.s. of. esults in the desied popety L T g L = g,. i.e., popety.4 holds fo the invese of L. This stipulates that O3, is, in fact, a goup.

5 .: Natual Repesentation of the Loentz Goup 9 Classification of Loentz Tansfomations We like to classify now the elements of L = O3,. Fo this pupose we conside fist the value of det L. A statement on this value can be made on account of popety.4. Using det AB = det A det B and det A T = det A yields det L = o det L = ±..3 One can classify Loentz tansfomations accoding to the value of the deteminant into two distinct classes. A second class popety follows fom.4 which we employ in the fomulation.. Consideing in. the case ρ =, σ = yields L L L L 3 =..4 o since L + L + L 3 it holds L. Fom this we can conclude L o L,.5 i.e., thee exist two othe distinct classes. Popeties.3 and.5 can be stated as follows: The set of all Loentz tansfomations L is given as the union L = L + L + L L.6 whee L +, L +, L, L ae disjunct sets defined as follows L + = { L, L O3,, det L =, L } ;.7 L + = { L, L O3,, det L =, L } ;.8 L = { L, L O3,, det L =, L } ;.9 L = { L, L O3,, det L =, L }..3 It holds g L and L as one can eadily veify testing fo popety.4. One can also veify that one can wite L = gl + = L + g ;.3 L + = L + ;.3 L = gl + = L + g.33 whee we used the definition am = {M, M, M M, M = a M }. The above shows that the set of pope Loentz tansfomations L + allows one to geneate all Loentz tansfomations, except fo the tivial factos g and. It is, hence, entiely suitable to investigate fist only Loentz tansfomations in L +. We stat ou investigation by demonstating that L + foms a goup. Obviously, L + contains. We can also demonstate that fo A, B L + holds C = AB L +. Fo this pupose we conside the value of C = A µb µ = 3 j= A jb j + A B. Schwatz s inequality yields 3 3 A jb j A 3 j B j..34 j= j= j=

6 9 Relativistic Quantum Mechanics Fom. follows B 3 j= Bj = o 3 j= Bj = B. Similaly, one can conclude fom. 3 j= A j = A..34 povides then the estimate 3 A jb j j= [ A ] [ B ] < A B..35 One can conclude, theefoe, 3 j= A jb j < A B. Since A and B, obviously A B. Using the above expession fo C one can state C >. In fact, since the goup popety of O3, ascetains C T gc = g it must hold C. The next goup popety of L + to be demonstated is the existence of the invese. Fo the invese of any L L + holds.6. This elationship shows L = L, fom which one can conclude L L +. We also note that the identity opeato has elements µ ν = δ µ ν.36 whee we defined δ µ ν = { fo µ = ν fo µ ν.37 It holds, = and, hence, L +. Since the associative popety holds fo matix multiplication we have veified that L + is indeed a subgoup of SO3,. L + is called the subgoup of pope, othochonous Loentz tansfomations. In the following we will conside solely this subgoup of SO3,. Infinitesimal Loentz tansfomations The tansfomations in L + have the popety that they ae continously connected to the identity, i.e., these tansfomations can be paametized such that a continuous vaiation of the paametes connects any element of L + with. This popety will be exploited now in that we conside fist tansfomations in a small neighbohood of which we paametize by infinitesimal paametes. We will then employ the Lie goup popeties to geneate all tansfomations in L +. Accodingly, we conside tansfomations L µ ν = δ µ ν + ɛ µ ν ; ɛ µ ν small..38 Fo these tansfomations, obviously, holds L > and the value of the deteminant is close to unity, i.e., if we enfoce.4 actually L and det L = must hold. Popety.4 implies + ɛ T g + ɛ = g.39 whee we have employed the matix fom ɛ defined as in.3. To ode Oɛ holds ɛ T g + g ɛ =..4 It should be noted that accoding to ou pesent definition holds δ µν = g µρδ ρ ν and, accodingly, δ = and δ = δ = δ 33 =.

7 .: Natual Repesentation of the Loentz Goup 93 Using.5 one can conclude which eads explicitly This elationship implies ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ 3 ɛ 3 ɛ 3 ɛ 3 3 ɛ T = g ɛ g.4 = ɛ µ µ = ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ 3 ɛ 3 ɛ 3 ɛ 3 3 ɛ j = ɛ j, j =,, 3..4 ɛ j k = ɛ k j, j, k =,, 3.43 Inspection shows that the matix ɛ has 6 independent elements and can be witten w w w 3 ɛϑ, ϑ, ϑ 3, w, w, w 3 = w ϑ 3 ϑ w ϑ 3 ϑ..44 w 3 ϑ ϑ This esult allows us now to define six geneatos fo the Loentz tansfomationsk =,, 3 The geneatos ae explicitly J = K = J k = ɛϑ k =, othe five paametes zeo.45 K k = ɛw k =, othe five paametes zeo..46 ; J = ; K = ; J 3 = ; K 3 = These commutatos obey the following commutation elationships [ J k, J l ] = ɛ klm J m.49 [ K k, K l ] = ɛ klm J m [ J k, K l ] = ɛ klm K m. The opeatos also obey J K = J J + J J + J 3 J 3 =.5

8 94 Relativistic Quantum Mechanics as can be eadily veified. Execise 7.: Demonstate the commutation elationships.49,.5. The commutation elationships.49 define the Lie algeba associated with the Lie goup L +. The commutation elationships imply that the algeba of the geneatos J k, K k, k =,, 3 is closed. Following the teatment of the otation goup SO3 one can expess the elements of L + though the exponential opeatos L ϑ, w = exp ϑ J + w K ; ϑ, w R 3.5 whee we have defined ϑ J = 3 k= ϑ kj k and w K = 3 k= w kk k. One can eadily show, following the algeba in Chapte 5, and using the elationship J k =.5 L k whee the 3 3 matices L k ae the geneatos of SO3 defined in Chapte 5, that the tansfomations.5 fo w = coespond to otations of the spatial coodinates, i.e., L ϑ, w = = R ϑ..53 Hee R ϑ ae the 3 3 otation matices constucted in Chapte 5. Fo the paametes ϑ k of the Loentz tansfomations holds obviously ϑ k [, π[, k =,, 3.54 which, howeve, constitutes an ovecomplete paametization of the otations see Chapte 5. We conside now the Loentz tansfomations fo ϑ = which ae efeed to as boosts. A boost in the x diection is L = expw K. To detemine the explicit fom of this tansfomation we evaluate the exponential opeato by Taylo expansion. In analogy to equation 5.35 it issufficient to conside in the pesent case the matix since L Using the idempotence popety = exp w exp w K = exp = = n= w n n! L n =.57

9 .: Natual Repesentation of the Loentz Goup 95 one can cay out the Taylo expansion above: L = n= w n n! + n= w n+ n +! = cosh w + sinh w = cosh w sinh w sinh w cosh w..58 The conventional fom. of the Loentz tansfomations is obtained though the paamete change v = sinh w cosh w = tanh w.59 Using cosh w sinh w = one can identify sinhw = cosh w and coshw = sinh w +. Coespondingly, one obtains fom.59 v = cosh w cosh w = sinh w sinh w +..6 These two equations yield cosh w = / v ; sinh w = v / v,.6 and.56,.59 can be witten exp w K = v v v v v v.6 Accoding to.3,.6,.5 the explicit tansfomation fo space time coodinates is then x = x v t v, t = t v x v, x = x, x 3 = x 3.63 which agees with.. The ange of the paametes w k can now be specified. v k defined in.59 fo the case k = coesponds to the elative velocity of two fames of efeence. We expect that v k can only assume values less than the velocity of light c which in the pesent units is c =. Accodingly, we can state v k ], [. This popety is, in fact, consistent with.59. Fom.59 follows, howeve, fo w k w k ], [..64 We note that the ange of w k -values is not a compact set even though the ange of v k -values is compact. This popety of the w k -values contasts with the popety of the paametes ϑ k specifying otational angles which assume only values in a compact ange.

10 96 Relativistic Quantum Mechanics. Scalas, 4 Vectos and Tensos In this Section we define quantities accoding to thei behaviou unde Loentz tansfomations. Such quantities appea in the desciption of physical systems and statements about tansfomation popeties ae often extemely helpful and usually povide impotant physical insight. We have encounteed examples in connection with otational tansfomations, namely, scalas like = x + x + x 3, vectos like = x, x, x 3 T, spheical hamonics Y lm ˆ, total angula momentum states of composite systems like Y lm l, l ˆ, ˆ and, finally, tenso opeatos T km. Some of these quantities wee actually defined with espect to epesentations of the otation goup in function spaces, not in the so-called natual epesentation associated with the 3 dimensional Euclidean space E 3. Pesently, we have not yet defined epesentations of Loentz tansfomations beyond the natual epesentation acting in the 4 dimensional space of position and time coodinates. Hence, ou definition of quantities with special popeties unde Loentz tansfomations pesently is confined to the natual epesentation. Nevetheless, we will encounte an impessive example of physical popeties. Scalas The quantities with the simplest tansfomation behaviou ae so-called scalas f R which ae invaiant unde tansfomations, i.e., f = f..65 An example is s defined in.4, anothe example is the est mass m of a paticle. Howeve, not any physical popety f R is a scala. Counteexamples ae the enegy, the chage density, the z component x 3 of a paticle, the squae of the electic field E, t o the scala poduct of two paticle positions. We will see below how tue scalas unde Loentz tansfomations can be constucted. 4-Vectos The quantities with the tansfomation behaviou like that of the position time vecto x µ defined in.3 ae the so-called 4 vectos a µ. These quantites always come as fou components a, a, a, a 3 T and tansfom accoding to Examples of 4-vectos beside x µ ae the momentum 4-vecto p µ = E, p, E = a µ = L µ ν a ν..66 m v, p = m v v.67 the tansfomation behaviou of which we will demonstate futhe below. A thid 4-vecto is the so-called cuent vecto J µ = ρ, J.68 whee ρ, t and J, t ae the chage density and the cuent density, espectively, of a system of chages. Anothe example is the potential 4-vecto A µ = V, A.69 whee V, t and A, t ae the electical and the vecto potential of an electomagnetic field. The 4-vecto chaacte of J µ and of A µ will be demonstated futhe below.

11 .: Scalas, 4 Vectos and Tensos 97 Scala Poduct then 4-vectos allow one to constuct scala quantities. If a µ and b µ ae 4-vectos is a scala. This popety follows fom.66 togethe with. a µ g µν b ν.7 a µ gµν b ν = L µ ρ g µν L ν σa ρ b σ = a ρ g ρσ b σ.7 Contavaiant and Covaiant 4-Vectos It is convenient to define a second class of 4-vectos. The espective vectos a µ ae associated with the 4-vectos a µ, the elationship being a µ = g µν a ν = a, a, a, a 3.7 whee a ν is a vecto with tansfomation behaviou as stated in.66. One calls 4-vectos a µ covaiant and 4-vectos a µ contavaiant. Covaiant 4-vectos tansfom like whee we defined a µ = g µν L ν ρg ρσ a σ.73 g µν = g µν..74 We like to point out that fom definition.7 of the covaiant 4-vecto follows a µ = g µν a ν. In fact, one can employ the tensos g µν and g µν to aise and lowe indices of L µ ν as well. We do not establish hee the consistency of the ensuing notation. In any case one can expess.73 a µ = L µ σ a σ..75 Note that accoding to.7 L σ µ is the tansfomation invese to L σ µ. In fact, one can expess [L T ] µ ν = L ν µ and, accodingly,.7 can be witten L ν µ = L µ ν..76 The 4-Vecto µ An impotant example of a covaiant 4-vecto is the diffeential opeato µ = x µ = t,.77 The tansfomed diffeential opeato will be denoted by µ def =..78 x µ To pove the 4-vecto popety of µ we will show that g µν ν tansfoms like a contavaiant 4- vecto, i.e., g µν ν = L µ ρg ρσ σ. We stat fom x µ = L µ νx ν. Multiplication and summation of x µ = L µ νx ν by L ρ σg ρµ yields, using., g σν x ν = L ρ σg ρµ x µ and g µσ g σν = δ µ ν, x ν = g νσ L ρ σg ρµ x µ..79 This is the invese Loentz tansfomation consistent with.6. We have duplicated the expession fo the invese of L µ ν to obtain the coect notation in tems of covaiant, i.e., lowe, and

12 98 Relativistic Quantum Mechanics contavaiant, i.e., uppe, indices..79 allows us to detemine the connection between µ and µ. Using the chain ule of diffeential calculus we obtain µ = 3 ν= x ν x µ x ν = gνσ L ρ σg ρµ ν = L ν µ ν.8 Multiplication by g λµ and summation ove µ togethe with g λµ g ρµ = δ λ ρ yields i.e., µ does indeed tansfom like a covaiant vecto. g λµ µ = L λ σg σν ν,.8 d Alembet Opeato We want to constuct now a scala diffeential opeato. Fo this pupose we define fist the contavaiant diffeential opeato µ = g µν ν = t,..8 Then the opeato µ µ = t.83 is a scala unde Loentz tansfomations. In fact, this opeato is equal to the d Alembet opeato which is known to be Loentz-invaiant. Poof that p µ is a 4-vecto We will demonstate now that the momentum 4-vecto p µ defined in.67 tansfoms like.66. Fo this pupose we conside the scala diffeential It holds fom which follows One can wite dτ = dx µ dx µ = dt d.84 p = E = The emaining components of p µ can be witten, e.g., dτ = v.85 dt d dτ = d v dt..86 m = m dt v v dt..87 p = m v = m dx v v dt..88 One can expess then the momentum vecto p µ = m dx µ v dt = m d dτ xµ..89

13 .3: Relativistic Electodynamics 99 The opeato m d dτ tansfoms like a scala. Since xµ tansfoms like a contavaiant 4-vecto, the.h.s. of.89 alltogethe tansfoms like a contavaiant 4-vecto, and, hence, p µ on the l.h.s. of.89 must be a 4-vecto. The momentum 4-vecto allows us to constuct a scala quantity, namely Evaluation of the.h.s. yields accoding to.67 o p µ p µ = p µ g µν p ν = E p.9 E p = which, in fact, is a scala. We like to ewite the last esult m v m v v = m.9 p µ p µ = m.9 E = p + m.93 o E = ± p + m..94 In the non-elativistic limit the est enegy m is the dominant contibution to E. Expansion in m should then be apidly convegent. One obtains E = ±m ± p m p p 3 4m 3 + O 4m This obviously descibes the enegy of a fee paticle with est enegy ±m, kinetic enegy ± p m and elativistic coections..3 Relativistic Electodynamics In the following we summaize the Loentz-invaiant fomulation of electodynamics and demonstate its connection to the conventional fomulation as povided in Sect. 8. Loentz Gauge In ou pevious desciption of the electodynamic field we had intoduced the scala and vecto potential V, t and A, t, espectively, and had chosen the so-called Coulomb gauge 8., i.e., A =, fo these potentials. This gauge is not Loentz-invaiant and we will adopt hee anothe gauge, namely, t V, t + A, t =..96 The Loentz-invaiance of this gauge, the so-called Loentz gauge, can be demonstated eadily using the 4-vecto notation.69 fo the electodynamic potential and the 4-vecto deivative.77 which allow one to expess.96 in the fom µ A µ =..97 We have poven aleady that µ is a contavaiant 4-vecto. If we can show that A µ defined in.69 is, in fact, a contavaiant 4-vecto then the l.h.s. in.97 and, equivalently, in.96 is a scala and, hence, Loentz-invaiant. We will demonstate now the 4-vecto popety of A µ.

14 3 Relativistic Quantum Mechanics Tansfomation Popeties of J µ and A µ The chage density ρ, t and cuent density J, t ae known to obey the continuity popety t ρ, t + J, t =.98 which eflects the pinciple of chage consevation. This pinciple should hold in any fame of efeence. Equation.98 can be witten, using.77 and.68, µ J µ x µ =..99 Since this equation must be tue in any fame of efeence the ight hand side must vanish in all fames, i.e., must be a scala. Consequently, also the l.h.s. of.99 must be a scala. Since µ tansfoms like a covaiant 4-vecto, it follows that J µ, in fact, has to tansfom like a contavaiant 4-vecto. We want to deive now the diffeential equations which detemine the 4-potential A µ in the Loentz gauge.97 and, theeby, pove that A µ is, in fact, a 4-vecto. The espective equation fo A = V can be obtained fom Eq Using t A, t = t A, t togethe with.96, i.e., A, t = t V, t, one obtains t V, t V, t = 4πρ, t.. Similaly, one obtains fo A, t fom 8.7 using the identity 8.8 and, accoding to.96, A, t = t V, t t A, t A, t = 4 π J, t.. Combining equations.,., using.83 and.69, yields µ µ A ν x σ = 4 π J ν x σ.. In this equation the.h.s. tansfoms like a 4-vecto. The l.h.s. must tansfom likewise. Since µ µ tansfoms like a scala one can conclude that A ν x σ must tansfom like a 4-vecto. The Field Tenso The electic and magnetic fields can be collected into an anti-symmetic 4 4 tenso E x E y E z F µν = E x B z B y E y B z B x..3 E z B y B x Altenatively, this can be stated F k = F k = E k, F mn = ɛ mnl B l, k, l, m, n =,, 3.4 whee ɛ mnl = ɛ mnl is the totally anti-symmetic thee-dimensional tenso defined in 5.3. One can eadily veify, using 8.6 and 8.9, that F µν can be expessed though the potential A µ in.69 and µ in.8 as follows F µν = µ A ν ν A µ..5

15 .3: Relativistic Electodynamics 3 The elationships.3,.4 establishe the tansfomation behaviou of E, t and B, t. In a new fame of efeence holds F µν = L µ α L ν β F αβ.6 In case that the Loentz tansfomation L µ ν is given by.6 o, equivalently, by.63, one obtains E x Ey v B z Ez+v B y v v F µν = E x B z v E y v E y v B z v E z +v B y v B y +v E z v B z v E y v B x B y+v E z B v x.7 Compaision with F µν = E x E y E z E x B z B y E y B z B x E z B y B x.8 yields then the expessions fo the tansfomed fields E and B. The esults can be put into the moe geneal fom E = E, E = E + v B v B = B, B = B v E v.9. whee E, B and E, B ae, espectively, the components of the fields paallel and pependicula to the velocity v which detemines the Loentz tansfomation. These equations show that unde Loentz tansfomations electic and magnetic fields convet into one anothe. Maxwell Equations in Loentz-Invaiant Fom One can expess the Maxwell equations in tems of the tenso F µν in Loentz-invaiant fom. Noting µ F µν = µ µ A ν µ ν A µ = µ µ A ν ν µ A µ = µ µ A ν,. whee we used.5 and.97, one can conclude fom. µ F µν = 4π J ν.. One can eadily pove that this equation is equivalent to the two inhomogeneous Maxwell equations 8., 8.. Fom the definition.5 of the tenso F µν one can conclude the popety σ F µν + µ F νσ + ν F σµ =.3 which can be shown to be equivalent to the two homogeneous Maxwell equations 8.3, 8.4.

16 3 Relativistic Quantum Mechanics Loentz Foce One impotant popety of the electomagnetic field is the Loentz foce acting on chaged paticles moving though the field. We want to expess this foce though the tenso F µν. It holds fo a paticle with 4-momentum p µ as given by.67 and chage q dp µ dτ = q m p ν F µν.4 whee d/dτ is given by.86. We want to demonstate now that this equation is equivalent to the equation of motion 8.5 whee p = m v/ v. To avoid confusion we will employ in the following fo the enegy of the paticle the notation E = m/ v [see.87] and etain the definition E fo the electic field. The µ = component of.4 eads then, using.4, o with.86 de dτ = q m p E.5 de dt Fom this one can conclude, employing.93, = q E p E..6 de dt = d p dt = q p E.7 This equation follows, howeve, also fom the equation of motion 8.5 taking the scala poduct with p p d p = q p dt E.8 whee we exploited the fact that accoding to p = m v/ v holds p v. Fo the spatial components, e.g., fo µ =,.4 eads using.3 dp x dτ = q m EE x + p y B z p z B y..9 Employing again.86 and.67, i.e., E = m/ v, yields dp x dt = q [ E x + v B x ]. which is the x-component of the equation of motion 8.5. We have, hence, demonstated that.4 is, in fact, equivalent to 8.5. The tem on the.h.s. of. is efeed to as the Loentz foce. Equation.4, hence, povides an altenative desciption of the action of the Loentz foce.

17 .4: Function Space Repesentation of Loentz Goup 33.4 Function Space Repesentation of Loentz Goup In the following it will be equied to decibe the tansfomation of wave functions unde Loentz tansfomations. In this section we will investigate the tansfomation popeties of scala functions ψx µ, ψ C 4. Fo such functions holds in the tansfomed fame ψ L µ νx ν = ψx µ. which states that the function values ψ x µ at each point x µ in the new fame ae identical to the function values ψx µ in the old fame taken at the same space time point x µ, i.e., taken at the pais of points x µ = L µ νx ν, x µ. We need to emphasize that. coves solely the tansfomation behaviou of scala functions. Functions which epesent 4-vectoso othe non-scala entities, e.g., the chage-cuent density in case of Sect..3 o the bi-spino wave function of electon-positon pais in Sect..7, obey a diffeent tansfomation law. We like to expess now ψ x µ in tems of the old coodinates x µ. Fo this pupose one eplaces x µ in. by L µ ν xν and obtains ψ x µ = ψl µ ν xν.. This esult gives ise to the definition of the function space epesentation ρl µ ν of the Loentz goup ρl µ νψx µ def = ψl µ ν xν..3 This definition coesponds closely to the function space epesentation 5.4 of SO3. In analogy to the situation fo SO3 we seek an expession fo ρl µ ν in tems of an exponential opeato and tansfomation paametes ϑ, w, i.e., we seek an expession which coesponds to.5 fo the natual epesentation of the Loentz goup. The esulting expession should be a genealization of the function space epesentation 5.48 of SO3, in as fa as SO3, is a genealization otation + boosts of the goup SO3. We will denote the intended epesentation by L ϑ, w def = ρl µ ν ϑ, ϑ w = ρ e J + w K.4 which we pesent in the fom L ϑ, w = exp ϑ J + w K..5 In this expession J = J, J, J 3 and K = K, K, K 3 ae the geneatos of L ϑ, w which coespond to the geneatos J k and K k in.47, and which can be constucted following the pocedue adopted fo the function space epesentation of SO3. Howeve, in the pesent case we exclude the facto i [cf and.5]. Accodingly, one can evaluate J k as follows J k = lim ϑ k [ ] ρ e ϑ kj k ϑ.6 and K k K k = lim w k w [ ρ e w k K k ]..7

18 34 Relativistic Quantum Mechanics One obtains J = x 3 x 3 ; K = x + x which we like to demonstate fo J and K. In ode to evaluate.6 fo J we conside fist J = x 3 x 3 ; K = x + x J 3 = x x ; K 3 = x 3 + x 3.8 e ϑ J = e ϑ J = cosϑ sinϑ sinϑ cosϑ.9 which yields fo small ϑ ρ e ϑ J ψx µ = ψx, x, cosϑ x + sinϑ x 3, sinϑ x + cosϑ x 3 = ψx µ + ϑ x 3 x 3 ψx µ + Oϑ..3 This esult, obviously, epoduces the expession fo J in.8. One can detemine similaly K stating fom e w K = e w K = coshw sinhw sinhw coshw..3 This yields fo small w ρ e w K ψx µ = ψcoshw x + sinhw x, sinhw x + coshw x, x, x 3 = ψx µ + w x + x ψx µ + Ow.3 and, obviously, the expession fo K in.6. The geneatos J, K obey the same Lie algeba.49 as the geneatos of the natual epesentation, i.e. [ J k, J l ] = ɛ klm J m [ K k, K l ] = ɛ klm J m [ J k, K l ] = ɛ klm K m..33 We demonstate this fo thee cases, namely [J, J ] = J 3, [K, K ] = J 3, and [J, K ] = K 3 : [ J, J ] = [x 3 x 3, x 3 x 3 ] = [x 3, x 3 ] [x 3, x 3 ] = x + x = J 3,.34

19 .4: Function Space Repesentation of Loentz Goup 35 [ K, K ] = [x + x, x + x ] = [x, x ] [x, x ] = x + x = J 3,.35 [ J, K ] = [x 3 x 3, x + x ] = [x 3, x ] [x 3, x ] One-Dimensional Function Space Repesentation = x 3 + x 3 = K The exponential opeato.5 in the case of a one-dimensional tansfomation of the type Lw 3 = exp w 3 K 3,.37 whee K 3 is given in.8, can be simplified consideably. Fo this pupose one expesses K 3 in tems of hypebolic coodinates R, Ω which ae connected with x, x 3 as follows a elationship which can also be stated x = R coshω, x 3 = R sinhω.38 R = { + x x 3 if x x x 3 if x <.39 and tanhω = x3 x, x cothω = x 3..4 The tansfomation to hypebolic coodinates closely esembles the tansfomation to adial coodinates fo the geneatos of SO3 in the function space epesentation [cf. Eqs ]. In both cases the adial coodinate is the quantity conseved unde the tansfomations, i.e., x + x + x 3 in the case of SO3 and x x 3 in case of tansfomation.37. In the following we conside solely the case x. The elationships.39,.4 allow one to expess the deivatives, 3 in tems of R, Ω. We note and The chain ule yields then = R x 3 = R x 3 R x Ω x 3 = Ω x = R + R + = x R, R x 3 = x R Ω tanhω tanhω x 3 = cosh Ω x Ω cothω.4 cothω x = sinh Ω x 3..4 Ω x Ω = x R R sinh Ω x 3 Ω Ω x 3 Ω = x3 R R + cosh Ω x Ω..43

20 36 Relativistic Quantum Mechanics Inseting these esults into the definition of K 3 in.8 yields K 3 = x 3 + x 3 = Ω..44 The action of the exponential opeato.37 on a function fω C is then that of a shift opeato Lw 3 fω = exp w 3 fω = fω + w Ω.5 Klein Godon Equation In the following Sections we will povide a heuistic deivation of the two most widely used quantum mechanical desciptions in the elativistic egime, namely the Klein Godon and the Diac equations. We will povide a deivation of these two equations which stem fom the histoical development of elativistic quantum mechanics. The histoic oute to these two equations, howeve, is not vey insightful, but cetainly is shot and, theefoe, extemely useful. Futhe below we will povide a moe systematic, epesentation theoetic teatment. Fee Paticle Case A quantum mechanical desciption of a elativistic fee paticle esults fom applying the coespondence pinciple, which allows one to eplace classical obsevables by quantum mechanical opeatos acting on wave functions. In the position epesentation the coespondence pinciple states E = Ê = i t p = ˆ p = i.46 which, in 4-vecto notation eads p µ = ˆp µ = i t, = i µ ; p µ = ˆp µ = i t, = i µ..47 Applying the coespondence pinciple to.9 one obtains the wave equation µ µ ψx ν = m ψx ν.48 o µ µ + m ψx ν =..49 whee ψx µ is a scala, complex-valued function. The latte popety implies that upon change of efeence fame ψx µ tansfoms accoding to.,.. The patial diffeential equation.5 is called the Klein-Godon equation. In the following we will employ so-called natual units = c =. In these units the quantities enegy, momentum, mass, length, and time all have the same dimension. In natual units the Klein Godon equation.5 eads µ µ + m ψx µ =.5

21 .5: Klein Godon Equation 37 o t + m ψx µ =..5 One can notice immeadiately that.5 is invaiant unde Loentz tansfomations. This follows fom the fact that µ µ and m ae scalas, and that as postulated ψx µ is a scala. Unde Loentz tansfomations the fee paticle Klein Godon equation.5 becomes µ µ + m ψ x µ =.5 which has the same fom as the Klein Godon equation in the oiginal fame. Cuent 4-Vecto Associated with the Klein-Godon Equation As is well-known the Schödinge equation of a fee paticle is associated with a consevation law fo paticle pobability whee i t ψ, t = m ψ, t.53 t ρ S, t + j S, t =.54 ρ S, t = ψ, t ψ, t.55 descibes the positive definite pobability to detect a paticle at position at time t and whee j S, t = mi [ ψ, t ψ, t ψ, t ψ, t ].56 descibes the cuent density connected with motion of the paticle pobability distibution. To deive this consevation law one ewites the Schödinge equation in the fom i t m ψ = and consides Im [ψ i t ] m ψ =.57 which is equivalent to.54. In ode to obtain the consevation law connected with the Klein Godon equation.5 one consides Im [ ψ µ µ + m ψ ] =.58 which yields which coesponds to whee ψ t ψ ψ t ψ ψ ψ + ψ ψ = t ψ t ψ ψ t ψ + ψ ψ ψ ψ =.59 t ρ KG, t + j KG, t =.6 ρ KG, t = i m ψ, t t ψ, t ψ, t t ψ, t j KG, t = mi ψ, t ψ, t ψ, t ψ, t..6

22 38 Relativistic Quantum Mechanics This consevation law diffes in one impotant aspect fom that of the Schödinge equation.54, namely, in that the expession fo ρ KG is not positive definite. When the Klein-Godon equation had been initially suggested this lack of positive definiteness woied physicists to a degee that the Klein Godon equation was ejected and the seach fo a Loentz invaiant quantum mechanical wave equation continued. Today, the Klein-Godon equation is consideed as a suitable equation to descibe spin paticles, fo example pions. The pope intepetation of ρ KG, t, it had been ealized late, is actually that of a chage density, not of paticle pobability. Solution of the Fee Paticle Klein Godon Equation Solutions of the fee paticle Klein Godon equation ae ψx µ = N e ip µx µ = N e i p E o t..6 Inseting this into the Klein Godon equation.5 yields E o p m ψ, t =.63 which esults in the expected [see.93] dispesion elationship connecting E, p, m The coesponding enegy is E = m + p o..64 E o p o, ± = ± m + p o.65 This esult togethe with.6 shows that the solutions of the fee paticle Klein-Godon e- quation.5 ae actually detemined by p o and by the choice of sign ±. We denote this by summaizing the solutions as follows µ µ + m ψ o p, λ x µ =.66 ψ o p, λ x µ = N λ,p e i p λe o pt E o p = m + p o, λ = ± The spectum of the Klein Godon equation.5 is a continuum of positive enegies E m, coesponding to λ = +, and of negative enegies E m, coesponding to λ =. The density ρ KG p, λ associated with the coesponding wave functions ψ o p, λ x µ accoding to.6 and.66 is ρ KG p, λ = λ E o p m ψ o p, λ x µ ψ o p, λ x µ.67 which is positive fo λ = + and negative fo λ =. The pope intepetation of the two cases is that the Klein Godon equation descibes paticles as well as anti-paticles; the anti-paticles cay a chage opposite to that of the associated paticles, and the density ρ KG p, λ actually descibes chage density athe than pobability. Geneating a Solution Though Loentz Tansfomation A paticle at est, i.e., with p =, accoding to?? is decibed by the independent wave function ψ o p =, λ x µ = N e iλmt, λ = ±..68

23 .6: Klein Godon Equation with Electomagnetic Field 39 We want to demonstate now that the wave functions fo p in?? can be obtained though appopiate Loentz tansfomation of.68. Fo this pupose we conside the wave function fo a paticle moving with momentum velocity v in the diection of the x 3 axis. Such wave function should be geneated by applying the Loentz tansfomation in the function space epesentation.45 choosing p m = sinhw3. This yields, in fact, fo the wave function.68, using.38 to eplace t = x by hypebolic coodinates R, Ω, Lw 3 ψ o p =, λ x µ = exp w 3 N e iλmrcoshω Ω = N e iλmrcoshω+w3..69 The addition theoem of hypebolic functions coshω+w 3 = coshω coshw 3 + sinhω sinhw 3 allows us to ewite the exponent on the.h.s. of.69 iλ m coshw 3 R coshω iλ m sinhw 3 R sinhω..7 The coodinate tansfomation.38 and the elationships.6 yield fo this expession m m v iλ v x iλ v x3..7 One can intepet then fo λ = +, i.e., fo positive enegy solutions, p = mv/ v.7 as the momentum of the paticle elative to the moving fame and m m = v v = m + m v v = m + p = E o p.73 as the enegy [c.f..66] of the paticle. In case of λ = + one obtains finally Lw 3 ψ o p =, λ = + x µ = N e ipx3 E o px.74 which agees with the expession given in.66. In case of λ =, i.e., fo negative enegy solutions, one has to intepete p = mv/ v.75 as the momentum of the paticle and one obtains Lw 3 ψ o p =, λ = x µ = N e ipx3 + E o px Klein Godon Equation fo Paticles in an Electomagnetic Field We conside now the quantum mechanical wave equation fo a spin paticle moving in an electomagnetic field descibed by the 4-vecto potential A µ x µ = V, t, A, t ; A µ x µ = V, t, A, t.77

24 3 Relativistic Quantum Mechanics fee classical fee quantum classical paticle in quantum paticle in paticle field V, A paticle field V, A enegy E E qv i t i t qv momentum p p q A ˆ p = i ˆ p q A = ˆ π 4-vecto p µ p µ qa µ i µ i µ qa µ = π µ Table.: Coupling of a paticle of chage q to an electomagnetic field descibed by the 4-vecto potential A µ = V, A o A µ = V, A. Accoding to the so-called minimum coupling pinciple the pesence of the field is accounted fo by alteing enegy, momenta fo classical paticles and the espective opeatos fo quantum mechanical paticles in the manne shown. See also Eq..47. To obtain the appopiate wave equation we follow the deivation of the fee paticle Klein Godon equation above and apply again the coespondence pinciple to.93, albeit in a fom, which couples a paticle of chage q to an electomagnetic field descibed though the potential A µ x ν. Accoding to the pinciple of minimal coupling [see.69] one eplaces the quantum mechanical opeatos, i.e., i t and i in.5, accoding to the ules shown in Table.. Fo this pupose one wites the Klein-Godon equation.5 g µν i µ i ν + m ψx µ =..78 Accoding to the eplacements in Table. this becomes which can also be witten g µν i µ qa µ i ν A ν ψx µ = m ψx µ.79 g µν π µ π ν ; m ψx µ =..8 In tems of space-time deivatives this eads [ i t qv, t ψ, t = i qa, ] t + m ψ, t..8 Non-Relativistic Limit of Fee Paticle Klein Godon Equation In ode to conside futhe the intepetation of the positive and negative enegy solutions of the Klein Godon equation one can conside the non-elativistic limit. Fo this pupose we split-off a facto exp imt which descibes the oscillations of the wave function due to the est enegy, and focus on the emaining pat of the wave function, i.e., we define ψ, t = e imt Ψ, t,.8 and seek an equation fo Ψ, t. We will also assume, in keeping withnthe non-elativistic limit, that the mass m of the paticle, i.e., it s est enegy, is much lage than all othe enegy tems, in

25 .6: Klein Godon Equation with Electomagnetic Field 3 paticula, lage than i t Ψ/Ψ and alge than qv, i.e., i tψ << m, q V << m..83 Ψ The tem on the l.h.s. of.8 can then be appoximated as follows: i t qv e imt Ψ = i t qv me imt Ψ + e imt i t Ψ qv e imt Ψ = m e imt Ψ + me imt i t Ψ qv e imt Ψ +me imt i t Ψ e imt Ψ qv e imt i t Ψ me imt qv Ψ e imt i t qv Ψ + q V e imt Ψ m e imt Ψ mqv e imt Ψ me imt i t Ψ.84 whee we neglected all tems which did not contain factos m. The appoximation is justified on the gound of the inequalities.83. The Klein-Godon equation.8 eads then [ [ˆ p qa, i t Ψ, t = ] t] + qv, t Ψ, t.85 m This is, howeve, identical to the Schödinge equation. of a non-elativistic spin- paticle moving in an electomagnetic field. Pionic Atoms To apply the Klein Godon equation.8 to a physical system we conside pionic atoms, i.e., atoms in which one o moe electons ae eplaced by π mesons. This application demonstates that the Klein Godon equation descibes spin zeo paticles, e.g., spin- mesons. To manufactue pionic atoms, π mesons ae geneated though inelastic poton poton scatteing p + p p + p + π + π +,.86 then ae slowed down, filteed out of the beam and finally fall as slow pions onto elements fo which a pionic vaiant is to be studied. The pocess of π meson captue involves the so-called Auge effect, the binding of a negative chage typically an electon while at the same time a lowe shell electon is being emitted π + atom atom e + π + e..87 We want to investigate in the following a desciption of a stationay state of a pionic atom involving a nucleus with chage +Ze and a π meson. A stationay state of the Klein Godon equation is descibed by a wave function ψx µ = ϕ e iɛt..88 Inseting this into.8 yields we assume now that the Klein Godon equation descibes a paticle with mass m π and chage e fo qv, t = Ze and A, t [ ] ɛ + Ze + m π ϕ =..89

26 3 Relativistic Quantum Mechanics Because of the adial symmety of the Coulomb potential we expess this equation in tems of spheical coodinates, θ, φ. The Laplacian is = + sin θ θsinθ θ + sin θ φ = ˆL..9 With this expession and afte expanding ɛ + Ze one obtains d d ˆL Z e 4 + ɛze + ɛ m π The opeato ˆL in this equation suggests to choose a solution of the type ϕ = R l φ =..9 Y lm θ, φ.9 whee the functions Y lm θ, φ ae spheical hamonics, i.e., the eigenfunctions of the opeato ˆL in.9 ˆL Y lm θ, φ = l l + Y lm θ, φ leads then to the odinay diffeential equation d d ll + Z e 4 + ɛze + ɛ m π R l =..94 Bound state solutions can be obtained eadily noticing that this equation is essentially identical to that posed by the Coulomb poblem potential Ze fo the Schödinge equation d ll + d + m πze + m π E R l =.95 The latte poblem leads to the well-known spectum E n = m π Ze n ; n =,,... ; l =,,... n..96 In this expession the numbe n defined though n = n l.97 counts the numbe of nodes of the wave function, i.e., this quantity definitely must be an intege. The similaity of.94 and.95 can be made complete if one detemines λ such that λl λl + = l l + Z e The suitable choice is λl = + l + Z e 4.99

27 .6: Klein Godon Equation with Electomagnetic Field 33 and one can wite.94 d λl λl + d + ɛze + ɛ m π R l =.. The bound state solutions of this equation should coespond to ɛ values which can be obtained fom.96 if one makes the eplacement One obtains E ɛ m π m π, l λl, e e ɛ m π.. ɛ m π m π m π Z e 4 ɛ m = π n + λl +.. Solving this fo ɛ choosing the oot which endes ɛ m π, i.e., which coesponds to a bound state yields ɛ = m π + Z e 4 n + λl+ ; n =,,... ; l =,, E KG n, l, m = Using.97,.99 and defining E KG = ɛ esults in the spectum m π + Z e 4 n l + l+ Z e 4 n =,,... l =,,..., n m = l, l +,..., +l.4 In ode to compae this esult with the spectum of the non-elativistic hydogen-like atom we expand in tems of the fine stuctue constant e to ode Oɛ 8. Intoducing α = Z e 4 and β = l +.4 eads + and one obtains the seies of appoximations + α n β + β α α n β + β α α n α β + Oα α n α β n + Oα

28 34 Relativistic Quantum Mechanics + α + α + Oα n βn α n + α βn 3 α n α 8n 4 + Oα 3 α βn 3 + α 8n 4 + α 4n 4 + Oα3..6 Fom this esults fo.4 [ E KG n, l, m m mz e 4 n mz4 e 8 n 3 l + ] 3 + OZ 6 e..7 4n Hee the fist tem epesents the est enegy, the second tem the non-elativistic enegy, and the thid tem gives the leading elativistic coection. The latte tem agees with obsevations of pionic atoms, howeve, it does not agee with obsevations of the hydogen spectum. The latte spectum shows, fo example, a splitting of the six n =, l = states into goups of two and fou degeneate states. In ode to descibe electon specta one must employ the Loentz-invaiant wave equation fo spin- paticles, i.e., the Diac equation intoduced below. It must be pointed out hee that ɛ does not denote enegy, but in the pesent case athe the negative of the enegy. Also, the π meson is a pseudoscala paticle, i.e., the wave function changes sign unde eflection..7 The Diac Equation Histoically, the Klein Godon equation had been ejected since it did not yield a positive-definite pobability density, a featue which is connected with the nd ode time deivative in this equation. This deivative, in tun, aises because the Klein Godon equation, though the coespondence pinciple, is elated to the equation E = m + p of the classical theoy which involves a tem E. A moe satisfactoy Loentz invaiant wave equation, i.e., one with a positive-definite density, would have only a fist ode time deivative. Howeve, because of the equivalence of space and time coodinates in the Minkowski space such equation necessaily can only have then fist ode deivatives with espect to spatial coodinates. It should featue then a diffeential opeato of the type D = iγ µ µ. Heuistic Deivation Stating fom the Klein-Godon Equation An obvious stating point fo a Loentz-invaiant wave equation with only a fist ode time deivative is E = ± m + p. Application of the coespondence pinciple.46 leads to the wave equation i t Ψ, t = ± m Ψ, t..8 These two equation can be combined i t + m i t m Ψ, t.9

29 .7: Diac Equation 35 which, in fact, is identical to the two equations.8. Equations.8,.9, howeve, ae unsatisfactoy since expansion of the squae oot opeato involves all powes of the Laplace opeato, but not an opeato i γ as suggested by the pinciple of elativity equivalence of space and time. Many attempts wee made by theoetical physicists to lineaize the squae oot opeato in.8,.9, but fo a long time to no avail. Finally, Diac succeeded. His solution to the poblem involved an ingenious step, namely, the ealization that the lineaization can be caied out only if one assumes a 4-dimensional epesentation of the coefficients γ µ. Initially, it was assumed that the 4-dimensional space intoduced by Diac could be linked to 4- vectos, i.e., quantities with the tansfomation law.66. Howeve, this was not so. Instead, the 4-dimensionsional epesentation discoveed by Diac involved new physical popeties, spin- and anti-paticles. The discovey by Diac, achieved though a beautiful mathematical theoy, stengthens the believe of many theoetical physicists today that the popeties of physical matte ultimately deive fom a, yet to be discoveed, beautiful mathematical theoy and that, theefoe, one oute to impotant discoveies in physics is the ceation of new mathematical desciptions of natue, these desciptions ultimately meging with the tue theoy of matte. Popeties of the Diac Matices Let us now tace Diac s steps in achieving the lineaization of the squae oot opeato in.8. Stating point is to boldly factoize, accoding to.9, the opeato of the Klein Godon equation µ µ + m = P + m P m. whee P = iγ µ µ.. Obviously, this would lead to the two wave equations P mψ = and P + mψ = which have a fist ode time deivative and, theefoe, ae associated with a positive-definite paticle density. We seek to identify the coefficients γ µ. Inseting. into. yields g µν µ ν m = iγ µ µ + miγ µ µ m = γ µ γ ν µ ν m = γµ γ ν µ ν + γ ν γ µ ν µ m = γµ γ ν + γ ν γ µ µ ν m. whee we have changed dummy summation indices, exploited µ ν = ν µ, but did not commute the, so fa, unspecified algebaic objects γ µ and γ ν. Compaing the left-most and the ight-most side of the equations above one can conclude the following popety of γ µ γ µ γ ν + γ ν γ µ = [ γ µ, γ ν ] + = g µν.3 We want to detemine now the simplest algebaic ealization of γ µ. It tuns out that no 4-vecto of eal o complex coefficients can satisfy these conditions. In fact, the quantities γ, γ, γ, γ 3 can only be ealized by d d matices equiing that the wave function Ψx µ is actually a d dimensional vecto of functions ψ x µ, ψ x µ,... ψ d x µ. Fo µ = ν condition.3 eads { γ µ µ = =..4 µ =,, 3

30 36 Relativistic Quantum Mechanics Fom this follows that γ has eal eigenvalues ± and γ j, j =,, 3 has imaginay eigenvalues ±i. Accodingly, one can impose the condition Fo µ ν.3 eads γ is hemitian ; γ j, j =,, 3 ae anti-hemitian..5 γ µ γ ν = γ ν γ µ,.6 i.e., the γ µ ae anti-commuting. Fom this one can conclude fo the deteminants of γ µ detγ µ γ ν = det γ ν γ µ = d detγ ν γ µ = d detγ µ γ ν..7 Obviously, as long as detγ µ the dimension d of the squae matices γ µ must be even so that d =. Fo d = thee exist only thee anti-commuting matices, namely the Pauli matices σ, σ, σ 3 fo which, in fact, holds σ j = ; σ j σ k = σ k σ j fo j k..8 The Pauli matices allow one, howeve, to constuct fou matices γ µ fo the next possible dimension d = 4. A pope choice is γ = ; γ j σ j = σ j,.9 Using popety.8 of the Pauli matices one can eadily pove that condition.3 is satisfied. We will ague futhe below that the choice f γ µ, except fo similaity tasnfomations, is unique. The Diac Equation Altogethe we have shown that the Klein Godon equation can be factoized fomally iγ µ µ + m iγ µ µ m Ψx µ =. whee Ψx µ epesents a 4-dimensional wave function, athe than a scala wave function. Fom this equation one can conclude that also the following should hold which is the celebated Diac equation. The Adjoint Diac Equation iγ µ µ m Ψx µ =. The adjoint equation is Ψ x µ iγ µ µ + m =. whee we have defined Ψ = ψ, ψ, ψ 3, ψ 4 and whee µ denotes the diffeential opeato µ opeating to the left side, athe than to the ight side. One can eadily show using the hemitian

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

### MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary

PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC

### 1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

### AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom

Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### Chapter 4: Fluid Kinematics

Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

### The transport performance evaluation system building of logistics enterprises

Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics

### Lab #7: Energy Conservation

Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### Seshadri constants and surfaces of minimal degree

Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth

### 12.1. FÖRSTER RESONANCE ENERGY TRANSFER

ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

### Saturated and weakly saturated hypergraphs

Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 6-7 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### A Glossary Of Complex Envelope Vectoization And Its Working Principle

Complex Envelope Vectoization fo the solution of mid-high fequency acoustic poblems A. Sestiei Depatment of Mechanical and Aeospace Engineeing Univesity of Rome la Sapienza Pesentation layout - Low fequency

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

### Explicit, analytical solution of scaling quantum graphs. Abstract

Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 06459-0155, USA E-mail: ydabaghian@wesleyan.edu (Januay 6, 2003)

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### Strength Analysis and Optimization Design about the key parts of the Robot

Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

### (a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of

Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the

### Approximation Algorithms for Data Management in Networks

Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute

### Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo

Chapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment

### The Effect of Modified Gravity on Solar System Scales

The Effect of Modified Gavity on Sola System Scales Dane Pittock Physics Depatment Case Westen Reseve Univesity Cleveland, Ohio 44106 USA May 3, 013 Abstact Duing my senio poject, I have exploed the effects

### METHODOLOGICAL APPROACH TO STRATEGIC PERFORMANCE OPTIMIZATION

ETHODOOGICA APPOACH TO STATEGIC PEFOANCE OPTIIZATION ao Hell * Stjepan Vidačić ** Željo Gaača *** eceived: 4. 07. 2009 Peliminay communication Accepted: 5. 0. 2009 UDC 65.02.4 This pape pesents a matix

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

### Chapter 2 Modelling of Fluid Flow and Heat Transfer in Rotating-Disk Systems

Chapte 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems 2.1 Diffeential and Integal Equations 2.1.1 Diffeential Navie Stokes and Enegy Equations We will conside hee stationay axisymmetic

### A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

### CHAPTER 10 Aggregate Demand I

CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

### Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

### Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

### Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

### Financing Terms in the EOQ Model

Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

### The Binomial Distribution

The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

### CONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS

CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.

### Over-encryption: Management of Access Control Evolution on Outsourced Data

Ove-encyption: Management of Access Contol Evolution on Outsouced Data Sabina De Capitani di Vimecati DTI - Univesità di Milano 26013 Cema - Italy decapita@dti.unimi.it Stefano Paaboschi DIIMM - Univesità

### Converting knowledge Into Practice

Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

### Symmetric polynomials and partitions Eugene Mukhin

Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

### Give me all I pay for Execution Guarantees in Electronic Commerce Payment Processes

Give me all I pay fo Execution Guaantees in Electonic Commece Payment Pocesses Heiko Schuldt Andei Popovici Hans-Jög Schek Email: Database Reseach Goup Institute of Infomation Systems ETH Zentum, 8092

### Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

### Comparing Availability of Various Rack Power Redundancy Configurations

Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance the availability

### Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

### How To Find The Optimal Stategy For Buying Life Insuance

Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

### An Introduction to Omega

An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

### Modeling and Verifying a Price Model for Congestion Control in Computer Networks Using PROMELA/SPIN

Modeling and Veifying a Pice Model fo Congestion Contol in Compute Netwoks Using PROMELA/SPIN Clement Yuen and Wei Tjioe Depatment of Compute Science Univesity of Toonto 1 King s College Road, Toonto,

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

### CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS

9. Intoduction CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS In this chapte we show how Keple s laws can be deived fom Newton s laws of motion and gavitation, and consevation of angula momentum, and

### NUCLEAR MAGNETIC RESONANCE

19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic

### 30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research

RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 29-45) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This

### Excitation energies for molecules by Time-Dependent. based on Effective Exact Exchange Kohn-Sham potential

Excitation enegies fo molecules by Time-Dependent Density-Functional Theoy based on Effective Exact Exchange Kohn-Sham potential Fabio Della Sala National Nanotechnology Laboatoies Lecce Italy A. Göling

### Determining solar characteristics using planetary data

Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

### An Efficient Group Key Agreement Protocol for Ad hoc Networks

An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,

### Valuation of Floating Rate Bonds 1

Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

### Chapter 2. Electrostatics

Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.

### Structure and evolution of circumstellar disks during the early phase of accretion from a parent cloud

Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The

### Channel selection in e-commerce age: A strategic analysis of co-op advertising models

Jounal of Industial Engineeing and Management JIEM, 013 6(1):89-103 Online ISSN: 013-0953 Pint ISSN: 013-843 http://dx.doi.og/10.396/jiem.664 Channel selection in e-commece age: A stategic analysis of

### INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE

1 INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE ANATOLIY A. YEVTUSHENKO 1, ALEXEY N. KOCHEVSKY 1, NATALYA A. FEDOTOVA 1, ALEXANDER Y. SCHELYAEV 2, VLADIMIR N. KONSHIN 2 1 Depatment of

### Software Engineering and Development

I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining

### Solution Derivations for Capa #8

Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass

### MATHEMATICAL SIMULATION OF MASS SPECTRUM

MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec

### A Capacitated Commodity Trading Model with Market Power

A Capacitated Commodity Tading Model with Maket Powe Victo Matínez-de-Albéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

### Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment

Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability

### Nontrivial lower bounds for the least common multiple of some finite sequences of integers

J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

### The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

### Supplementary Material for EpiDiff

Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module

### Uncertain Version Control in Open Collaborative Editing of Tree-Structured Documents

Uncetain Vesion Contol in Open Collaboative Editing of Tee-Stuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecom-paistech.f Talel Abdessalem

### Instituto Superior Técnico Av. Rovisco Pais, 1 1049-001 Lisboa E-mail: virginia.infante@ist.utl.pt

FATIGUE LIFE TIME PREDICTIO OF POAF EPSILO TB-30 AIRCRAFT - PART I: IMPLEMETATIO OF DIFERET CYCLE COUTIG METHODS TO PREDICT THE ACCUMULATED DAMAGE B. A. S. Seano 1, V. I. M.. Infante 2, B. S. D. Maado

### It is required to solve the heat-condition equation for the excess-temperature function:

Jounal of Engineeing Physics and Themophysics. Vol. 73. No. 5. 2 METHOD OF PAIED INTEGAL EQUATIONS WITH L-PAAMETE IN POBLEMS OF NONSTATIONAY HEAT CONDUCTION WITH MIXED BOUNDAY CONDITIONS FO AN INFINITE