prevous ndex next Lnear Algebra for Quantum Mechancs Mchael Fowler 0/4/08 Introducton We ve seen that n quantum mechancs, the state of an electron n some potental s gven by a ψ x t, and physcal varables are represented by operators on ths wave wave functon (, ) functon, such as the momentum n the x-drecton p = / x The Schrödnger wave equaton s a lnear equaton, whch means that f ψ and ψ are solutons, then so s cψ + cψ, where c, c are arbtrary complex numbers Ths lnearty of the sets of possble solutons s true generally n quantum mechancs, as s the representaton of physcal varables by operators on the wave functons The mathematcal structure ths descrbes, the lnear set of possble states and sets of operators on those states, s n fact a lnear algebra of operators actng on a vector space From now on, ths s the language we ll be usng most of the tme To clarfy, we ll gve some defntons What s a Vector Space? The prototypcal vector space s of course the set of real vectors n ordnary three-dmensonal v, v, v measurng the space, these vectors can be represented by tros of real numbers ( ) components n the x, y and z drectons respectvely The basc propertes of these vectors are: x 3 any vector multpled by a number s another vector n the space, a( v, v, v ) ( av, av, av ) = ; 3 3 the sum of two vectors s another vector n the space, that gven by just addng the correspondng components together:( v + w, v + w, v + w ) 3 3 These two propertes together are referred to as closure : addng vectors and multplyng them by numbers cannot get you out of the space A further property s that there s a unque null vector ( 0,0,0 ) and each vector has an addtve nverse ( v, v, v ) whch added to the orgnal vector gves the null vector 3 Mathematcans have generalzed the defnton of a vector space: a general vector space has the propertes we ve lsted above for three-dmensonal real vectors, but the operatons of addton and multplcaton by a number are generalzed to more abstract operatons between more general enttes The operators are, however, restrcted to beng commutatve and assocatve
Notce that the lst of necessary propertes for a general vector space does not nclude that the vectors have a magntude that would be an addtonal requrement, gvng what s called a normed vector space More about that later To go from the famlar three-dmensonal vector space to the vector spaces relevant to quantum mechancs, frst the real numbers (components of the vector and possble multplyng factors) are to be generalzed to complex numbers, and second the three-component vector goes an n component vector The consequent n-dmensonal complex space s suffcent to descrbe the quantum mechancs of angular momentum, an mportant subject But to descrbe the wave functon of a partcle n a box requres an nfnte dmensonal space, one dmenson for each Fourer component, and to descrbe the wave functon for a partcle on an nfnte lne requres the set of all normalzable contnuous dfferentable functons on that lne Fortunately, all these generalzatons are to fnte or nfnte sets of complex numbers, so the mathematcans vector space requrements of commutatvty and assocatvty are always trvally satsfed We use Drac s notaton for vectors,, and call them kets, so, n hs language, f, belong to the space, so does c + c for arbtrary complex constants c, c Snce our vectors are made up of complex numbers, multplyng any vector by zero gves the null vector, and the addtve nverse s gven by reversng the sgns of all the numbers n the vector Clearly, the set of solutons of Schrödnger s equaton for an electron n a potental satsfes the requrements for a vector space: ψ ( x, t) s just a complex number at each pont n space, so only complex numbers are nvolved n formng cψ + cψ, and commutatvty, assocatvty, etc, follow at once Vector Space Dmensonalty The vectors,,3 are lnearly ndependent f mples c + c + c 3 = 0 3 c = c = c3 = 0 A vector space s n-dmensonal f the maxmum number of lnearly ndependent vectors n the space s n Such a space s often called V n (C), or V n (R) f only real numbers are used Now, vector spaces wth fnte dmenson n are clearly nsuffcent for descrbng functons of a contnuous varable x But they are well worth revewng here: as we ve mentoned, they are fne for descrbng quantzed angular momentum, and they serve as a natural ntroducton to the nfnte-dmensonal spaces needed to descrbe spatal wavefunctons
3 A set of n lnearly ndependent vectors n n-dmensonal space s a bass any vector can be wrtten n a unque way as a sum over a bass: V = v You can check the unqueness by takng the dfference between two supposedly dstnct sums: t wll be a lnear relaton between ndependent vectors, a contradcton Snce all vectors n the space can be wrtten as lnear sums over the elements of the bass, the sum of multples of any two vectors has the form: a V + b W = ( av + bw ) Inner Product Spaces The vector spaces of relevance n quantum mechancs also have an operaton assocatng a number wth a par of vectors, a generalzaton of the dot product of two ordnary threedmensonal vectors, ab = ab Followng Drac, we wrte the nner product of two ket vectors V, to ths W as WV Drac refers form as a bracket made up of a bra and a ket Ths means that each ket vector V has an assocated bra V For the case of a real n-dmensonal vector, V, dentcal but we requre for the more general case that V are WV = VW where * denotes complex conjugate Ths mples that for a ket ( v v ),, n the bra wll be ( v,, vn ) (Actually, bras are usually wrtten as rows, kets as columns, so that the nner product follows the standard rules for matrx multplcaton) Evdently for the n-dmensonal complex vector VV s real and postve except for the null vector: n VV = v For the more general nner product spaces consdered later we requre VV to be postve, except for the null vector (These requrements do restrct the classes of vector spaces we are consderng no Lorentz metrc, for example but they are all satsfed by the spaces relevant to nonrelatvstc quantum mechancs)
4 The norm of V s then defned by V = V V If V s a member of V n (C), so s av, for any complex number a We requre the nner product operaton to commute wth multplcaton by a number, so W ( a V ) = a W V The complex conjugate of the rght hand sde s a V W For consstency, the bra correspondng to the ket av must therefore be V a n any case obvous from the defnton of the bra n n complex dmensons gven above It follows that f j j V = v, W = w, then V W = v w Constructng an Orthonormal Bass: the Gram-Schmdt Process To have somethng better resemblng the standard dot product of ordnary three vectors, we need j = δ j, that s, we need to construct an orthonormal bass n the space There s a straghtforward procedure for dong ths called the Gram-Schmdt process We begn wth a lnearly ndependent set of bass vectors,,,3, We frst normalze by dvdng t by ts norm Call the normalzed vector I Now cannot be parallel to I, because the orgnal bass was of lnearly ndependent vectors, but n general has a nonzero component parallel to I, equal to I I, snce I s normalzed Therefore, the vector I I s perpendcular to I, as s easly verfed It s also easy to compute the norm of ths vector, and dvde by t to get II, the second member of the orthonormal bass Next, we take 3 and subtract off ts components n the drectons I and II, normalze the remander, and so on In an n-dmensonal space, havng constructed an orthonormal bass wth members, any vector V can be wrtten as a column vector,
5 V v v 0 = v =, where = and so on v n 0 The correspondng bra s V =, whch we wrte as a row vector wth the elements v complex conjugated, V ( v, v, vn ) = Ths operaton, gong from columns to rows and takng the complex conjugate, s called takng the adjont, and can also be appled to matrces, as we shall see shortly The reason for representng the bra as a row s that the nner product of two vectors s then gven by standard matrx multplcaton: w VW = ( v, v,, vn ) wn (Of course, ths only works wth an orthonormal base) The Schwartz Inequalty The Schwartz nequalty s the generalzaton to any nner product space of the result ab a b (or cos θ ) for ordnary three-dmensonal vectors The equalty sgn n that result only holds when the vectors are parallel To generalze to hgher dmensons, one mght just note that two vectors are n a two-dmensonal subspace, but an llumnatng way of understandng the nequalty s to wrte the vector a as a sum of two components, one parallel to b and one perpendcular to b The component parallel to b s just b( a b) / b, so the component perpendcular to b s the vector a = a b( a b) / b Substtutng ths expresson nto a a 0, the nequalty follows Ths same pont can be made n a general nner product space: f V, W are two vectors, then Z W W V = V W
6 s the component of V perpendcular to W, as s easly checked by takng ts nner product wth W Then Z Z 0 gves mmedately V W V W Lnear Operators A lnear operator A takes any vector n a lnear vector space to a vector n that space, A V = V, and satsfes wth c, c arbtrary complex constants ( ) A c V c V c A V c A V The dentty operator I s (obvously!) defned by: + = +, I V = V for all V For an n-dmensonal vector space wth an orthonormal bass,, n, snce any vector n the space can be expressed as a sum V = v, the lnear operator s completely determned by ts acton on the bass vectors ths s all we need to know It s easy to fnd an expresson for the dentty operator n terms of bras and kets Takng the nner product of both sdes of the equaton V = v, so V v wth the bra gves = V = v = V Snce ths s true for any vector n the space, t follows that that the dentty operator s just n I = Ths s an mportant result: t wll reappear n many dsguses To analyze the acton of a general lnear operator A, we just need to know how t acts on each bass vector Begnnng wth A, ths must be some sum over the bass vectors, and snce they are orthonormal, the component n the drecton must be just A That s,
7 n A = A = A, wrtng A A n = So f the lnear operator A actng on lnearty tells us that V v gves V = = v, that s, A V = V, the j j v = V = A V = v A j = v A j = v A j j, j, j where n the fourth step we just nserted the dentty operator Snce the s are all orthogonal, the coeffcent of a partcular on the left-hand sde of the equaton must be dentcal wth the coeffcent of the same on the rght-hand sde That s, v = Ajvj Therefore the operator A s smply equvalent to matrx multplcaton: v A A A n v v A A v = na na nan v v n n Evdently, then, applyng two lnear operators one after the other s equvalent to successve matrx multplcaton and, therefore, snce matrces do not n general commute, nor do lnear operators (Of course, f we hope to represent quantum varables as lnear operators on a vector space, ths has to be true the momentum operator p = d / dxcertanly doesn t commute wth x!) Projecton Operators It s mportant to note that a lnear operator appled successvely to the members of an orthonormal bass mght gve a new set of vectors whch no longer span the entre space To gve an example, the lnear operator appled to any vector n the space pcks out the vector s component n the drecton It s called a projecton operator The operator ( + ) projects a vector nto ts components n the subspace spanned by the vectors and, and so on f we extend the sum to be over the whole bass, we recover the dentty operator
Exercse: prove that the 8 n n matrx representaton of the projecton operator ( + ) has all elements zero except the frst two dagonal elements, whch are equal to one There can be no nverse operator to a nontrval projecton operator, snce the nformaton about components of the vector perpendcular to the projected subspace s lost The Adjont Operator and Hermtan Matrces As we ve dscussed, f a ket V n the n-dmensonal space s wrtten as a column vector wth n (complex) components, the correspondng bra s a row vector havng as elements the complex conjugates of the ket elements WV = VW then follows automatcally from standard matrx multplcaton rules, and on multplyng V by a complex number a to get av (meanng that each element n the column of numbers s multpled by a) the correspondng bra goes to V a = a V But suppose that nstead of multplyng a ket by a number, we operate on t wth a lnear operator What generates the parallel transformaton among the bras? In other words, f A V = V, what operator sends the bra V to V? It must be a lnear operator, because A s lnear, that s, f under A V V, V V and V 3 = V V, + then under A 3 3 V s requred to go to V = V + V Consequently, under the parallel bra transformaton we must have V V, V V and V3 V 3, the bra transformaton s necessarly also lnear Recallng that the bra s an n-element row vector, the most general lnear transformaton sendng t to another bra s an n n matrx operatng on the bra from the rght Ths bra operator s called the adjont of A, wrtten bra A That s, the ket A V has correspondng V A In an orthonormal bass, usng the notaton A to denote the bra correspondng to the ket A = A, say, ( ) A A j Aj ja = = = = A j j So the adjont operator s the transpose complex conjugate Important: for a product of two operators (prove ths!), A ( AB) = B A An operator equal to ts adjont A= A s called Hermtan As we shall fnd n the next lecture, Hermtan operators are of central mportance n quantum mechancs An operator equal to mnus ts adjont, A= A, s ant Hermtan (sometmes termed skew Hermtan) These two
9 operator types are essentally generalzatons of real and magnary number: any operator can be expressed as a sum of a Hermtan operator and an ant Hermtan operator, A = ( A+ A ) + ( A A ) The defnton of adjont naturally extends to vectors and numbers: the adjont of a ket s the correspondng bra, the adjont of a number s ts complex conjugate Ths s useful to bear n mnd when takng the adjont of an operator whch may be partally constructed of vectors and numbers, such as projecton-type operators The adjont of a product of matrces, vectors and numbers s the product of the adjonts n reverse order (Of course, for numbers the order doesn t matter) Untary Operators An operator s untary f UU= Ths mples frst that U operatng on any vector gves a vector havng the same norm, snce the new norm VUUV = VV Furthermore, nner products are preserved, WUUV = WV Therefore, under a untary transformaton the orgnal orthonormal bass n the space must go to another orthonormal bass Conversely, any transformaton that takes one orthonormal bass nto another one s a untary transformaton To see ths, suppose that a lnear transformaton A sends the members of the orthonormal bass (,,, n )to the dfferent orthonormal set (,,, n ), so A =,etc Then the vector V = v wll go to V = A V v, = havng the same norm, V V = V V = v A matrx elememt W V = W V = w * v, but also W V = W A A V That s, WV = W AAV for arbtrary kets V, W Ths s only possble f AA= I, so A s untary A untary operaton amounts to a rotaton (possbly combned wth a reflecton) n the space Evdently, snce UU=, the adjont U rotates the bass back t s the nverse operaton, and so UU = also, that s, U and U commute Determnants We revew n ths secton the determnant of a matrx, a functon closely related to the operator propertes of the matrx Let s start wth matrces: A a a = a a The determnant of ths matrx s defned by: det A= A = a a a a
0 Wrtng the two rows of the matrx as vectors: a = a a a a a (, ) (, ) R R = R R (R denotes row), det A = a a s just the area (wth approprate sgn) of the parallelogram havng the two row vectors as adjacent sdes: ( a, a ) ( a, a ) Ths s zero f the two vectors are parallel (lnearly dependent) and s not changed by addng any R R multple of a to a (because the new parallelogram has the same base and the same heght as the orgnal check ths by drawng) Let s go on to the more nterestng case of 3 3matrces: a a a A a a a a a a 3 = 3 3 3 33 The determnant of A s defned as det A = εjkaa ja3k where ε jk = 0 f any two are equal, + f jk = 3, 3 or 3 (that s to say, an even permutaton of 3) and f jk s an odd permutaton of 3 Repeated suffxes, of course, mply summaton here Wrtng ths out explctly, det A= a a a + a a a + a a a a a a a a a a a a 33 3 3 3 3 3 3 33 3 3
Just as n two dmensons, t s worth lookng at ths expresson n terms of vectors representng the rows of the matrx so R a = ( a, a, a3) R a = ( a, a, a3) a = ( a, a, a ) R 3 3 3 33 R a R R R R A = a, and we see that det A= ( a a ) a3 R a 3 Ths s the volume of the parallelopped formed by the three vectors beng adjacent sdes (meetng at one corner, the orgn) Ths parallelepped volume wll of course be zero f the three vectors le n a plane, and t s not changed f a multple of one of the vectors s added to another of the vectors That s to say, the determnant of a matrx s not changed f a multple of one row s added to another row Ths s because the determnant s lnear n the elements of a sngle row, a + λa a a det det det R R R R R R R a = a + λ a R R R a3 a3 a3 and the last term s zero because two rows are dentcal so the trple vector product vanshes A more general way of statng ths, applcable to larger determnants, s that for a determnant wth two dentcal rows, the symmetry of the two rows, together wth the antsymmetry of ε jk, ensures that the terms n the sum all cancel n pars Snce the determnant s not altered by addng some multple of one row to another, f the rows are lnearly dependent, one row could be made dentcally zero by addng the rght multples of
the other rows Snce every term n the expresson for the determnant has one element from each row, the determnant would then be dentcally zero For the three-dmensonal case, the lnear dependence of the rows means the correspondng vectors le n a plane, and the parallelepped s flat The algebrac argument generalzes easly to n n determnants: they are dentcally zero f the rows are lnearly dependent The generalzaton from 3 3 to n n determnants s that det A = ε jkaa ja3k becomes: det A = ε aa a a jk p j 3k np where jk p s summed over all permutatons of 3 n, and the ε symbol s zero f any two of ts suffxes are equal, + for an even permutaton and for an odd permutaton (Note: any permutaton can be wrtten as a product of swaps of neghbors Such a representaton s n general not unque, but for a gven permutaton, all such representatons wll have ether an odd number of elements or an even number) An mportant theorem s that for a product of two matrces A, B the determnant of the product s the product of the determnants, det AB = det A det B Ths can be verfed by brute force for matrces, and a proof n the general case can be found n any book on mathematcal physcs (for example, Byron and Fuller) It can also be proved that f the rows are lnearly ndependent, the determnant cannot be zero (Here s a proof: take an n n matrx wth the n row vectors lnearly ndependent Now consder the components of those vectors n the n dmensonal subspace perpendcular to (, 0,,0) These n vectors, each wth only n components, must be lnearly dependent, snce there are more of them than the dmenson of the space So we can take some combnaton of the rows below the frst row and subtract t from the frst row to leave the frst row (a, 0, 0,,0), and a cannot be zero snce we have a matrx wth n lnearly ndependent rows We can then subtract multples of ths frst row from the other rows to get a determnant havng zeros n the frst column below the frst row Now look at the n by n determnant to be multpled by a Its rows must be lnearly ndependent snce those of the orgnal matrx were Now proceed by nducton) To return to three dmensons, t s clear from the form of det A= aaa33 + aa3a3 + a3aa3 aa3a3 aaa33 a3aa3 that we could equally have taken the columns of A as three vectors, A = ( a C, a C, a C 3 ) n an C C C obvous notaton, det A = ( a a ) a3, and lnear dependence among the columns wll also ensure the vanshng of the determnant so, n fact, lnear dependence of the columns ensures lnear dependence of the rows
3 Ths, too, generalzes to n n: n the defnton of determnant det A = ε jk paa ja3k anp, the row suffx s fxed and the column suffx goes over all permssble permutatons, wth the approprate sgn but the same terms would be generated by havng the column suffxes kept n numercal order and allowng the row suffx to undergo the permutatons An Asde: Recprocal Lattce Vectors It s perhaps worth mentonng how the nverse of a 3 3 matrx operator can be understood n terms of vectors For a set of lnearly ndependent vectors ( a, a, a3), a recprocal set ( b, b, b3) can be defned by a a3 b = a a a 3 and the obvous cyclc defntons for the other two recprocal vectors We see mmedately that from whch t follows that the nverse matrx to a b = δ j j a A = a B = b b b 3 a R R C C C s R 3 ( ) (These recprocal vectors are mportant n x-ray crystallography, for example If a crystallne lattce has certan atoms at postons na + na + na 3 3, where n, n, n 3 are ntegers, the recprocal vectors are the set of normals to possble planes of the atoms, and these planes of atoms are the mportant elements n the dffractve x-ray scatterng) Egenkets and Egenvalues If an operator A operatng on a ket V gves a multple of the same ket, A V = λ V then V s sad to be an egenket (or, just as often, egenvector, or egenstate!) of A wth egenvalue λ Egenkets and egenvalues are of central mportance n quantum mechancs: dynamcal varables are operators, a physcal measurement of a dynamcal varable yelds an egenvalue of the operator, and forces the system nto an egenket In ths secton, we shall show how to fnd the egenvalues and correspondng egenkets for an operator A We ll use the notaton A a = a a for the set of egenkets a wth
4 correspondng egenvalues a (Obvously, n the egenvalue equaton here the suffx s not summed over) The frst step n solvng A V = λ V s to fnd the allowed egenvalues a Wrtng the equaton n matrx form: A λ A A n v A A λ v = 0 An Ann λ v n Ths equaton s actually tellng us that the columns of the matrx A λi are lnearly dependent! To see ths, wrte the matrx as a row vector each element of whch s one of ts columns, and the equaton becomes v C C C ( M, M,, Mn ) = 0 v n whch s to say C C C vm + vm = + vm = 0, the columns of the matrx are ndeed a lnearly dependent set We know that means the determnant of the matrx A λi s zero, n n A λ A A n A = 0 A n A λ A nn λ Evaluatng the determnant usng det A = ε jk paa ja3k anp gves an n th order polynomal n λ sometmes called the characterstc polynomal Any polynomal can be wrtten n terms of ts roots: C λ a λ a λ a n ( )( )( ) = 0
5 where the a s, the roots of the polynomal, and C s an overall constant, whch from nspecton of the determnant we can see to be ( ) n n (It s the coeffcent of λ ) The polynomal roots (whch we don t yet know) are n fact the egenvalues For example, puttng λ = a n the matrx, det A a I 0, whch means that A a I V = has a nontrval soluton V, and ( ) = ( ) ths s our egenvector a 0 Notce that the dagonal term n the determnant ( A λ)( A λ) ( Ann λ) leadng two orders n the polynomal ( ) n n n λ ( + + ) λ terms too) Equatng the coeffcent of generates the ( A A nn ), (and some lower order n λ n here wth that n ( ) ( λ )( λ )( λ ), n n a A = = = A = Tr a a a n Puttng λ = 0 n both the determnantal and the polynomal representatons (n other words, equatng the λ -ndependent terms), n = a = det A So we can fnd both the sum and the product of the egenvalues drectly from the determnant, and for a matrx ths s enough to solve the problem For anythng bgger, the method s to solve the polynomal equaton ( A λi) det = 0 to fnd the set of egenvalues, then use them to calculate the correspondng egenvectors Ths s done one at a tme Labelng the frst egenvalue found as a, the correspondng equaton for the components v of the egenvector a s A a A A n v A A a v = 0 An Ann a v n Ths looks lke n equatons for the n numbers v, but t sn t: remember the rows are lnearly dependent, so there are only n ndependent equatons However, that s enough to determne the ratos of the vector components v,, v n, then fnally the egenvector s normalzed The process s then repeated for each egnevalue (Extra care s needed f the polynomal has concdent roots we ll dscuss that case later)
6 Egenvalues and Egenstates of Hermtan Matrces For a Hermtan matrx, t s easy to establsh that the egenvalues are always real (Note: A basc postulate of Quantum Mechancs, dscussed n the next lecture, s that physcal observables are represented by Hermtan operators) Takng (n ths secton) A to be hermtan, A= A, and labelng the egenkets by the egenvalue, that s, A a = a a the nner product wth the bra a gves a A a = a a a But the nner product of the adjont equaton (rememberng A= A ) a A a a = wth a gves a A a a a a, = so a = a, and all the egenvalues must be real They certanly don t have to all be dfferent for example, the unt matrx I s Hermtan, and all ts egenvalues are of course But let s frst consder the case where they are all dfferent It s easy to show that the egenkets belongng to dfferent egenvalues are orthogonal If A a = a a A a a a =, take the adjont of the frst equaton and then the nner product wth a the nner product of the second equaton wth a :, and compare t wth a A a = a a a = a a a so a a = 0 unless the egenvalues are equal (If they are equal, they are referred to as degenerate egenvalues) Let s frst consder the nondegenerate case: A has all egenvalues dstnct The egenkets of A, approprately normalzed, form an orthonormal bass n the space Wrte v v v v n v v v vn a, and consder the matrx V = = = ( a a a n ) v v v v n n n nn
7 Now so ( n ) ( n n ) AV = A a a a = a a a a a a a a 0 0 a 0 a 0 = ( n n ) = a n 0 0 an V AV a a a a a a Note also that, obvously, V s untary: a 0 0 a 0 0 ( n ) = = a n 0 0 VV a a a We have establshed, then, that for a Hermtan matrx wth dstnct egenvalues (nondegenerate case), the untary matrx V havng columns dentcal to the normalzed egenkets of A dagonalzes A, that s, V AV s dagonal Furthermore, ts (dagonal) elements equal the correspondng egenvalues of A Another way of sayng ths s that the untary matrx V s the transformaton from the orgnal orthonormal bass n ths space to the bass formed of the normalzed egenkets of A Proof that the Egenvectors of a Hermtan Matrx Span the Space We ll now move on to the general case: what f some of the egenvalues of A are the same? In ths case, any lnear combnaton of them s also an egenvector wth the same egenvalue Assumng they form a bass n the subspace, the Gram Schmdt procedure can be used to make t orthonormal, and so part of an orthonormal bass of the whole space However, we have not actually establshed that the egenvectors do form a bass n a degenerate subspace Could t be that (to take the smplest case) the two egenvectors for the sngle egenvalue turn out to be parallel? Ths s actually the case for some matrces for example,, we need to prove t s not true for Hermtan matrces, and nor are the analogous 0 statements for hgher-dmensonal degenerate subspaces A clear presentaton s gven n Byron and Fuller, secton 47 We follow t here The procedure s by nducton from the case The general Hermtan matrx has the form
8 a b b c where a, c are real It s easy to check that f the egenvalues are degenerate, ths matrx becomes a real multple of the dentty, and so trvally has two orthonormal egenvectors Snce we already know that f the egenvalues of a Hermtan matrx are dstnct t can be dagonalzed by the untary transformaton formed from ts orthonormal egenvectors, we have establshed that any Hermtan matrx can be so dagonalzed To carry out the nducton process, we now assume any (n ) (n ) Hermtan matrx can be dagonalzed by a untary transformaton We need to prove ths means t s also true for an n n Hermtan matrx A (Recall a untary transformaton takes one complete orthonormal bass to another If t dagonalzes a Hermtan matrx, the new bass s necessarly the set of orthonormalzed egenvectors Hence, f the matrx can be dagonalzed, the egenvectors do span the n-dmensonal space) T Choose an egenvalue a of A, wth normalzed egenvector a = ( v, v,, v n ) (We put n T for transpose, to save the awkwardness of fllng the page wth a few column vectors) We construct a untary operator V by makng ths the frst column, then fllng n wth n other normalzed vectors to construct, wth a, an n-dmensonal orthonormal bass Now, snce A a = a a, the frst column of the matrx AV wll just be a a, and the rows of the matrx V V = wll be a followed by n normalzed vectors orthogonal to t, so the frst column of the matrx V AV wll be a followed by zeros It s easy to check that V AV s Hermtan, snce A s, so ts frst row s also zero beyond the frst dagonal term Ths establshes that for an n n Hermtan matrx, a untary transformaton exsts to put t n the form: V AV a 0 0 0 M Mn = 0 0 0 Mn M nn But we can now perform a second untary transformaton n the (n ) (n ) subspace orthogonal to a (ths of course leaves a nvarant), to complete the full dagonalzaton that s to say, the exstence of the (n ) (n ) dagonalzaton, plus the argument above, guarantees the exstence of the n n dagonalzaton: the nducton s complete
9 Dagonalzng a Hermtan Matrx As dscussed above, a Hermtan matrx s dagonal n the orthonormal bass of ts set of egenvectors: a, a,, an, snce a A a = a a a = a a a = a δ j j j j j j j If we are gven the matrx elements of A n some other orthonormal bass, to dagonalze t we need to rotate from the ntal orthonormal bass to one made up of the egenkets of A Denotng the ntal orthonormal bass n the standard fashon 0 0 0 0 0 = 0 = 0 = n = 0 0 0 0 th,, ( n place down), the elements of the matrx are Aj = Aj A transformaton from one orthonormal bass to another s a untary transformaton, as dscussed above, so we wrte t V V = U V Under ths transformaton, the matrx element W AV W A V = W U AU V So we can fnd the approprate transformaton matrx U by requrng that U AU be dagonal wth respect to the orgnal set of bass vectors (Transformng the operator n ths way, leavng the vector space alone, s equvalent to rotatng the vector space and leavng the operator alone Of course, n a system wth more than one operator, the same transformaton would have to be appled to all the operators) In fact, just as we dscussed for the nondegenerate (dstnct egenvalues) case, the untary matrx U we need s just composed of the normalzed egenkets of the operator A, And t follows as before that (,,, n ) U = a a a
( ) j 0 U AU = a a a = δ a j j j j (The repeated suffxes here are of course not summed over), a dagonal matrx If some of the egenvalues are the same, the Gram Schmdt procedure may be needed to generate an orthogonal set, as mentoned earler Functons of Matrces The same untary operator U that dagonalzes an Hermtan matrx A wll also dagonalze A, because = = U A U U AAU U AUU AU so a 0 0 0 0 a 0 0 U AU = 0 0 a 3 0 0 a n Evdently, ths same process works for any power of A, and formally for any functon of A expressble as a power seres, but of course convergence propertes need to be consdered, and ths becomes trcker on gong from fnte matrces to operators on nfnte spaces Commutng Hermtan Matrces From the above, the set of powers of an Hermtan matrx all commute wth each other, and have a common set of egenvectors (but not the same egenvalues, obvously) In fact t s not dffcult to show that any two Hermtan matrces that commute wth each other have the same set of egenvectors (after possble Gram Schmdt rearrangements n degenerate subspaces) If two n n Hermtan matrces A, B commute, that s, AB = BA, and A has a nondegenerate set of egenvectors A a = a a, then AB a = BAa = Ba a = aba, that s, Ba s an egenvector of A wth egenvalue a Snce A s nondegenerate, Ba must be some multple of, and we conclude that A, B have the same set of egenvectors a Now suppose A s degenerate, and consder the m msubspace S a spanned by the egenvectors a,, a,, of A havng egenvalue Applyng the argument n the paragraph above, a Ba,, Ba,, must also le n ths subspace Therefore, f we transform B wth the same untary transformaton that dagonalzed A, B wll not n general be dagonal n the subspace S a,
but t wll be what s termed block dagonal, n that f B operates on any vector n vector n S a S a t gves a B can be wrtten as two dagonal blocks: one m m these dagonal blocks, for example, for m =, n = 5:, one ( n m) ( n m), wth zeroes outsde b b b b 0 0 0 0 0 0 0 0 0 0 0 0 b b b 33 34 35 b b b 43 44 45 b b b 53 54 55 And, n fact, f there s only one degenerate egenvalue that second block wll only have nonzero terms on the dagonal: b b 0 0 0 b b 0 0 0 0 0 b3 0 0 0 0 0 b4 0 0 0 0 0 b 5 B therefore operates on two subspaces, one m-dmensonal, one (n m)-dmensonal, ndependently a vector entrely n one subspace stays there Ths means we can complete the dagonalzaton of B wth a untary operator that only operates on the m m block S Such an operator wll also affect the egenvectors of A, but that doesn t a matter, because all vectors n ths subspace are egenvectors of A wth the same egenvalue, so as far as A s concerned, we can choose any orthonormal bass we lke the bass vectors wll stll be egenvectors Ths establshes that any two commutng Hermtan matrces can be dagonalzed at the same tme Obvously, ths can never be true of noncommutng matrces, snce all dagonal matrces commute Dagonalzng a Untary Matrx Any untary matrx can be dagonalzed by a untary transformaton To see ths, recall that any matrx M can be wrtten as a sum of a Hermtan matrx and an ant Hermtan matrx, M + M M M M = + = A+ B
where both A, B are Hermtan Ths s the matrx analogue of wrtng an arbtrary complex number as a sum of real and magnary parts If A, B commute, they can be smultaneously dagonalzed (see the prevous secton), and therefore M can be dagonalzed Now, f a untary matrx s expressed n ths form U = A+ Bwth A, B Hermtan, t easly follows fromuu = U U = that A, B commute, so any untary matrx U can be dagonalzed by a untary transformaton More generally, f a matrx M commutes wth ts adjont M, t can be dagonalzed (Note: t s not possble to dagonalze M unless both A, B are smultaneously dagonalzed Ths follows from U AU, U BU beng Hermtan and anthermtan for any untary operator U, so ther off-dagonal elements cannot cancel each other, they must all be zero f M has been dagonalzed by U, n whch case the two transformed matrces U AU, U BU are dagonal, therefore commute, and so do the orgnal matrces A, B) It s worthwhle lookng at a specfc example, a smple rotaton of one orthonormal bass nto another n three dmensons Obvously, the axs through the orgn about whch the bass s rotated s an egenvector of the transformaton It s less clear what the other two egenvectors mght be or, equvalently, what are the egenvectors correspondng to a two dmensonal rotaton of bass n a plane? The way to fnd out s to wrte down the matrx and dagonalze t The matrx cosθ snθ U ( θ ) = snθ cosθ Note that the determnant s equal to unty The egenvalues are gven by solvng cosθ λ snθ ± = 0 to gve λ = e θ snθ cosθ λ The correspondng egenvectors satsfy ± ± cosθ snθ u ± θ u e ± = ± snθ cosθ u u The egenvectors, normalzed, are: ± u ± = u ± Note that, n contrast to a Hermtan matrx, the egenvalues of a untary matrx do not have to be real In fact, from UU=, sandwched between the bra and ket of an egenvector, we see that any egenvalue of a untary matrx must have unt modulus t s a complex number on the unt crcle Wth hndsght, we should have realzed that one egenvalue of a two-dmensonal
3 rotaton had to be e θ, the product of two two-dmensonal rotatons s gven be addng the angles of rotaton, and a rotaton through π changes all sgns, so has egenvalue - Note that the egenvector tself s ndependent of the angle of rotaton the rotatons all commute, so they must have common egenvectors Successve rotaton operators appled to the plus egenvector add ther angles, when appled to the mnus egenvector, all angles are subtracted prevous ndex next