Chapter 10 Represetatios of State Vectors ad Operators I the precedig Chapters, the mathematical ideas uderpiig the quatum theory have bee developed i a fairly geeral (though, admittedly, ot a mathematically rigorous) fashio However, much that has bee preseted, particularly the cocept of a operator, ca be developed i aother way that is, i some respects, less abstract tha what has bee used so far This alterate form of presetatio ivolves workig with the compoets of the state vectors ad operators, leadig to their beig represeted by colum ad row vectors, ad matrices This developmet of the theory is completely aalogous to the way i which this is doe whe dealig with the positio vector i ordiary three dimesioal space Below, we will look at how the idea of writig the positio vectors i two dimesioal space ca be writte i terms of colum ad row vectors We will the use the ideas developed there to show how state vectors ad operators ca be expressed i a similar fashio This alterate route offers aother way of itroducig such cocepts as addig, multiplyig ad takig the iverse of operators through their represetatios as matrices, ad further provides aother way to itroduce the idea of the Hermitea adjoit of a operator, ad of a Hermitia operator 101 Represetatio of Vectors I Euclidea Space as Colum ad Row Vectors Whe writig dow a vector, we have so far made explicit the basis vectors whe writig a expressio such as r xî + yĵ for a positio vector, or S a + + b for the state of a spi half system But the choice of basis vectors is ot uique, ie we could use ay other pair of orthoormal uit vectors î ad ĵ, ad express the vector r i terms of these ew basis vectors, though of course the compoets of r will chage The same is true for the spi basis vectors ±, ie we ca express the state S i terms of some other basis vectors, such as the states for which the x compoet of spi has the values S x 1, though oce agai, the compoets of S will ow be differet But it is typically the case that oce the choice of basis vectors have bee decided o, it should ot be ecessary to always write them dow whe writig dow a vector, ie it would be just as useful to just write dow the compoets of a vector Doig this leads to a coveiet otatio i
Chapter 10 Represetatios of State Vectors ad Operators 14 which vectors are writte i terms of colum ad row vectors It also provides a direct route to some of the importat mathematical etities ecoutered i quatum mechaics such as bra vectors ad operators that are more rigorously itroduced i a more abstract way 1011 Colum Vectors To illustrate the ideas, we will use the example of a positio vector i two dimesioal space The poit that is to be established here is that a positio vector is a idepedetly existig geometrical object suspeded i space, much as a pecil held i the air with a steady positio ad orietatio has a fixed legth ad orietatio Oe ed of the pecil, say where the eraser is, ca be take to be the origi O, ad the other ed (the sharp ed) the positio of a poit P The the positio ad orietatio of the pecil defies a positio vector r of P with respect to the origi O This vector ca be represeted by a sigle arrow joiig O to P whose legth ad orietatio specify the positio of P with respect to O As such, the vector r also has a idepedet existece as a geometrical object sittig i space, ad we ca work with this vector r ad others like it by, for istace, performig vector additios by usig the triagle law of vector additio as illustrated i Fig (81), or performig scalar products by use of the defiitio Eq (8) I what was just described, we work oly with the whole vector itself This is i cotrast with a very useful alterate way of workig with vectors, that is to express ay vector as a liear combiatio of a pair of basis vectors (i two dimesios), which amouts to buildig aroud these vectors some sort of scaffoldig, a coordiate system such as a pair of X ad Y axes, ad describe the vector i terms of its compoets with respect to these axes More to the poit, what is provided is a pair of basis vectors such as the familiar uit vectors î ad ĵ ad write r xî + yĵ We see that ay positio vector ca be writte i this way, ie the uit vectors costitute a pair of orthoormal basis vectors, ad x ad y are kow as the compoets of r with respect to the basis vectors î ad ĵ We ca the work out how to add vectors, calculate scalar products ad so o workig solely with these compoets For istace, if we have two vectors rˆ 1 ad rˆ give by rˆ 1 x 1 î + y 1 ĵ ad rˆ x î + y ĵ the rˆ 1 + rˆ (x 1 + x )î + (y 1 + y )ĵ It is importat to ote that while a vector ˆr is a uique geometrical object, there is o uique choice of basis vectors, ad correspodigly the compoets of the vector will chage depedig o the choice of basis vectors Thus we could equally well have chose the basis vectors î ad ĵ, as illustrated i Fig (101) so that the same vector r ca be writte r x î + y ĵ x î + y ĵ (101) with x x ad y y ĵ ĵ O P x y x Figure 101: The positio vector r writte as a liear combiatio of two differet pairs of orthoormal basis vectors Although the coordiates of r are differet with respect to the two sets of basis vectors, the vector r remais the same î r î y
Chapter 10 Represetatios of State Vectors ad Operators 143 Oce a choice of basis vectors has bee made, it proves to be very coveiet to work solely with the coordiates There is a useful otatio by which this ca be doe I this otatio, the vector r is writte as x r y (10) what is kow as a colum vector We the say that this colum vector is a represetatio of the vector r with respect to the basis vectors î ad ĵ It is importat to ote that we do ot say that r equals the colum vector, i fact it is ot a equal sig that is used i Eq (10), rather the symbol is used, which is to be read as is represeted by The reaso for this is that, as metioed above, while the vector r is a uique geometrical object, its compoets are ot they deped o the choice of basis vectors We could have equally chose basis vectors î ad ĵ, ad sice the compoets x ad y will be, i geeral, differet from x ad y, we ed up with a differet colum vector represetig the same vector: r x y (103) ie two apparetly differet colum vectors represetig the same vector r Equivaletly, if we had two colum vectors with exactly the same umbers i the two positios, we could ot coclude that they represet the same vector uless we were told that the basis vectors were the same i each case Thus if there is ay chace of ambiguity, we eed to make it clear whe we use the colum vector otatio, exactly what the basis vectors are The termiology the is to say that the vector r is give by the colum vector i Eq (10) i the {î, ĵ} represetatio Oce a choice of basis vectors has bee settled o, ad cosistetly used, we ca proceed with vector calculatios usig the ew otatio Thus, for istace, we ca add two vectors: r r 1 + r (104) which becomes, usig the {î, ĵ} represetatio x y x 1 y 1 + x y x 1 + x (105) y 1 + y 101 Row Vectors The scalar product (r 1, r ) r 1 r ca be calculated by usig the usual rule r 1 r r 1 r cos θ, but it ca also be expressed i terms of the compoets of r 1 ad r i, say, the {î, ĵ} represetatio, though ote that the same umerical result is obtaied whaatever represetatio is used The result is simply (r 1, r ) r 1 r x 1 x + y 1 y (106)
Chapter 10 Represetatios of State Vectors ad Operators 144 At this poit we ote that, if we use the rules of matrix multiplicatio, this last result ca be writte (r 1, r ) r 1 r ( x 1 y 1 ) x y (107) where we ote the appearace of the colum vector represetig the vector r, but r 1, the first factor i the scalar product, has bee represeted by a row vector If the compoets of r 1 ad r were complex, the we would write the ier product as (r 1, r ) r 1 r x 1 x + y 1 y ( x 1 y 1 ) x y (108) The use of a row vector to represet r 1 ca be looked o here as a coveiece so that the rules of matrix multiplicatio ca be applied, but there is a deeper sigificace to its use 1 that will become apparet whe we look at the colum ad row vector represetatios of ket ad bra vectors 10 Represetatios of State Vectors ad Operators The procedure here is idetical to that which was followed i the case of the positio vector, ie we itroduce a complete set of orthoormal basis states { ϕ ; 1,, } that spa the state space of the quatum system, ad the work with the compoets of the ket ad bra vectors, ad the operators Of course, we ow do ot have the luxury of iterpretig these basis vectors as represetig physical directios i real space rather they are abstract vectors i a multi-dimesioal complex vector space, but much of what has bee said above i coectio with vectors i ordiary Euclidea space ca be carried over to this more abstract situatio 101 Row ad Colum Vector Represetatios for Spi Half State Vectors To set the scee, we will look at the particular case of spi half state vectors for which, as we have see earlier, Sec 8, a arbitrary state S ca be writte S S + + + S, ie the state S is expressed as a liear combiatio of the two basis states ± We further saw that the ket vectors as +, could be put ito direct correspodece with the (complex) uit vectors û 1 ad û respectively, ad that the probability amplitudes 1 Effectively, what is goig o is that correspodig to ay vector r represeted by a colum vector, there correspods aother vector r kow as its dual which is represeted by a row vector The origial vector is the physical vector while its dual is a abstract mathematical compaio The origial vector ad its dual belog to two differet vector spaces
Chapter 10 Represetatios of State Vectors ad Operators 145 ± S are the compoets of S i the directio of the basis states ± We ca complete the aalogy with the case of ordiary vectors by otig that we could the write this ket vector as a colum vector, ie S S (109) + S If we pursue this lie further, we ca get a idea of how to iterpret bra vectors To do this, cosider the more geeral probability amplitudes S S This we ca write as If we ow use this becomes which we ca write as S S S S + S + + S (1010) ± S S ± (1011) S S S S + + S + S (101) S S ( S + S ) S (1013) + S I other words, the bra vector S is represeted by the row vector S ( S + S ) (1014) This shows that a bra vector is more tha just the complex cojugate of a ket vector, sice a row vector is ot the same as a colum vector We ca ow exted the idea to the more geeral situatio of a state space of dimesio > 10 Represetatio of Ket ad Bra Vectors I terms of the basis states { ϕ ; 1,, }, a arbitrary state vector ψ ca be writte as ψ ϕ ϕ ψ (1015) Let us ow write ϕ ψ ψ (1016) We the have, by aalogy with the positio vector: ψ 1 ψ ψ ψ 3 (1017)
Chapter 10 Represetatios of State Vectors ad Operators 146 This is a represetatio of ψ as a colum vector with respect to the set of basis states { ϕ ; 1,, } I particular, the basis state ϕ m will have compoets (ϕ m ) ϕ ϕ m δ m (1018) ad so they will be represeted by colum vectors of the form 1 0 ϕ 1 0 0 ϕ 1 0 (1019) ie the m th compoet ϕ m of ϕ is zero except i the m th positio where ϕ mm 1 Now form the ier product χ ψ : χ ψ χ ϕ ϕ ψ (100) We kow that χ ϕ ( ϕ χ ), ad followig o from the otatio itroduce above, we write χ ϕ χ so that χ ϕ χ (101) ad hece which we ca write as χ ψ χ ψ (10) ψ 1 ( ) χ ψ χ 1 χ χ 3 ψ ψ 3 (103) which we evaluate by the usual rules of matrix multiplicatio Note that here we have made the idetificatio of the bra vector χ as a row vector: ( ) χ χ 1 χ χ 3 (104) with respect to the set of basis states { ϕ ; 1,, } This ca be compared with the
Chapter 10 Represetatios of State Vectors ad Operators 147 represetatio of the ket vector χ as a colum vector: χ 1 χ χ χ 3 (105) This differece i appearace of the represetatio of a bra ad a ket vector, the first as a row vector, the secod as a colum vector, perhaps emphasizes the poit made i Sectio 85 that the bra vectors form a vector space, the dual Hilbert space H related to, but distict from, the Hilbert space H of the ket vectors I a sese, a bra vector ca be thought of as somethig aki to beig the complex cojugate of its correspodig ket vector 103 Represetatio of Operators Now tur to the operator equatio  ψ φ (106) which we ca write as φ  ψ  ϕ ϕ ψ  ϕ ϕ ψ (107) The which we ca write as where ϕ m φ ϕ m  ϕ ϕ ψ (108) φ m A m ψ (109) A m ϕ m  ϕ (1030) We ca write this as a matrix equatio: φ 1 A 11 A 1 A 13 ψ 1 φ A 1 A A 3 ψ (1031) φ 3 A 31 A 3 A 33 ψ 3
Chapter 10 Represetatios of State Vectors ad Operators 148 where the operator  is represeted by a matrix: A 11 A 1 A 13  A 1 A A 3 A 31 A 3 A 33 (103) The quatities A m are kow as the matrix elemets of the operator  with respect to the basis states { ϕ ; 1,, } It is importat to keep i mid that the colum vectors, row vectors, ad matrices above are costructed with respect to a particular set of basis states If a differet set of basis states are used, the the state vectors ad operators remai the same, but the colum or row vector, or matrix represetig the state vector or operator respectively will chage Thus, to give ay meaig to a row vector, or a colum vector, or a matrix, it is essetial that the basis states be kow A importat part of quatum mechacis is the mathematical formalism that deals with trasformig betwee differet sets of basis states However, we will ot be lookig at trasformatio theory here Ex 101 Cosider two state vectors 1 1 [ i + ] 1 [ + i + ] where ± are the usual base states for a spi half system We wat to represet these ket vectors as colum vectors with respect to the set of basis states { +, } Firstly, we ote that i the geeral developmet described above, we assumed that the basis states were amed ϕ 1, ϕ ad so o But here we are usig a differet way of labellig the basis states, which meas we have a choice as to which of ± we idetify with ϕ 1 ad ϕ If makes o differece what we choose: we make the choice to suit ourselves, ad provided we use it cosistetly the o problems should arise Thus, here, we will choose ϕ 1 + ad ϕ Thus we ca write write 1 + 0 ad 0 1 We ca the express the states 1 ad i colum vector otatio as which ca also be writte as i 1 1 1 1 i 1 0 + 1 0 1
Chapter 10 Represetatios of State Vectors ad Operators 149 The correspodig bra vectors are 1 1 [ + i + ] 1 [ i + ] or, as row vectors 1 ( i 1 ) ad ( i 1 ) We ca calculate ier products as follows: ad so o 1 1 ( ) i i 1 1 0, 1 1 1 ( ) i i 1 1 1 Ex 10 We ca also look at the operator  defied by  ± ± 1 i which ca be writte out i matrix form as +  +   + +  0 1 i  1 i 0 so that, for istace 0 1  1 i 1 i 0 1i 1 1 i i 1 i 1 Thus we have  1 1 1, which icidetally shows that 1 is a eigestate of Â
Chapter 10 Represetatios of State Vectors ad Operators 150 Usig the represetatios of bra vectors ad operators, it is straightforward to see what the actio of a operator o a bra vector is give by Thus, we have: A 11 A 1 A 13 ψ  ( ψ 1 ψ ψ 3 ) A 1 A A 3 A 31 A 3 A 33 ( ψ 1 A 11 + ψ A 1 + ψ A 1 + ψ A + ψ 1 A 13 + ψ A 3 + ) (1033) The fial result ca the be writte i the correspodig bra vector otatio if desired This ca be illustrated by example Ex 103 Evaluate  usig the represetatios of the bra vector ad operator Â:  1 ( i 1 ) which ca be writte as  1 0 1 i 1 i 1 ( 1 i 1 ) 1 1 ( i 1 ) 104 Properties of Matrix Represetatios of Operators May of the properties of operators ca be expressed i terms of the properties of their represetative matrices Most of these properties are straightforward, ad will be preseted below without commet Equality Two operators are equal if their correspodig operator matrix elemets are equal, ie  ˆB if A m B m Uit ad Zero Operator The uit operator ˆ1 is the operator such that ˆ1 ψ ψ for all states ψ It has the the matrix elemets ˆ1 m δ m, ie the diagoal elemets are all uity, ad the off-diagoal elemets are all zero The uit operator has the same form i all represetatios, ie irrespective of the choice of basis states The zero operator ˆ0 is the operator such that ˆ0 ψ 0 for all states ψ Its matrix elemets are all zero
Chapter 10 Represetatios of State Vectors ad Operators 151 Additio of Operators Give two operators  ad ˆB with matrix elemets A m ad B m, the the matrix elemets of their sum Ŝ  + ˆB are give by S m A m + B m (1034) Multiplicatio by a Complex Number If λ is a complex umber, the the matrix elemets of the operator Ĉ λâ are give by C m λa m (1035) Product of Operators Give two operators  ad ˆB with matrix elemets A m ad B m, the the matrix elemets of their product ˆP  ˆB are give by P m A mk B k (1036) k ie the usual rule for the multiplicatio of two matrices Matrix multiplicatio, ad hece operator multiplicatio, is ot commutative, ie i geeral  ˆB ˆB The differece,  ˆB ˆBÂ, kow as the commutator of  ad ˆB ad writte [Â, ˆB], ca be readily evaluated usig the matrix represetatios of  ad ˆB Ex 104 Three operators ˆσ 1, ˆσ ad ˆσ 3, kow as the Pauli spi matrices, that occur i the theory of spi half systems (ad elsewhere) have the matrix represetatios with respect to the { +, } basis give by 0 1 ˆσ 1 1 0 0 i ˆσ i 0 1 0 ˆσ 3 0 1 The commutator [ ˆσ 1, ˆσ ] ca be readily evaluated usig these matrices: [ ˆσ 1, ˆσ ] ˆσ 1 ˆσ ˆσ ˆσ 1 1 0 i 0 1 0 1 0 i 0 i 0 1 1 0 i 0 i 0 1 0 i 0 i 0 0 i 0 i
Chapter 10 Represetatios of State Vectors ad Operators 15 The fial matrix ca be recogized as the represetatio of ˆσ 3, so that overall we have show that [ ˆσ 1, ˆσ ] i ˆσ 3 Cyclic permutatio of the subscripts the gives the other two commutators Fuctios of Operators If we have a fuctio f (x) which we ca expad as a power series i x: f (x) a 0 + a 1 x + a x + a x (1037) 0 the we defie f (Â), a fuctio of the operator Â, to be also give by the same power series, ie f (Â) a 0 + a 1  + a  + a  (1038) Oce agai, usig the matrix represetatio of Â, it is possible, i certai cases, to work out what the matrix represetatio is of f (Â) Ex 105 Oe of the most importat fuctios of a operator that is ecoutered is the expoetial fuctio To illustrate what this meas, we will evaluate here the expoetial fuctio exp(iφ ˆσ 3 ) where ˆσ 1 is oe of the Pauli spi matrices itroduced above, for which 0 1 ˆσ 1 1 0 ad φ is a real umber Usig the power series expasio of the expoetial fuctio, we have e iφ ˆσ 1 iφ! ˆσ 1 It is useful to ote that 0 ˆσ 3 0 1 1 0 0 1 0 0 1 ie ˆσ 1 ˆ1, the idetity operator Thus we ca always write ˆσ 1 ˆ1 ˆσ +1 1 ˆσ 1 Thus, if we separate the ifiite sum ito two parts: e iφ ˆσ 1 0 (iφ) ()! ˆσ 1 + (iφ) +1 ( + 1)! ˆσ+1 1 0
Chapter 10 Represetatios of State Vectors ad Operators 153 where the first sum is over all the eve itegers, ad the secod over all the odd itegers, we get e iφ ˆσ 1 ˆ1 ( 1) φ ()! + i ˆσ 1 0 cos φ + i ˆσ 1 si φ 0 cos φ 0 0 i si φ + 0 cos φ i si φ 0 cos φ i si φ i si φ cos φ ( 1) φ +1 ( + 1)! Iverse of a Operator Fidig the iverse of a operator, give its matrix represetatio, amouts to fidig the iverse of the matrix, provided, of course, that the matrix has a iverse Ex 106 The iverse of exp(iφ ˆσ 1 ) ca be foud by takig the iverse of its represetative matrix: ( e iφ ˆσ 1 ) 1 cos φ i si φ i si φ cos φ 1 cos φ i si φ i si φ cos φ This iverse ca be recogized as beig just cos φ i si φ cos( φ) i si φ cos φ i si( φ) i si( φ) cos( φ) which meas that ( e iφ ˆσ 1 ) 1 e iφ ˆσ 1 a perhaps usurprisig result i that it is a particular case of the fact that the iverse of exp(â) is just exp( Â), exactly as is the case for the expoetial fuctio of a complex variable 105 Eigevectors ad Eigevalues Operators act o states to map them ito other states Amogst the possible outcomes of the actio of a operator o a state is to map the state ito a multiple of itself: Â φ a φ φ (1039)
Chapter 10 Represetatios of State Vectors ad Operators 154 where φ is, i geeral, a complex umber The state φ is the said to be a eigestate or eigeket of the operator  with a φ the associated eigevalue The fact that operators ca possess eigestates might be thought of as a mathematical fact icidetal to the physical cotet of quatum mechaics, but it turs out that the opposite is the case: the eigestates ad eigevalues of various kids of operators are essetial parts of the physical iterpretatio of the quatum theory, ad hece warrat close study Notatioally, is is ofte useful to use the eigevalue associated with a eigestate to label the eigevector, ie the otatio  a a a (1040) This otatio, or mior variatios of it, will be used almost exclusively here Determiig the eigevalues ad eigevectors of a give operator Â, occasioally referred to as solvig the eigevalue problem for the operator, amouts to fidig solutios to the eigevalue equatio  φ a φ φ Writte out i terms of the matrix represetatios of the operator with respect to some set of orthoormal basis vectors { ϕ ; 1,, }, this eigevalue equatio is A 11 A 1 φ 1 φ 1 A 1 A φ a φ (1041) This expressio is equivalet to a set of simultaeous, homogeeous, liear equatios: A 11 a A 1 φ 1 A 1 A a φ 0 (104) which have to be solved for the possible values for a, ad the associated values for the compoets φ 1, φ, of the eigevectors The procedure is stadard The determiat of coefficiets must vaish i order to get o-trivial solutios for the compoets φ 1, φ, : A 11 a A 1 A 1 A a 0 (1043) which yields a equatio kow as the secular equatio, or characteristic equatio, that has to be solved to give the possible values of the eigevalues a Oce these are kow, they have to be resubstituted ito Eq (1041) ad the compoets φ 1, φ, of the associated eigevectors determied The details of how this is doe properly belogs to a text o liear algebra ad will ot be cosidered ay further here, except to say that the eigevectors are typically determied up to a ukow multiplicative costat This costat is usually fixed by the requiremet that these eigevectors be ormalized to uity I the
Chapter 10 Represetatios of State Vectors ad Operators 155 case of repeated eigevalues, ie whe the characteristic polyomial has multiple roots (otherwise kow as degeerate eigevalues), the determiatio of the eigevectors is made more complicated still Oce agai, issues coected with these kids of situatios will ot be cosidered here I geeral, for a state space of fiite dimesio, it is foud that the operator  will have oe or more discrete eigevalues a 1, a, ad associated eigevectors a 1, a, The collectio of all the eigevalues of a operator is called the eigevalue spectrum of the operator Note also that more tha oe eigevector ca have the same eigevalue Such a eigevalue is said to be degeerate For the preset we will be cofiig our attetio to operators that have discrete eigevalue spectra Modificatios eeded to hadle cotiuous eigevalues will be itroduced later 106 Hermitea Operators Apart from certai calculatioal advatages, the represetatio of operators as matrices makes it possible to itroduce i a direct fashio Hermitea operators, already cosidered i a more abstract way i Sectio 931, which have a cetral role to play i the physical iterpretatio of quatum mechaics To begi with, suppose we have a operator  with matrix elemets A m with respect to a set of orthoormal basis states { ϕ ; 1,, } From the matrix represetig this operator, we ca costruct a ew operator by takig the traspose ad complex cojugate of the origial matrix: A 11 A 1 A 13 A 11 A 1 A 31 A 1 A A 3 A 31 A 3 A 33 A 1 A A 3 A 13 A 3 A 33 (1044) The ew matrix will represet a ew operator which is obviously related to Â, which we will call Â, ie (A ) 11 (A ) 1 (A ) 13 A 11 A 1 A 31  (A ) 1 (A ) (A ) 3 (A ) 31 (A ) 3 (A ) 33 A 1 A A 3 A 13 A 3 A 33 (1045) ie ϕ m  ϕ ( ϕ  ϕ m ) (1046)
Chapter 10 Represetatios of State Vectors ad Operators 156 The ew operator that we have created, Â, ca be recogized as the Hermitea adjoit of  The Hermitea adjoit has a useful property which we ca most readily see if we use the matrix represetatio of the operator equatio  ψ φ (1047) which we earlier showed could be writte as φ m ϕ m  ϕ ψ (1048) If we ow take the complex cojugate of this expressio we fid φ m ψ ϕ  ϕ m (1049) which we ca write i row vector form as ( φ 1 φ ) ( ψ 1 ψ ) (A ) 11 (A ) 1 (A ) 1 (A ) (1050) which is the matrix versio of φ ψ  (1051) I other words we have show that if  ψ φ, the ψ  φ, a result that we used earlier to motivate the defiitio of the Hermitea adjoit i the first place Thus, there are two ways to approach this cocept: either through a geeral abstract argumet, or i terms of matrix represetatios of a operator Ex 107 Cosider the operator  which has the represetatio i some basis 1 i  0 1 The 1 0  i 1 To be oticed i this example is that   Ex 108 Now cosider the operator 0 i  i 0
Chapter 10 Represetatios of State Vectors ad Operators 157 The 0 i  i 0 ie   This is a example of a situatio i which  ad  are idetical I this case, the operator is said to be selfadjoit, or Hermitea Oce agai, we ecoutered this idea i a more geeral cotext i Sectio 94, where it was remarked that Hermitea operators have a umber of very importat properties which leads to their playig a cetral role i the physical iterpretatio of quatum mechaics These properties of Hermitea operators lead to the idetificatio of Hermitea operators as represetig the physically observable properties of a physical system, i a way that will be discussed i the ext Chapter