5 1.4 Hermitian conjugation and Hermitian operators then we have the results AB lan）=bnA lan}=bnan lam）, (1.23) BA lan）=anB lan）=anbn lan） (1.24) If the above two relations hold for all values of n then AB=BA. (1.25) Thus under the special conditions just outlined the two operators will commute. 1.4 Hermitian conjugation and Hermitian operators We now define the"Hermitian conjugate",ofan operator A and discuss a particularclass of operators called"Hermitian"operators which play a central role in quantum mechanics. (i)In the same manner as we defined the complex conjugate operation for the state vectors, we define 4 through the following complex conjugation [4 la)]=(al4t (1.26 and (B14la)"=(al4t 1B). (1.27) If on the left-hand side of (126),la)is replaced by cl)where c is a complex constant,then on the right-hand side one must include a factor c". (ii)From(1.26)and (1.27)it follows that if A=la)(BI (1.28) then 4'1B)(al. (1.29 At this stage it is important to emphasize that ),()l)(Bl,and l)B)are four totally different mathematical quantities which should not be mixed up:the first is a state vector.the second is an ordinary number.the third is an operator,and the fourth describes a product of two states. (iii)The Hermitian conjugate of the product operator B is found tobe (AB)=B'A'. (1.30) This can be proved by first noting from(1.27)that for an arbitrary state [(4B)la)]=(al(4B)t (1.31)
5 1.4 Hermitian conjugation and Hermitian operators then we have the results AB |αn = bnA |αn = bnan |αn, (1.23) BA |αn = anB |αn = anbn |αn. (1.24) If the above two relations hold for all values of n then AB = BA. (1.25) Thus under the special conditions just outlined the two operators will commute. 1.4 Hermitian conjugation and Hermitian operators We now define the “Hermitian conjugate” A†, of an operator A and discuss a particular class of operators called “Hermitian” operators which play a central role in quantum mechanics. (i) In the same manner as we defined the complex conjugate operation for the state vectors, we define A† through the following complex conjugation [A |α] ∗ = α| A† (1.26) and β| A |α ∗ = α| A† |β. (1.27) If on the left-hand side of (1.26), |α is replaced by c |α where c is a complex constant, then on the right-hand side one must include a factor c∗. (ii) From (1.26) and (1.27) it follows that if A = |α β| (1.28) then A† = |β α| . (1.29) At this stage it is important to emphasize that |α, β| α, |α β|, and |α |β are four totally different mathematical quantities which should not be mixed up: the first is a state vector, the second is an ordinary number, the third is an operator, and the fourth describes a product of two states. (iii) The Hermitian conjugate of the product operator AB is found to be (AB) † = B†A†. (1.30) This can be proved by first noting from (1.27) that for an arbitrary state |α [(AB)|α] ∗ = α|(AB) † . (1.31)
6 Basic formalism If we take B la)=1B) (1.32) where)is another state vector,then the left-hand side of(1.31)can be written as [(4B)la)]=[4 1B)] (1.33) From the definition given in(126)we obtain [4B)]=(B14t (al Bt 4t (a1B'4t (1.34) where we have used the fact that (BI =[B la)]*=(al B'.Since la)is an arbitrary vector,comparing(1.31)and (1.34),we obtain(1.30). (iv)Finally,if 4=4t (1.359) then the operator 4 is called"Hermitian." 1.5 Hermitian operators:their eigenstates and eigenvalues Hermitian operators play a central role in quantum mechanics.We show below that the eigenstates of Hermitian operators are orthogonal and have real eigenvalues. Consider the eigenstates lam)of an operator A, Alan）=an lan (136) where lan)'s have a unit norm.By multiplying both sides of (1.36)by (anl we obtain an=(anl A lan） (1.37 Taking the complex conjugate of both sides we find ai=(anl A lan）)°=(anl At lan）=(an.A lan) (1.38) The last equalit y follows from the fact that A is Hermitian (=4).Equating(1.37)and (1.38)we conclude that=Therefore,the eigenvalues of a Hermitian operator must be real. An important postulate based on this result says that since physically observable quantities are expected to be real,the operators representing these observables must be Hermitian
6 Basic formalism If we take B |α = |β (1.32) where |β is another state vector, then the left-hand side of (1.31) can be written as [(AB)|α] ∗ = [A |β] ∗ . (1.33) From the definition given in (1.26) we obtain [A |β] ∗ = β| A† = α| B† A† = α| B†A† (1.34) where we have used the fact that β| = [B |α] ∗ = α| B†. Since |α is an arbitrary vector, comparing (1.31) and (1.34), we obtain (1.30). (iv) Finally, if A = A† (1.35) then the operator A is called “Hermitian.” 1.5 Hermitian operators: their eigenstates and eigenvalues Hermitian operators play a central role in quantum mechanics. We show below that the eigenstates of Hermitian operators are orthogonal and have real eigenvalues. Consider the eigenstates |an of an operator A, A |an = an |an (1.36) where |an’s have a unit norm. By multiplying both sides of (1.36) by an| we obtain an = an| A |an. (1.37) Taking the complex conjugate of both sides we find a∗ n = an| A |an ∗ = an| A† |an = an| A |an. (1.38) The last equality follows from the fact that A is Hermitian (A† = A). Equating (1.37) and (1.38) we conclude that a∗ n = an. Therefore, the eigenvalues of a Hermitian operator must be real. An important postulate based on this result says that since physically observable quantities are expected to be real, the operators representing these observables must be Hermitian
7 1.6 Superposition principle We now show that the eigenstates are orthogonal.We consider two eigenstates la)and lamofA, Alan}=anlan》, (1.39) Alan）=amam） (1.40) Taking the complex conjugate of the second equation we have (amlA=am (aml (1.41) where we have used the Hermitian property of A,and the fact that the eigenvalue am is real. Multiplying (1.39)on the left by and(1.41)on the right by la)and subtracting.we obtain (am-an)(am an）=0. (1.42) Thus,if the eigenvalues am and a are different we have (am an）=0, (1.43) which shows that the two eigenstates are orthogonal.Using the fact that the ket vectors are normalized,we can write the general orthonormality relation between them as (aml an)=8mn (1.44 called the Kronecker delta,which has the property 8mn 1 for m=n (1.45) =0form≠n. For those cases where there is a degeneracy in the eigenvalues,i.e if two different states have the same eienvalue,the treatment is slightly different and will be ater chapters. We note that the operators need not be Hermitian in order to have eigenvalues.However, in these cases none of the above results will hold.For example,the eigenvalues will not necessarily be real.Unless otherwise stated,we will assume the eigenvalues to be real 1.6 Superposition principle A basic theorem in quantum mechanics based on linear vector algebra is that an arbitrary vector in a given vector space can be expressed as a linear combination-a superposition- of a complete set of eigenstates of any operator in that space.A complete set is defined to
7 1.6 Superposition principle We now show that the eigenstates are orthogonal. We consider two eigenstates |an and |am of A, A |an = an |an, (1.39) A |am = am |am. (1.40) Taking the complex conjugate of the second equation we have am| A = am am| (1.41) where we have used the Hermitian property of A, and the fact that the eigenvalue am is real. Multiplying (1.39) on the left by am| and (1.41) on the right by |an and subtracting, we obtain (am − an)am| an = 0. (1.42) Thus, if the eigenvalues αm and αn are different we have am| an = 0, (1.43) which shows that the two eigenstates are orthogonal. Using the fact that the ket vectors are normalized, we can write the general orthonormality relation between them as am| an = δmn (1.44) where δmn is called the Kronecker delta, which has the property δmn = 1 for m = n (1.45) = 0 for m = n. For those cases where there is a degeneracy in the eigenvalues, i.e., if two different states have the same eigenvalue, the treatment is slightly different and will be deferred until later chapters. We note that the operators need not be Hermitian in order to have eigenvalues. However, in these cases none of the above results will hold. For example, the eigenvalues will not necessarily be real. Unless otherwise stated, we will assume the eigenvalues to be real. 1.6 Superposition principle A basic theorem in quantum mechanics based on linear vector algebra is that an arbitrary vector in a given vector space can be expressed as a linear combination – a superposition – of a complete set of eigenstates of any operator in that space. A complete set is defined to
8 Basic formalism be the set of all possible eigenstates of an operator.Expressing this result for an arbitrary state vector)in terms of the eigenstates a)of the operator A,we have lp)=∑cn lan) (1.46 where the summation index n goes over all the eigenstates withn=1,2,....If we multiply (1.46)by (aml then the orthonormality relation (1.44)between the lan)'s yields cm=(am中). (1.47) It is then postulated that cm2is the probability that )contains m).That is.m is the probability that)has the eigenvalue If is normalized to unity,(=1,then ∑lcP=l (1.48) That is,the probability of finding in state la),summed over all possible values of n, is one Since(16)is true for any arbitrary state we can express another state)as ly=∑lod (1.49 The scalar product()can then be written,using the orthonormality property of the eigenstates,as (l)=c篇cw (1.50) with c =(aml v)and cm =(am). The above relations express the fact that the state vectors can be expanded in terms of set of basis states. 1.7 Completeness relation We consider now the operators a)(,where the a)'s are the eigenstates ofan operator .A very important result in quantum mechanics involving the sum eover the posib infinite mer ft ∑lan)anl=1 (1.51) where the I on the right-hand side is a unit operator.This is the so called"completeness relation
8 Basic formalism be the set of all possible eigenstates of an operator. Expressing this result for an arbitrary state vector |φ in terms of the eigenstates |an of the operator A, we have |φ = n cn |an (1.46) where the summation index n goes over all the eigenstates with n = 1, 2, ... . If we multiply (1.46) by am| then the orthonormality relation (1.44) between the |an’s yields cm = am| φ. (1.47) It is then postulated that |cm| 2 is the probability that |φ contains |am. That is, |cm| 2 is the probability that |φ has the eigenvalue am. If |φ is normalized to unity, φ| φ = 1, then n |cn| 2 = 1. (1.48) That is, the probability of finding |φ in state |an, summed over all possible values of n, is one. Since (1.46) is true for any arbitrary state we can express another state |ψ as |ψ = n c n |an. (1.49) The scalar product ψ |φ can then be written, using the orthonormality property of the eigenstates, as ψ |φ = m c ∗ m cm (1.50) with c m = am| ψ and cm = am| φ. The above relations express the fact that the state vectors can be expanded in terms of the eigenstates of an operator A. The eigenstates |an are then natural candidates to form a set of basis states. 1.7 Completeness relation We consider now the operators |an an| , where the |an’s are the eigenstates of an operator A, with eigenvalues an. A very important result in quantum mechanics involving the sum of the operators |an an| over the possibly infinite number of eigenvalues states that n |an an| = 1 (1.51) where the 1 on the right-hand side is a unit operator. This is the so called “completeness relation
1.8 Unitary operators To prove this relation we first multiply the sum on the left hand of the above equality by an arbitrary eigenvector m)to obtain ∑a,laa）-∑-∑wi=tad (1.52) where we have used the orthonormality of the eigenvectors.Since this relation holds for any arbitrary state lam),the operator in the square bracket on the left-hand side acts as a unit operator,thus reproducing the completeness relation. If we designate Pn=lan)(anl (1.53) then Pn lam）=Snm lam）. (1.54) Thus Pn projects out the state la)whenever it operates on an arbitrary state.For this reason P is called the projection operator,in terms of which one can write the completeness relation as ∑P.=1 (1.55) One can utilize the completeness relation to simplify the scalar product (where )and)are given above,if we write,using(1.51) 1)=11)=(∑la)all)=∑1 an)()=∑分c:1.56 This is the same result as the one we derived previously as(1.50). 1.8 Unitary operators If two state vectors la)and a')have the same norm then (ala)=(a'a'). (1.57) Expressing each of these states in terms of a complete set of eigenstates we obtain la)=cn lan)and la)=c lan) (1.58) The equality in(1.57)leads to the relation ∑P=∑ (1.59
9 1.8 Unitary operators To prove this relation we first multiply the sum on the left hand of the above equality by an arbitrary eigenvector |am to obtain n |an an| |am = n |an an| am = n |an δnm = |am (1.52) where we have used the orthonormality of the eigenvectors. Since this relation holds for any arbitrary state |am, the operator in the square bracket on the left-hand side acts as a unit operator, thus reproducing the completeness relation. If we designate Pn = |an an| (1.53) then Pn |am = δnm |am. (1.54) Thus Pn projects out the state |an whenever it operates on an arbitrary state. For this reason Pn is called the projection operator, in terms of which one can write the completeness relation as n Pn = 1. (1.55) One can utilize the completeness relation to simplify the scalar product ψ| φ, where |φ and |ψ are given above, if we write, using (1.51) ψ| φ = ψ| 1 |φ = ψ| n |an an| |φ = n ψ| anan| φ = n c ∗ n cn. (1.56) This is the same result as the one we derived previously as (1.50). 1.8 Unitary operators If two state vectors |α and α have the same norm then α| α = α α . (1.57) Expressing each of these states in terms of a complete set of eigenstates |an we obtain |α = n cn |an and α = n c n |an. (1.58) The equality in (1.57) leads to the relation n |cn| 2 = n c n 2 , (1.59)