10 Basic formalism which signifies that,even though cm may be different from c the total sum of the probabilities remains the same. Consider now an operator A such that Ala)=la) (1.60) if la)and have the same norm,then (ala)=(a'a')=(al4tA la) (1.61) This implies that AA=1. (1.62) The operator A is then said to be"unitary."From relation(1.60)it is clear that A can change the basis from one set to another.Unitary operators play a fundamental role in quantum mechanics. 1.9 Unitary operators as transformation operators Let us define the following operator in term of the eigenstates la)of operator A and eigenstates b)of operator B, U=>Iba)(anl (1.63) This is a classic example of a unitary operator as we show below.We first obtain the Hermitian conjugate ofU, Ut=∑Ian)bnl (1.64) Therefore, Uut=[∑lb)anl]∑)(bl=∑∑b)(ld (bal=∑lbbl=l (1.65) normality relation (m)= We note in passing that ∑lan)(anl (1.66 is a unit operator which is a special case of a unitary operator when (=
10 Basic formalism which signifies that, even though cn may be different from c n, the total sum of the probabilities remains the same. Consider now an operator A such that A |α = α . (1.60) If |α and α have the same norm, then α| α = α α = α| A†A |α. (1.61) This implies that A†A = 1. (1.62) The operator A is then said to be “unitary.” From relation (1.60) it is clear that A can change the basis from one set to another. Unitary operators play a fundamental role in quantum mechanics. 1.9 Unitary operators as transformation operators Let us define the following operator in term of the eigenstates |an of operator A and eigenstates |bn of operator B, U = n |bn an| . (1.63) This is a classic example of a unitary operator as we show below. We first obtain the Hermitian conjugate of U, U† = n |an bn| . (1.64) Therefore, UU† = n |bn an| m |am bm| = |bn an| ambm| = |bn bn| = 1 (1.65) where we have used the orthonormality relation an| am = δnm and the completeness relation for the state vectors |bn discussed in the previous section. Hence U is unitary. We note in passing that n |an an| (1.66) is a unit operator which is a special case of a unitary operator when bn| = an|
11 1.9 Unitary operators as transformation operators We also note that Ulam）=lbn）(an dm）=lbn〉8nm=lbm） 1.67 Hence U transforms the eigenstate lam)into )In other words,if we use lan)'s as the basis for the expansion of a state vector then U will convert this basis set to a new basis formed by the b)'s.Thus U allows one to transform from the"old basis"given by la)'s basis"given by ca do the conversion in the reverse order by multiplying both sides of(167)byon the left: U'U lam)=Uf bm). (1.68) Hence,from the unitary property of U,we find Ut Ibm）=lam) (1.69) Furthermore,the matrix element of an operator,A,in the old basis set can be related to its matrix elements in the new basis as follows (bnl A lbm)=(bal UU'AUU'Ibm)=(anl U'AU lam) (1.70 where we have used the property=1 and the relations(1.69).This relation will be relation in terms of the"transformed"operator 4.We then write AT U'AU. (1.71) Finally,if )'s are the eigenstates of the operator 4, A lan)=an lan) (1.72) and relation (1.67)connecting la)and Ib)holds,where Ib)'s are the eigenstates of an operator B,then we can multiply (1.72)by U on both sides to obtain UA a》=anU an》 (1.73) and write UAUt[Ulan】=an[Ulan] (1.74 Hence UAUT Ibn)an lbn). (1.75) Thus )is the eigenstate of UAU with the same eigenvalues as the eigenvalues of 4. However,since)'s are eigenstates of the operator B we find that UAU and B are in some sense equivalent
11 1.9 Unitary operators as transformation operators We also note that U |am = n |bn an| am = n |bn δnm = |bm. (1.67) Hence U transforms the eigenstate |am into |bm. In other words, if we use |an’s as the basis for the expansion of a state vector then U will convert this basis set to a new basis formed by the |bn’s. Thus U allows one to transform from the “old basis” given by |an’s to the “new basis” given by the |bn’s. One can do the conversion in the reverse order by multiplying both sides of (1.67) by U† on the left: U†U |am = U† |bm. (1.68) Hence, from the unitary property of U, we find U† |bm = |am. (1.69) Furthermore, the matrix element of an operator, A, in the old basis set can be related to its matrix elements in the new basis as follows bn| A |bm = bn| UU†AUU† |bm = an| U†AU |am (1.70) where we have used the property U†U = 1 and the relations (1.69). This relation will be true for all possible values of |an’s and |bn’s. Therefore, it can be expressed as an operator relation in terms of the “transformed” operator AT . We then write AT = U†AU. (1.71) Finally, if |an’s are the eigenstates of the operator A, A |an = an |an (1.72) and relation (1.67) connecting |an and |bn holds, where |bn’s are the eigenstates of an operator B, then we can multiply (1.72) by U on both sides to obtain UA |an = anU |an (1.73) and write UAU† [U |an] = an [U |an] . (1.74) Hence UAU† |bn = an |bn. (1.75) Thus |bn is the eigenstate of UAU† with the same eigenvalues as the eigenvalues of A. However, since |bn’s are eigenstates of the operator B we find that UAU† and B are in some sense equivalent
12 Basic formalism 1.10 Matrix formalism We define the"matrix element"of an operator A between states )and B)as (). (1.76 which is a complex number.To understand the meaning of this matrix element we note that when operateson)it gives another ket vector.This state when multiplied on the left by (BI gives a number.When there are many such la)'s and (BI's then we have a whole array of numbers that can be put into the form of a matrix.Specifically,the matrix formed by the matrix elements of A between the basis states b),with n=1,2....,N depending on the dimensionality of the space,is then a square matrix()written as follows 「AA2A3 A21A22A23 {40=A31A32A33· (1.77 ANN where Amn (bm LAl bn) (1.78) gives,inasense,aprofile of the operator A and describes what is an abs act object in terms of a matrix of complex numbers.The matrix representation will look different if basis sets formed by eigenstates of some other operator are used. The matrices follow the normal rules of matrix algebra.Some of the important properties (i)The relation between the matrix elements of 4 and A is given by (xA1B)=(BAla) (1.79) Thus the matrix representation oft can be written as [A A A52 。 4)=Ai (1.80 where A is represented by the matrix(1.77). (ii)If the operator A is Hermitian then (B14la)=(B14 la). 1.81)
12 Basic formalism 1.10 Matrix formalism We define the “matrix element” of an operator A between states |α and |β as β |A| α, (1.76) which is a complex number. To understand the meaning of this matrix element we note that when A operates on |α it gives another ket vector. This state when multiplied on the left by β| gives a number. When there are many such |α’s and β|’s then we have a whole array of numbers that can be put into the form of a matrix. Specifically, the matrix formed by the matrix elements of A between the basis states |bn, with n = 1, 2, ... , N depending on the dimensionality of the space, is then a square matrix bm |A| bn written as follows: {A} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 A13 . A1N A21 A22 A23 . . A31 A32 A33 . . . . .. . AN1 . .. ANN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1.77) where Amn = bm |A| bn. (1.78) The matrix (1.77) is then called the matrix representation of A in terms of the states |bn. It gives, in a sense, a profile of the operator A and describes what is an abstract object in terms of a matrix of complex numbers. The matrix representation will look different if basis sets formed by eigenstates of some other operator are used. The matrices follow the normal rules of matrix algebra. Some of the important properties are given below, particularly as they relate to the Hermitian and unitary operators. (i) The relation between the matrix elements of A† and A is given by α| A† |β = β| A |α ∗ . (1.79) Thus the matrix representation of A† can be written as {A†} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A∗ 11 A∗ 21 A∗ 31 . A∗ N1 A∗ 12 A∗ 22 A∗ 23 . . A∗ 13 A∗ 23 A∗ 33 . . . . .. . A∗ 1N . .. A∗ NN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1.80) where A is represented by the matrix (1.77). (ii) If the operator A is Hermitian then β| A |α = β| A† |α. (1.81)
13 1.10 Matrix formalism Using the property (1.79)we find (B14la)=(a141B)" (1.82) In particular,the matrix elements with respect to the eigenstates)satisfy (bm4bn）=(bm4bm) (1.83) A Hermitian operator will,therefore,have the following matrix representation: A2A22A23 {A=AAA3 (1.84) ANN We note that the same result is obtained by equating the matrices 4 and 4'given by (1.77)and(1.80)respectively.We also note that the diagonal elements 11.422....of a Hermitian operator are real since the matrix elements satisfy the relation =An (iii)The matrix representation of an operator in terms of its eigenstates is a diagona matrix because of the orthonormality property(1.44).It can,therefore,be written as An 00 0 0 A220 {A= 0 0 A33 (1.85) 0 ·ANN where Am=am where am's are the eigenvalues of 4.The matrix representation of A in terms of eigenstates b)of an operator B that is different from A and that does not share the same eigenstates is then written as A21 A22 A23 {4= A31A32A33 (1.86 where=(b).If the operator B has the same eigenstates as A then the above matrix will,once again,be diagonal. (iv)We now illustrate the usefulness of the completeness relation by considering several operator relations.First let us consider the matrix representation of the product of two operators (4B)by writing (bm 14BI ba)=(bm 141BI)=>(bm 4[lbp)(Bp]B bn).(1.87)
13 1.10 Matrix formalism Using the property (1.79) we find β| A |α = α| A |β ∗ . (1.82) In particular, the matrix elements with respect to the eigenstates |bn satisfy bm |A| bn = bn |A| bm ∗ . (1.83) A Hermitian operator will, therefore, have the following matrix representation: {A} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 A13 . A1N A∗ 12 A22 A23 . . A∗ 13 A∗ 23 A33 . . . . .. . A∗ 1N . .. ANN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (1.84) We note that the same result is obtained by equating the matrices A and A† given by (1.77) and (1.80) respectively. We also note that the diagonal elements A11, A22, ... of a Hermitian operator are real since the matrix elements satisfy the relation A∗ mn = Anm. (iii) The matrix representation of an operator A in terms of its eigenstates is a diagonal matrix because of the orthonormality property (1.44). It can, therefore, be written as {A} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 0 0. 0 0 A22 0. . 0 0 A33 . . . . .. . 0 . .. ANN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1.85) where Amm = am where am’s are the eigenvalues of A. The matrix representation of A in terms of eigenstates |bn of an operator B that is different from A and that does not share the same eigenstates is then written as {A} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 A13 . A1N A21 A22 A23 . . A31 A32 A33 . . . . .. . AN1 . .. ANN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1.86) where Amn = bm |A| bn. If the operator B has the same eigenstates as A then the above matrix will, once again, be diagonal. (iv) We now illustrate the usefulness of the completeness relation by considering several operator relations. First let us consider the matrix representation of the product of two operators {AB} by writing bm |AB| bn=bm |A1B| bn = p bm A bp bp B bn. (1.87)
14 Basic formalism In the second equality we have inserted a unit operator between the operators A and B and then replaced it by the sum of the complete set of states.Hence the matrix representation ofB is simply the product of two matrices: BIN (4B)= 1.88) Next we consider the operator relation Aa)=B). (1.89) It can be written as a matrix relation if we multiply both sides of the equation by the eigenstates(bl of an operator B and then insert a complete set of states bp)with p=1,2,. (bmlA∑lbp)pla)=∑(bp)(bp la)=(bmlB). (1.90) This is a matrix equation in which Ais represented by the matrix in(1.86),and )and B)are represented by the column matrices (bI la) (b11B)1 (b2 la) and (b2IB) (1.91) L(bx la) L(bwIB)」 respectively,and hence the above relation can be written in terms of matrices as A12 (bla） (biIB) 2 (b2la)》 (b2I) (1.92) (bN la) (bN IB) We can now follow the rules of matrix multiplications and write down N simultaneous (v)Amatrix element such as ()can itselfbe expressed in terms ofa matrix relation by using the completeness relation (1.93) =∑∑1blb)b
14 Basic formalism In the second equality we have inserted a unit operator between the operators A and B and then replaced it by the sum of the complete set of states. Hence the matrix representation of AB is simply the product of two matrices: {AB} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 . . A1N A21 A22 .. . . . .. . . . .. . AN1 . .. ANN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ B11 B12 . . B1N B21 B22 .. . . . .. . . . .. . BN1 . .. BNN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (1.88) Next we consider the operator relation A |α = |β. (1.89) It can be written as a matrix relation if we multiply both sides of the equation by the eigenstates bm| of an operator B and then insert a complete set of states bp with p = 1, 2, ... : bm| A ⎡ ⎣ p bp bp ⎤ ⎦ |α = p bm| A bp bp |α = bm |β. (1.90) This is a matrix equation in which A is represented by the matrix in (1.86), and |α and |β are represented by the column matrices ⎡ ⎢ ⎢ ⎣ b1 |α b2 |α . bN |α ⎤ ⎥ ⎥ ⎦ and ⎡ ⎢ ⎢ ⎣ b1 |β b2 |β . bN |β ⎤ ⎥ ⎥ ⎦ (1.91) respectively, and hence the above relation can be written in terms of matrices as ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 . . A1N A21 A22 .. . . . .. . . . .. . AN1 . .. ANN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ b1 |α b2 |α . . bN |α ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ b1 |β b2 |β . . bN |β ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (1.92) We can now follow the rules of matrix multiplications and write down N simultaneous equations. (v) A matrix element such as ψ| A |φ can itself be expressed in terms of a matrix relation by using the completeness relation ψ| A |φ = ψ| m |bm bm| A n |bn bn| |φ (1.93) = m n ψ |bm bm| A |bn bn| φ