Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all the possible wave- functions of a given system. Y1(x), Y2(x), ..., are wavefunctions in the representation of x. x can be general. For example, in the coordinate representation x = (~q;s), where ~q represents the set of coordinates (~q1;:::~qN), and s represents the set of particle spin (s1;z;:::;s2;z). The inner product between two wavefunctions are defined as Z Z (Y ;Y ) = dxY∗(x)Y(x) = d~q :::d~q Y∗ (~q ;::~q )Y (~q ;::;~q ): (1) A B ∑ 1 N s1s2:::sN 1 N s1s2:::sN 1 N s1;z;:::;sN;z Space of state vectors (right-vectors); the Hilbert space More conveniently, we use the notation of the state vector jYi (right/ket-vector) to represent a wavefunction Y(x) of a given quantum system. The advantage of the state vector notation is that it does not depend on concrete representations. All the state vectors jYi span the linear space denoted as the Hilbert space H of a given quantum system. The following correspondence between a wavefunction and a state vector is defined as: 1) Y(x) !jYi; 2) c1Y1(x) + c2Y2(x) ! c1jY1i + c2jY2i; 3) The inner product: (Y1;Y2) ! (jY1i;jY2i). The orthonormal complete bases for the space of state vectors For an orthonormal complete bases Ya, we have ´ ´ ∗ ´ ´ (Ya;Ya) = d(a;a ); ∑Ya(x)Ya(x ) = d(x − x ): (2) a For any wavefunction Y, we can expand it as Y(x) = ∑Ya(x)(Ya(x);Y); (3) a and the inner product as ´ ´ ∗ (Y ;Y) = ∑(Ya;Y ) (Ya;Y); (4) a Using the formalism of right-vectors, we use jYai to represent the wavefunction of 1 Ya(x), and rewrite the above equations as ´ ´ (jYai;jYai = d(a;a ); jYi = ∑jYai(jYai;jYi); a ´ ´ ∗ (jY i;jYi) = ∑(jYa;ijY i) (jYai;jYi); (5) a where (jYai;jYi) is the coordinate of the state vector jYi projection to the basis jYai. The dual space (space of left-vectors) The right-vector space H is a linear space. All the linear mappings from the right-vector space H to the complex number field C also form a linear space, which is denoted as the dual space. We use left-vector (bra-vector) to denote an element in the dual space. Let us consider a linear mapping denoted by hAj in the dual space, which can be determined by its operation on the orthonormal bases Fa in the Hilbert space H . ∗ hA : jFai ! aa; (6) ∗ where aa is a complex number, and a is the index to mark the orthonormal bases. Then for any right-vector jBi = ∑a bajYai where ba = (jBi;jYai), the operation of hAj on jBi is represented as ∗ hAj : jBi ! ∑aaba: (7) a We can identify a one-to-one correspondence between a right-vector and a linear mapping (a left-vector) as hAj !jAi = aajFai; (8) such that the operation of hAj on any right-vector jBi is expressed as ∗ hAj : jBi ! ∑aaba = (jAi;jBi): (9) a Below we will simply use the notation hAjBi to denote the mapping hAj : jBi. Using these notations, we can rewrite Eq. 5 as ´ ´ hYajYai = d(a;a ); jYi = ∑jYaihYajYi: (10) a We define the conjugation operation for left and right vectors, and complex numbers as jYi = hYj; hfj = jfi;a ¯ = a∗; (11) thus we have ajAi = hAja¯; ahBj = jBia¯: (12) 2 2 Operator of an observable We define the linear operator L acting on the right-vectors in H , which satisfies 1) LjBi is still a right-vector in H , 2) L(bjBi + cjCi) = bLjBi + cLjCi. L can be determined through its operation on the orthonormal basis jFbi of H as LjFbi = ∑LabjFai; (13) a where Lab = hYajLjYbi is the matrix element of L for the basis of Fa. For any state-vector jBi = ∑a bajFai, the operation of L is LjBi = ∑bbLjFbi = ∑LabbbjFai: (14) b ab The operation of L on the left vectors can also be defined. For a given hAj, the operation of hAjL is defined through the following equation hAjL : jBi = hAjLjBi; (15) where jBi is an arbitrary right-vector. Thus hAjL is a linear mapping from the right-vector space to complex numbers, thus it should be represented by a left-vector hfj = hAjL, such that hAjL : jBi = hfjBi. The conjugation operations We further extend the definition of conjugation operation be- low hY1jLjY2i = hY2jLjY1i; L1L2 = L2 L1; LjYi = hYjL; hAjBi = hBjAi: (16) In the following, we denote L as L†. † In the orthonormal basis Ya, the matrix elements of L reads † † ∗ Lab = hYajL jYbi = hYbjLjYai = Lba: (17) † ∗ If the operators L = L , i.e., Lab = Lba, we call that L is Hermitian. 3 3 Outer product between left and right-vectors as opera- tors We define jAihBj as a linear operator. When acting on a right-vector jYi, it behaves as (jAihBj)jYi = jAihBjYi: (18) Corollary: 1)hYj(jAihBj) = hYjAihBj; 2) jAihBj = jBihAj, 3) jAihAj is Hermitian. 4) jAihBjYi = hYjBihAj = jYi jAihBj. 5) For a set of orthonormal bases jYai, from Eq. 5, we have I = ∑jYaihYaj; (19) a where I is the identity operator. 6) Expansion of a linear operator L as L = ∑ jYaihYajLjYa´ ihYa´ j = ∑ jYaihYa´ jLaa´ ; (20) aa´ aa´ where Laa´ = hYajLjYa´ i is the matrix element under the bases of jYai. Examples: 1)For a single spinless particle, we denote j~ri as the eigenstate of the coordinate operator ~r, which satisfy the orthonormal condition h~rj~r´i = d(~r −~r´). We have Z d~rj~rih~rj = I; (21) thus Z d~rj~rih~rjYi = jYi; (22) and Z d~r´h~rj~r´ih~r´jYi = h~rjYi = Y(~r): (23) 4 Similarly, for an orthonormal basis Ya, we have ∗ ´ ∗ ´ ´ ´ ∑Ya(~r)Ya(~r ) = ∑h~rjYai h~r jYai = ∑h~r jYaihYaj~ri = h~r jf∑jYaihYagj~ri a a a a = h~r´j~ri = d(~r −~r´): (24) 4 Representations and transformation of representations When we fix a set of orthonormal bases jYai for the Hilbert space, it means that we are using a specific representation. We can express a state vector jYi and a linear operator L as matrices as jAi = ∑jYaihYajAi; a ´ ´ L = ∑ jYaihYajhYajLjYai; (25) aa´ and hAjBi = ∑hAjYaihYajBi a ∗ hAjLjBi = ∑hYajAi LabhYajBi: (26) a´ Using the matrix notation, we denote Aa = hYajAi, then in the representation of jYai, jAi is represented by a column vector of Aa, and L is represented by a matrix Lab. In the matrix notation, we have ∗ ∗ hAjBi = ∑AaBa; hAjLjBi = ∑AaLabBb: (27) a ab Let us choose another set of orthonormal basis jjli, which satisfy ∑l jjlihjlj = I. The transformation matrix U between these two sets of bases is defined as jjli = ∑jYaihYajjli = ∑jYaiUal; (28) a a where Ual = hYajjli. U is an unitary matrix, which satisfies the following relation U†U = UU† = I: (29) For an arbitrary state vector jAi, its coordinate hYajAi in the jYi representation can be ex- pressed in terms of its coordinates in the jji representation through the transformation matrix 5 U as hYajAi = ∑hYajjlihjljAi = ∑UalhjljAi: (30) l l And for the matrix element Laa´ in the jYi-representation can also be related to that in the in the jji representation as † h jLj ´ i = h j ih jLj ´ ih ´ j ´ i = U h jLj ´ iU : (31) Ya Ya ∑ Ya jl jl jl jl Ya al jl jl la´ ll´ 6.