Matrix Representation

Transcription

1 Matrix Representation Matrix Rep. Same basics as introduced already. Convenient method of working with vectors. Superposition Complete set of vectors can be used to express any other vector. Complete set of N orthonormal vectors can form other complete sets of N orthonormal vectors. Can find set of vectors for Hermitian operator satisfying Au u. Eigenvectors and eigenvalues Matrix method Find superposition of basis states that are eigenstates of particular operator. Get eigenvalues.

2 Orthonormal basis set in N dimensional vector space e j basis vectors Any N dimensional vector can be written as x N j1 x e j j with j x j e x To get this, project out j j e e from x j j j j x e e e x piece of x that is e, j j then sum over all e.

3 Operator equation y Ax N N j j j1 j1 y e A x e j j Substituting the series in terms of bases vectors. N j1 Left mult. by x Ae j i e j N i j yi e A e xj j1 The N 2 scalar products e i A e j are completely determined by A j and the basis set e. N values of j for each y i ; and N different y i

4 Writing i a e A e ij j gives for the linear transformation Matrix elements of A in the basis e j y y N ax j 1, 2, N i ij j j1 In terms of the vector representatives x1 y1 x 2 x y 2 y x N yn (Set of numbers, gives you vector when basis is known.) Ax Know the a ij because we know A and e The set of N linear algebraic equations can be written as double underline means matrix j Q 7xˆ 5yˆ 4zˆ vector 7 5 vector representative, must know basis 4

5 A array of coefficients - matrix A a a a a a a a a a N N aij N1 N2 NN The a ij are the elements of the matrix A. y Ax vector representatives in particular basis y1 a11 a12 a1n x1 y 2 a21 a 22 x 2 y a a a x N N1 N2 NN N A The product of matrix and vector representative x is a new vector representative y with components y N ax i ij j j1

6 Matrix Properties, Definitions, and Rules Two matrices, A and B are equal A B if a ij = b ij. The unit matrix ij 1 Gives identity transformation N y x x i ij j i j1 Corresponds to ones down principal diagonal The zero matrix x y 1 x x

7 Matrix multiplication Consider y Ax z By operator equations z BA x Using the same basis for both transformations z N b y z By k ki i i1 B matrix ( b ki ) z By BAx Cx C BA has elements Example c N b a kj ki ij i1 Law of matrix multiplication

8 Multiplication Associative ABC ABC Multiplication NOT Commutative except in special cases. AB BA Matrix addition and multiplication by complex number A B C c a b ij ij ij

9 Reciprocal of Product AB B A For matrix defined as A ( a ij ) Transpose A a ji interchange rows and columns * * A a ij Complex Conjugate Hermitian Conjugate * A a ji Inverse of a matrix inverse of A A 1 A A CT complex conjugate of each element complex conjugate transpose A 1 A 1 1 AA A A 1 identity matrix transpose of cofactor matrix (matrix of signed minors) determinant A If A A is singular

10 Rules AB BA transpose of product is product of transposes in reverse order A A determinant of transpose is determinant * ( AB) * * A B complex conjugate of product is product of complex conjugates * * A A determinant of complex conjugate is complex conjugate of determinant ( AB) B A Hermitian conjugate of product is product of Hermitian conjugates in reverse order A A * determinant of Hermitian conjugate is complex conjugate of determinant

11 Definitions A A Symmetric A A Hermitian A A * Real A A * Imaginary A 1 A Unitary a a ij ij ij Diagonal Powers of a matrix 1 2 A 1 A A A AA e A A 1 A 2! 2

12 Column vector representative one column matrix x1 x 2 x vector representatives in particular basis xn then y Ax becomes y1 a11 a12 a1n x1 y 2 a21 a 22 x 2 y a a a x N N1 N2 NN N row vector x x x x 1, 2 N transpose of column vector y Ax y xa y Ax y x A transpose Hermitian conjugate

13 Change of Basis orthonormal basis i e then i j e e i, j 1,2, N ij Superposition of a new basis i e i e can form N new vectors linearly independent N i k e u e i 1, 2, N k1 ik complex numbers

14 New Basis is Orthonormal j i e e ij if the matrix U U U u ik meets the condition 1 coefficients in superposition N i k e u e i 1, 2, N k1 ik U 1 U U is unitary Hermitian conjugate = inverse Important result. The new basis will be orthonormal UU U U i e if U, the transformation matrix, is unitary (see book and Errata and Addenda, linear algebra book ). 1

15 Unitary transformation substitutes orthonormal basis for orthonormal basis. Vector x e e x x xi e i i i i xi e vector line in space (may be high dimensionality abstract space) written in terms of two basis sets x Same vector different basis. The unitary transformation U can be used to change a vector representative of x in one orthonormal basis set to its vector representative in another orthonormal basis set. x vector rep. in unprimed basis x' vector rep. in primed basis x Ux x U x change from unprimed to primed basis change from primed to unprimed basis

16 Example Consider basis y xˆ, yˆ, zˆ s z x Vector s - line in real space. In terms of basis s 7xˆ 7yˆ 1zˆ Vector representative in basis xˆ, yˆ, zˆ s 7 7 1

17 Change basis by rotating axis system 45 around ẑ. Can find the new representative of s, s' s U s U is rotation matrix U x y z cosθ sinθ sinθ cosθ 1 For 45 rotation around z U 2/2 2/2 2/2 2/2 1

18 Then 2/2 2/ s 2/2 2/ s 1 vector representative of s in basis e Same vector but in new basis. Properties unchanged. Example length of vector ss 1/2 1/2 ss 1/2 * 1/2 1/2 1/2 ( s s) ( ) (99) ss s s * 1/2 1/2 1/2 ( ) (2 49 1) (99)

19 Can go back and forth between representatives of a vector x by change from unprimed to primed basis change from primed to unprimed basis x Ux x U x components of x in different basis

20 Consider the linear transformation y Ax operator equation y Ax In the basis e can write or y a x i ij j j Change to new orthonormal basis e using U yuyuaxuau x or y Ax A U AU with the matrix A given by Because U is unitary A U AU 1

21 Extremely Important A U AU 1 Can change the matrix representing an operator in one orthonormal basis into the equivalent matrix in a different orthonormal basis. Called Similarity Transformation

22 y Ax ABC ABC In basis e Go into basis e y Ax AB C A B C Relations unchanged by change of basis. Example AB C UABU UCU Can insert U U between AB because UAU UBU UCU 1 U U U U 1 Therefore AB C

23 Isomorphism between operators in abstract vector space and matrix representatives. Because of isomorphism not necessary to distinguish abstract vectors and operators from their matrix representatives. The matrices (for operators) and the representatives (for vectors) can be used in place of the real things.

24 Hermitian Operators and Matrices Hermitian operator x Ay yax Hermitian operator Hermitian Matrix A A + - complex conjugate transpose - Hermitian conjugate

25 Theorem (Proof: Powell and Craseman, P , or linear algebra book) For a Hermitian operator A in a linear vector space of N dimensions, there exists an orthonormal basis,, 1 2 N U U U relative to which A is represented by a diagonal matrix 1 2 A N. i The vectors, U, and the corresponding real numbers, i, are the solutions of the Eigenvalue Equation AU U and there are no others.

26 Application of Theorem Operator A represented by matrix A i in some basis e. The basis is any convenient basis. In general, the matrix will not be diagonal. There exists some new basis eigenvectors U i in which A represents operator and is diagonal eigenvalues. i To get from to i i U U e. e i U unitary transformation. A U AU 1 Similarity transformation takes matrix in arbitrary basis into diagonal matrix with eigenvalues on the diagonal.

27 Matrices and Q.M. Previously represented state of system by vector in abstract vector space. Dynamical variables represented by linear operators. Operators produce linear transformations. y Ax Real dynamical variables (observables) are represented by Hermitian operators. Observables are eigenvalues of Hermitian operators. AS S Solution of eigenvalue problem gives eigenvalues and eigenvectors.

28 Matrix Representation Hermitian operators replaced by Hermitian matrix representations. A A In proper basis, A is the diagonal Hermitian matrix and the diagonal matrix elements are the eigenvalues (observables). 1 A suitable transformation UAU takes A (arbitrary basis) into A (diagonal eigenvector basis) A U AU 1. U takes arbitrary basis into eigenvectors. Diagonalization of matrix gives eigenvalues and eigenvectors. Matrix formulation is another way of dealing with operators and solving eigenvalue problems.

29 All rules about kets, operators, etc. still apply. Example Two Hermitian matrices A and B can be simultaneously diagonalized by the same unitary transformation if and only if they commute. All ideas about matrices also true for infinite dimensional matrices.

30 Example Harmonic Oscillator Have already solved use occupation number representation kets and bras (already diagonal). 1 H P x 2 an 1 2 aa a a 2 2 nn1 a n n1 n1 matrix elements of a a a 1 1 a 2 1 a 1 a 1 1 a a a

31 a H aa a a 2 aa

32 1 H aa a a 2 a a

33 Adding the matrices aa and a a and multiplying by ½ gives H H The matrix is diagonal with eigenvalues on diagonal. In normal units the matrix would be multiplied by. This example shows idea, but not how to diagonalize matrix when you don t already know the eigenvectors.

34 Diagonalization Eigenvalue equation eigenvalue Au u Auu matrix representing operator representative of eigenvector In terms of the components N aij ij uj i 1,2 N j1 This represents a system of equations a u a u a u a u a u a u a u a u a u We know the a ij. We don't know - the eigenvalues u i - the vector representatives, one for each eigenvalue.

35 Besides the trivial solution u u u N 1 2 A solution only exists if the determinant of the coefficients of the u i vanishes. a11 a12 a13 a21 a22 a23 a31 a32 a33 Expanding the determinant gives N th degree equation for the the unknown 's (eigenvalues). know a ij, don't know 's Then substituting one eigenvalue at a time into system of equations, the u i (eigenvector representatives) are found. N equations for u's gives only N - 1 conditions. Use normalization. uu uu u u * * * N N 1

36 Example - Degenerate Two State Problem Basis - time independent kets orthonormal. H E H E and not eigenkets. Coupling. These equations define H. The matrix elements are H E H H H E And the Hamiltonian matrix is H E E

37 The corresponding system of equations is ( E ) ( E ) These only have a solution if the determinant of the coefficients vanish. E E Take the matrix E E Make into determinant. Subtract λ from the diagonal elements. Expanding 2 2 E 2E E E 2 Dimer Splitting Excited State Energy Eigenvalues E E Ground State E =

38 To obtain Eigenvectors Use system of equations for each eigenvalue. a b Eigenvectors associated with + and -. a b a, b and a, b are the vector representatives of and in the, basis set. We want to find these.

39 First, for the E write system of equations. eigenvalue ( H ) a H b H a ( H ) b H H ; H H ; H H ; H H Matrix elements of H ( E E ) a b a ( E E ) b The result is a b a b The matrix elements are H E H H H E

40 An equivalent way to get the equations is to use a matrix form. E a E b Substitute E E E a E E b a b Multiplying the matrix by the column vector representative gives equations. a b a b

41 a b a b The two equations are identical. a b Always get N 1 conditions for the N unknown components. Normalization condition gives necessary additional equation. a b Then a and b Eigenvector in terms of the basis set.

42 For the eigenvalue E using the matrix form to write out the equations E a E b Substituting E a b a a b b These equations give Using normalization 1 1 Therefore 2 2 a a b 1 1 b 2 2

43 Can diagonalize by transformation H U HU 1 diagonal not diagonal Transformation matrix consists of representatives of eigenvectors in original basis. U a a 1/ 2 1/ 2 b b 1/ 2 1/ 2 1 1/ 2 1/ 2 U 1/ 2 1/ 2 complex conjugate transpose

44 Then H H 1/ 2 1/ 2 E 1/ 2 1/ 2 E 1/ 2 1/ 2 1/ 2 1/ 2 Factoring out 1/ 2, one from each matrix. H E E after matrix multiplication E E E E more matrix multiplication E H E diagonal with eigenvalues on diagonal