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 ID: 1001575
Download Presentation The PPT/PDF document "Chapter 13 Matrix Representation" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
1. Chapter 13
2. Matrix RepresentationMatrix 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 Eigenvectors and eigenvalues Matrix method Find superposition of basis states that are eigenstates of particular operator. Get eigenvalues.Copyright – Michael D. Fayer, 2018
3. Orthonormal basis set in N dimensional vector spacebasis vectorsAny N dimensional vector can be written aswithTo get this, project outCopyright – Michael D. Fayer, 2018
4. Operator equationSubstituting the series in terms of bases vectors.Left mult. by The N2 scalar productsare completely determined byN values of j for each yi; and N different yi Copyright – Michael D. Fayer, 2018
5. WritingMatrix elements of A in the basis gives for the linear transformationKnow the aij because we know A and The set of N linear algebraic equations can be written asdouble underline means matrixIn terms of the vector representatives(Set of numbers, gives you vector when basis is known.)vectorvector representative,must know basisCopyright – Michael D. Fayer, 2018
6. array of coefficients - matrixThe aij are the elements of the matrix . The product of matrix and vector representative xis a new vector representative y with componentsvector representatives in particular basisCopyright – Michael D. Fayer, 2018
7. Matrix Properties, Definitions, and RulesTwo matrices, and are equalif aij = bij.The unit matrixones down principal diagonalGives identity transformationCorresponds toThe zero matrixCopyright – Michael D. Fayer, 2018
8. Matrix multiplicationConsiderUsing the same basis for both transformationshas elementsLaw of matrix multiplicationoperator equationsExampleCopyright – Michael D. Fayer, 2018
9. Multiplication AssociativeMultiplication NOT Commutative except in special cases.Matrix addition and multiplication by complex numberCopyright – Michael D. Fayer, 2018
10. Reciprocal of ProductFor matrix defined as interchange rows and columnsTransposeComplex Conjugatecomplex conjugate of each elementHermitian Conjugatecomplex conjugate transposeCopyright – Michael D. Fayer, 2018Inverse of a matrix inverse of identity matrixtranspose of cofactor matrix (matrix of signed minors)determinant
11. Rulestranspose of product is product of transposes in reverse orderdeterminant of transpose is determinantcomplex conjugate of product is product of complex conjugatesdeterminant of complex conjugate is complex conjugate of determinantHermitian conjugate of product is product of Hermitian conjugates in reverse orderdeterminant of Hermitian conjugate is complex conjugateof determinantCopyright – Michael D. Fayer, 2018
12. DefinitionsSymmetricHermitianRealImaginaryUnitaryDiagonalPowers of a matrixCopyright – Michael D. Fayer, 2018
13. Column vector representative one column matrixthenvector representatives in particular basisbecomesrow vector transpose of column vectortransposeHermitian conjugateCopyright – Michael D. Fayer, 2018
14. Change of Basisorthonormal basisthenSuperposition of can form N new vectors linearly independenta new basiscomplex numbersCopyright – Michael D. Fayer, 2018
15. New Basis is Orthonormalif the matrixcoefficients in superpositionmeets the conditionis unitary – Hermitian conjugate = inverseImportant result. The new basis will be orthonormal if , the transformation matrix, is unitary (see book and Errata and Addenda, linear algebra book ).Copyright – Michael D. Fayer, 2018
16. Unitary transformation substitutes orthonormal basis for orthonormal basis .VectorSame vector – different basis.vector – line in space (may be high dimensionality abstract space)written in terms of two basis setsThe unitary transformation can be used to change a vector representativeof in one orthonormal basis set to its vector representative in another orthonormal basis set.x – vector rep. in unprimed basisx' – vector rep. in primed basischange from unprimed to primed basischange from primed to unprimed basisCopyright – Michael D. Fayer, 2018
17. ExampleConsider basisVector - line in real space.In terms of basisVector representative in basisCopyright – Michael D. Fayer, 2018
18. Change basis by rotating axis system 45° around . Can find the new representative of , s' is rotation matrixFor 45° rotation around z Copyright – Michael D. Fayer, 2018
19. Thenvector representative of in basisSame vector but in new basis.Properties unchanged.Example – length of vectorCopyright – Michael D. Fayer, 2018
20. Can go back and forth between representatives of a vector bychange from unprimed to primed basischange from primed to unprimed basiscomponents of in different basisCopyright – Michael D. Fayer, 2018
21. Consider the linear transformationoperator equationIn the basis can writeorChange to new orthonormal basis usingorwith the matrix given byBecause is unitaryCopyright – Michael D. Fayer, 2018
22. Extremely ImportantCan change the matrix representing an operator in one orthonormal basisinto the equivalent matrix in a different orthonormal basis.CalledSimilarity TransformationCopyright – Michael D. Fayer, 2018
23. In basis Go into basis Relations unchanged by change of basis.ExampleCan insert between because ThereforeCopyright – Michael D. Fayer, 2018
24. Isomorphism between operators in abstract vector spaceand matrix representatives.Because of isomorphism not necessary to distinguishabstract vectors and operatorsfrom their matrix representatives.The matrices (for operators) and the representatives (for vectors)can be used in place of the real things.Copyright – Michael D. Fayer, 2018
25. Hermitian Operators and MatricesHermitian operatorHermitian operator Hermitian Matrix+ - complex conjugate transpose - Hermitian conjugateCopyright – Michael D. Fayer, 2018
26. Theorem (Proof: Powell and Craseman, P. 303 – 307, or linear algebra book)For a Hermitian operator A in a linear vector space of N dimensions,there exists an orthonormal basis,relative to which A is represented by a diagonal matrix .The vectors, , and the corresponding real numbers, ai, are thesolutions of the Eigenvalue Equationand there are no others.Copyright – Michael D. Fayer, 2018
27. Application of TheoremOperator A represented by matrixin some basis . The basis is any convenient basis. In general, the matrix will not be diagonal.There exists some new basis eigenvectorsin which represents operator and is diagonal eigenvalues.To get from to unitary transformation.Similarity transformation takes matrix in arbitrary basisinto diagonal matrix with eigenvalues on the diagonal.Copyright – Michael D. Fayer, 2018
28. 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.Real dynamical variables (observables) are represented by Hermitian operators.Observables are eigenvalues of Hermitian operators.Solution of eigenvalue problem gives eigenvalues and eigenvectors.Copyright – Michael D. Fayer, 2018
29. Matrix RepresentationHermitian operators replaced by Hermitian matrix representations.In proper basis, is the diagonal Hermitian matrix and the diagonal matrix elements are the eigenvalues (observables).A suitable transformation takes (arbitrary basis) into (diagonal – eigenvector basis)Diagonalization of matrix gives eigenvalues and eigenvectors.Matrix formulation is another way of dealing with operators and solving eigenvalue problems.takes arbitrary basis into eigenvectors.Copyright – Michael D. Fayer, 2018
30. All rules about kets, operators, etc. still apply.Example Two Hermitian matricescan be simultaneously diagonalized by the same unitarytransformation if and only if they commute.All ideas about matrices also true for infinite dimensional matrices.Copyright – Michael D. Fayer, 2018
31. Example – Harmonic OscillatorHave already solved – use occupation number representation kets and bras (already diagonal).matrix elements of aCopyright – Michael D. Fayer, 2018
32. Copyright – Michael D. Fayer, 2018
33. Copyright – Michael D. Fayer, 2018
34. Adding the matrices and and multiplying by ½ gives The matrix is diagonal with eigenvalues on diagonal. In normal unitsthe matrix would be multiplied by . This example shows idea, but not how to diagonalize matrix when youdon’t already know the eigenvectors.Copyright – Michael D. Fayer, 2018
35. DiagonalizationEigenvalue equationmatrix representingoperatorrepresentative of eigenvectoreigenvalueIn terms of the componentsThis represents a system of equationsWe know the aij.We don't knowa - the eigenvaluesui - the vector representatives, one for each eigenvalue.Copyright – Michael D. Fayer, 2018
36. Besides the trivial solutionA solution only exists if the determinant of the coefficients of the ui vanishes.Expanding the determinant gives Nth degree equation for the unknown a's (eigenvalues).know aij, don't know a'sCopyright – Michael D. Fayer, 2018Then substituting one eigenvalue at a time into system of equations, the ui (eigenvector representatives) are found. N equations for u's gives only N - 1 conditions.Use normalization.
37. Example - Degenerate Two State ProblemBasis - time independent kets orthonormal. a and b not eigenkets.Coupling g.These equations define H.The matrix elements areAnd the Hamiltonian matrix isCopyright – Michael D. Fayer, 2018
38. The corresponding system of equations isThese only have a solution if the determinant of the coefficients vanish.Ground StateE = 0E02gExcited StateDimer SplittingExpandingEnergy EigenvaluesTake thematrixMake into determinant.Subtract λ from the diagonalelements.Copyright – Michael D. Fayer, 2018
39. To obtain Eigenvectors Use system of equations for each eigenvalue.Eigenvectors associated with l+ and l-.and are the vector representatives of andin theWe want to find these.Copyright – Michael D. Fayer, 2018
40. First, for the eigenvaluewrite system of equations. The result isMatrix elements of The matrix elements areCopyright – Michael D. Fayer, 2018
41. An equivalent way to get the equations is to use a matrix form.Substitute Multiplying the matrix by the column vector representative gives equations.Copyright – Michael D. Fayer, 2018
42. The two equations are identical.Always get N – 1 conditions for the N unknown components.Normalization condition gives necessary additional equation.ThenandEigenvector in terms of thebasis set. Copyright – Michael D. Fayer, 2018
43. For the eigenvalueusing the matrix form to write out the equationsSubstitutingThese equations giveUsing normalizationThereforeCopyright – Michael D. Fayer, 2018
44. Can diagonalize by transformationdiagonal not diagonalTransformation matrix consists of representatives of eigenvectorsin original basis.complex conjugate transposeCopyright – Michael D. Fayer, 2018
45. ThenFactoring out , one from each matrix.after matrix multiplicationmore matrix multiplicationdiagonal with eigenvalues on diagonalCopyright – Michael D. Fayer, 2018