10 - 15.73 Lecture #10updated 9/18/02 8:55 AMMatrix Mechanicsshould have read CDTL pages 94-121read CTDL pages 121-144 ASAPLast time: Numerov-Cooley Integration of 1-D Schr. Eqn. Defined on a Grid.2-sided boundary conditionsnonlinear system - iterate to eigenenergies (Newton-Raphson)So far focussed on ψ(x) and Schr. Eq. as differential equation.Variety of methods {Ei, ψi(x)} ↔ V(x)Often we want to evaluate integrals of the formψφ*() ()x x dx aii=∫overlap of specialψ on standardfunctions {φ}a is “mixing coefficient”{φ} is complete set of “basisfunctions”ORφi*ˆxnφjdx ≡ xn()ij∫expectation valuestransition momentsThere are going to be elegant tricks for evaluating these integrals and relating oneintegral to others that are already known. Also “selection” rules for knowingautomatically which integrals are zero: symmetry, commutation rulesToday: begin matrix mechanics - deal with matrices composed of these integrals -focus on manipulating these matrices rather than solving a differentialequation - find eigenvalues and eigenvectors of matrices instead(COMPUTER “DIAGONALIZATION”)* Perturbation Theory: tricks to find approximate eigenvalues of infinite matrices* Wigner-Eckart Theorem and 3-j coefficients: use symmetry to identify and inter-relate values of nonzero integrals* Density Matrices: information about state of system as separate frommeasurement operatorsHIGHLIGHTS10 - 25.73 Lecture #10updated 9/18/02 8:55 AMFirst Goal: Dirac notation as convenient NOTATIONAL simplificationIt is actually a new abstract picture(vector spaces) — but we will stress the utility (ψ ↔ | 〉 relationships)rather than the philosophy!Find equivalent matrix form of standard ψ(x) concepts and methods.1. Orthonormality 2. completeness ψ(x) is an arbitrary function A. Always possible to expand ψ(x) uniquely in a COMPLETE BASIS SET {φ}ψi*ψjdx =δij∫ψ(x) = aiφi(x)i∑ai=φi*ψdx∫mixing coefficient — how to get it?Always possible to expand in { } since we can write in terms of { }.So simplify the question we are asking to What are the bj Multiply by j*ˆˆ?BBibjjjψφ ψ φφφ φ=∑{}∫B.φj*∫bj=φj*ˆBφidx ≡ Bji∫ˆBφi= Bjiφjj∑note counter-intuitive pattern ofindices. We will return to this.left multiply by* The effect of any operator on ψi is to give a linear combination of ψj’s.(expand ψ)(expand Bψ)*10 - 35.73 Lecture #10updated 9/18/02 8:55 AM3. Products of OperatorsˆAˆB()φi=ˆAˆBφi()=ˆABjiφjj∑can move numbers (but not operators) around freely= Bjij∑ˆAφj= BjiAkjk∑φkj∑= AkjBji()j,k∑φk= AB()kik∑φk* Thus product of 2 operators follows the rules of matrix multiplication:ˆAˆB acts like A BRecall rules for matrix multiplication:indices areArow,columnmust match # of columns on left to # of rows on rightNN NN NNNNNNNN×()⊗×()→×()×()⊗×()→×()×()⊗×()→×()a matrixa numbera matrix!" row vector" " column vector"columnvectorrowvector111111ordermatters!Need a notation that accomplishes all of this memorably and compactly.note repeated index10 - 45.73 Lecture #10updated 9/18/02 8:55 AMDirac’s bra and ket notationHeisenberg’s matrix mechanicsket is a column matrix, i.e. a vectora1a2MaNcontains all of the “mixing coefficients” for ψ expressed in some basis set.[These are projections onto unit vectors in N-dimensional vector space.]Must be clear what state is being expanded in what basisψ(x) =φi*ψdx∫[]i∑φi(x)ψ=φ1*ψdx∫φ2*ψdx∫MφN*ψdx∫* ψ expressed in φ basis* a column of complex #s* nothing here is a function of xφ ← bookkeeping device (RARE)OR, a pure state in its own basisφ2=010M0φone 1, all others 0bra is a row matrix contains all mixing coefficients for in basis set(This is * of (x) above)bb bxdxxNiii12,*( ) * ( )***…(){}=[]∫∑ψφψφψφ ψThe * stuff is needed to make sure ψψ = 1 even though φiψ is complex.ai10 - 55.73 Lecture #10updated 9/18/02 8:55 AMThe symbol 〈a|b〉, a bra–ket, is defined in the sense of product of(1 × N) ⊗ (N × 1) matrices → a 1 × 1 matrix: a number!Box Normalization in both ψ and 〈 | 〉 pictures1 =ψ* ψdx∫ψ= φi*ψdx∫()i∑φiψ* =φjψ *dx()φj*j∑expand both inφ basis112==()=∫∫∫∑∫∫∑ψ ψ φψ φψ φφφψ***,**dx dx dx dxdxjiijjijjreal, positive #’sδijc.c.now in picturesame resultrow vector: “bra”column vector “ket”jjψψ φψ φψφψφψφψ==∫∫∫∫∫∑12122*****dx dxdxdxdxL12444444 3444444M12434[CTDL talks about dual vector spaces — best to walk before you run. Alwaystranslate 〈 〉 into ψ picture to be sure you understand the notation.]forces 2 sums tocollapse into 1pWe have proved that sum of |mixing coefficients|2 = 1. These are called“mixing fractions” or “fractional character”.10 - 65.73 Lecture #10updated 9/18/02 8:55 AMAny symbol 〈 〉 is a complex number.Any symbol | 〉 〈 | is a square matrix. again ψψ ψφ ψφφψφψψφ φψ ψψ=…()===∑12121Miii1 unit matrix what is φ1φ1=1 0 … 0()10M0=10L00MOwhat is φiφii∑ =11001Ounit or identitymatrix = 1“completeness” or “closure” involves insertion of 1 between any two symbols.a shorthand forspecifying onlythe importantpart of aninfinite matrix10 - 75.73 Lecture #10updated 9/18/02 8:55 AMUse 1 to evaluate matrix elements of product of 2 operators, AB (we know howto do this in ψ picture). φφφφ φφφφijjjijij ikkkjik kjijkAAAABAAAB A BAB=……()()=……()====()∑∑01001001012::Ma numberj-th position – picksout j-th column of A1i-thsquare matrixi-th〈x ψ 〉 is the same thing as ψ(x) ↑x is continuously variable ↔ δ(x)overlap of state vector ψ with δ(x) – a complex number. ψ(x) is a complexfunction of a real variable.i.e., ∫′()′()′=δψ ψxx x dx x,* ()In Heisenberg picture, how do we get exact equivalent of ψ(x)?basis set δ(x,x0) for all x0 – this is a complete basis (eigenbasisfor x^, eigenvalue x0) - perfect localization at any x010 - 85.73 Lecture #10updated