5.73 F2018 Lecture 10: Matrix Mechanics

5.73 Lecture #10 10 - 1 Matrix Mechanics should have read CDTL pages 94-121 read CTDL pages 121-144 ASAP Last time: * Numerov-Cooley Integration of 1-D Schr. Eqn. Defined on a Grid. * 2-sided boundary conditions (two different kinds of boundary condition) * nonlinear 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 form overlap of special ψwith standard functions {φ} expectation values and transition moments ∫ * ψ (x)φi (x)dx = a i a is “mixing coefficient” φi is a member of a “complete” set of basis functions, {�} OR ∫ * n ( n) φi x̂ φ jdx ≡ x ij There are going to be elegant tricks for evaluating these integrals and relating one integral to others that are already known. Also “selection” rules for knowing automatically which integrals are zero: symmetry, commutation rules Today: begin matrix mechanics - deal with matrices composed of these integrals focus on manipulating these matrices rather than solving a differential equation - find eigenvalues and eigenvectors of matrices instead (COMPUTER “DIAGONALIZATION”). LINEAR ALGEBRA. H I G * Perturbation Theory: tricks to find approximate eigenvalues of infinite matrices H L * Wigner-Eckart Theorem and 3-j coefficients: use symmetry to identify and interrelate values of nonzero integrals I G H * Density Matrices: information about “state of system” as separate from “measurement operators” T S 1 5.73 Lecture #10 First Goal: 10 - 2 Dirac notation as convenient NOTATIONAL simplification It is actually a new abstract picture (vector spaces) — but we will stress the utility of ψ ↔ |〉 relationships rather than the philosophy! Find equivalent matrix form of standard ψ (x) concepts and methods. 1. Orthonormality ∫ ψ i ψ j dx = δ ij 2. Completeness ψ(x) is an arbitrary function (expand ψ) A. Always possible to expand ψ(x) uniquely in a COMPLETE BASIS SET {φ} ψ(x) = a iφ i (x) i mixing coefficient — how to get it? ∑ * (expand B̂ ψ) * B. left multiply by �*i and integrate over x a i = ∫ φ*i ψdx Always possible to expand B̂ψ in {φ} since we can write ψ in terms of {φ}. So simplify the question we are asking to B̂φ = ∑ b φ i j j j ⎧ ⎫ What are the ⎨ b ⎬ ? Multiply by ∫ φ* j ⎩ j⎭ b j = ∫ φ*j B̂φi dx ≡ B ji B̂ φ i = ∑ B ji φ j j note counter-intuitive pattern of indices. We will return to this. * The effect of any operator on �i is to give a linear combination of �j’s. 2 5.73 Lecture #10 10 - 3 3. Products of Operators (ÂB̂)φi = Â(B̂φ i ) = Â∑ B jiφ j j can move numbers (but not operators) around freely = ∑ B jiÂφ j = ∑ ∑ Bji Akjφ k j = note repeated j-index j k ∑( AkjB ji )φ k = ∑( AB) kiφ k j,k note repeated k-index k * Thus the product of 2 operators follows the rules of matrix multiplication: ÂB̂ acts like A B Recall rules for matrix multiplication: ⎛ ⎜ ⎝ ⎞⎛ ⎟⎜ ⎠⎝ ⎞ indices of a matrix are Arow ,column ⎟ ⎠ must match # of columns on left to # of rows on right the order of matrices matters! ( N × N) (1 × N ) ⊗ ⊗ " row vector" ( N ×1) column vector ( N × N) ( N ×1) → → ( N × N) (1× 1) a matrix a number → ( N × N) a matrix! " column vector" ⊗ (1 × N ) row vector Need a notation that accomplishes all of this memorably and compactly. 3 5.73 Lecture #10 10 - 4 Dirac’s bra and ket notation Heisenberg’s matrix mechanics ket is a column matrix , i.e. a vector ⎛ a1 ⎞ ⎜ a2 ⎟ ⎜⎜ ! ⎟⎟ ⎝ aN⎠ The ket contains all of the “mixing coefficients” for ψ expressed in some (implicit) basis set. [These are projections onto unit vectors in N-dimensional vector space.] Must be clear what state is being expanded in what basis ai [ ] ψ(x) = ∑ ∫ φi*ψdx φ i (x) i express {ψ} basis in {φi} basis ⎛ ∫ φ1* ψdx ⎞ ⎛ a1 ⎞ ⎜ φ * ψdx ⎟ ⎜a ⎟ ∫ 2 ⎟ =⎜ 2⎟ ψ =⎜ ⎜ ⎟ ⎜ ! ⎟ ! ⎜ * ⎟ ⎜ ⎟ ⎝ ∫ φ N ψdx⎠ φ ⎝ a N ⎠ * ψ expressed in φ basis * a column of complex #s * nothing here is a function of x bookkeeping device (RARELY USED) to specify basis set OR, a pure state in its own basis ⎛ 0⎞ ⎜ 1⎟ φ 2 = ⎜ 0⎟ one 1, all others 0 (often expressed as ∣2〉) ⎜⎜ ! ⎟⎟ ⎝ 0⎠ φ ⎛ ⎜ ψ = a1 ⎜ ⎜ ⎜⎝ 1 0 ! 0 ⎛ ⎞ ⎜ ⎟ ⎟ + a2 ⎜ ⎜ ⎟ ⎜⎝ ⎟⎠ bra 0 1 ! 0 ⎛ ⎞ ⎜ ⎟ ⎟ + …aN ⎜ ⎜ ⎟ ⎟⎠ ⎜⎝ 0 ! ! 1 ⎞ ⎟ ⎟ ⎟ ⎟⎠ a weighted sum of unit vectors is a row matrix ( b1, b2 …bN ) { } contains all mixing coefficients for ψ * in φ * basis set ψ * (x) = ∑ ⎡⎣ ∫ φiψ * dx ⎤⎦φi* (x) (This is * of ψ (x) above) i The * stuff is needed to make sure ψ ψ = 1 even though φ i ψ is complex. 4 5.73 Lecture #10 10 - 5 The symbol 〈a|b〉, a bra–ket, is defined in the sense of a product of (1 × N) ⊗ (N × 1) matrices → a 1 × 1 matrix: a number! Box Normalization in both ψ and 〈 | 〉 pictures 1 = ∫ ψ * ψ dx ( ∫ φ ψ dx )φ ψ* = ∑ ( ∫ φ ψ * dx ) φ ψ=∑ * i i i take complex conjugate of � equation * j j j 1 = ∫ ψ * ψ dx = ∑ i, j 1 = ∑ ∫ φ ψ dx * j expand both in ortho-normal φ basis ( ∫ φ ψ * dx )( ∫ φ ψ dx )∫ φ φ dx * i j * j i c.c. 2 δij forces the 2 sums (over i and j) to collapse into 1 sum (over j) j real, positive #’s We have proved that sum of |mixing coefficients |2 = 1. These mixing coefficients “squared” are called “mixing fractions” or “fractional characters”. picture now in ⎞ ⎛ ⎜ φ*ψ dx ⎟ ∫ 1 ⎟ ⎞⎜ ⎛ ⎟ ⎜ * ψ ψ = ⎜ ∫ φ1ψ * dx φ2 ψ * dx !⎟ ⎜ ∫ φ2 ψ dx ⎟ = 1× 1 matrix ∫ ⎜⎝ "$$$$$#$$$$$% ⎟⎠ ⎟ ⎜ row vector: “bra” & ⎜ "$#$% ⎟ column ⎟ ⎜⎝ vector “ket” ⎠ =∑ j ∫ φ ψ dx * j 2 same result as in wavefunction representation [CTDL talks about “dual vector spaces” — best to walk before you run. Always translate 〈 〉 into ψ picture until you are sure you understand the notation.] 5 5.73 Lecture #10 Any symbol 〈 〉 Any symbol ∣ 〉 〈 ∣ 10 - 6 is a complex number. is a square matrix. ⎛ ⎜ ψ φ2 …) ⎜ ⎜ ⎜⎝ again ψ ψ = ( ψ φ1 φ1 ψ ⎞ ⎟ φ2 ψ ⎟ ⎟ ! ⎟⎠ = ∑ ψ φi φi ψ = ψ ψ = 1 i unit matrix � ⎛ ⎜ what is φ1 φ1 = 10…0 ⎜ ⎜ ⎜⎝ ( ) 1 0 ! 0 ⎞ ⎛ ⎞ ⎟ ⎜ 1 0 " ⎟ ⎟ =⎜ 0 0 ⎟ ⎟ ⎜ ! # ⎟⎠ ⎟⎠ ⎝ ⎛ 1 0 ⎜ 1 0 what is ∑ φi φi = ⎜ 0 1 ⎜ i ⎜⎝ 0 # ⎞ ⎟ ⎟ ⎟ ⎟⎠ three dots are shorthand for specifying only the important part of an infinite matrix unit or identity matrix = � Large zero (0) denotes a lot of zeroes. “completeness” or “closure” involves insertion of � between any two symbols. 6 5.73 Lecture #10 10 - 7 Use 1 to evaluate the matrix elements of the product of 2 operators, AB (we know how to do this inψpicture). i-th ⎛ ⎜ ⎜ φi A φ j = ( 0…1…0 )( A ) ⎜ ⎜ ⎜ ⎝ i-th 0 : 1 : 0 ⎛ A1 j ⎜ = ( 0…1…0 ) ⎜ A2 j ⎜ ⎝ ! ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ = Aij ⎟ ⎠ square matrix j-th position – picks out j-th column of A picks out the i-th element of a column vector φi AB φ j = ∑ φi A φ k φ k B φ j k = ∑ Aik Bkj = ( AB )ij a number (obtained by matrix multiplication) k In the Heisenberg picture, how do we get an exact equivalent of ψ(x)? Use basis set δ(x,x0) for all x0 – this is a “complete” basis (eigenbasis for ^ x, eigenvalue x0) - perfect localization at any x0 This 〈x ⎜ψ 〉 symbol is the same thing as ψ(x) ⎛ i.e., ↑, ∫ δ (x , x′) * ψ( x′ ) dx′ = ψ(x) ⎞⎠ ⎝ x is continuously variable ↔ δ(x) Overlap of state vector ψ with δ(x) – a complex number. ψ(x) is a complex function of a real variable. 7 5.73 Lecture #10 10 - 8 other ψ ↔ 〈 ⎜ 〉 relationships 1. All observable quantities are represented by a Hemitian operator (Why – because the expectation values of a Hermitian operator are always real). Definition of Hermitian operator: † Easy to prove that if all For a matrix: A ij = A*ji or A = A expectation values of A are real, then A = A† and 2. Change of basis set vice-versa A φ ↔ Au {φ} to {u } Aijφ ≡ φi A φ j = φi 1A1 φ j = ∑ φi uk uk A uℓ uℓ φ j k ,ℓ † S*ki ≡ Sik ↑, u Sℓj ← φ, S is Frequently used to denote an ↑, “overlap” integral u This is the j-th column of S φ, u basis ( † u = ∑ Sik Akℓ Sℓj = S† Au S k ,ℓ φ † u A =S A S ) ij φ basis ≡ Aijφ S† … S is a special kind of transformation (unitary) (different from more-familiar T–1AT “similarity” transformation) 8 5.73 Lecture #10 10 - 9 For a state vector (ket): N φ j = ∑ uℓ uℓ φ j = ∑ ℓ ℓ=1 ⎛ S ℓj ⎜ =⎜ " ⎜ S Nj ⎝ φj ⎛ S1 j ⎞ u ℓ S ℓj = ⎜ " ⎟ ⎜ ⎟ ⎜⎝ S ⎟⎠ Nj ⎞ ⎟ ⎟ This is the j-th column of S. ⎟ ⎠ The linear combination of ui for each φ j is the j-th column of S. Also, the linear combination of φ j for each ui is the i-th column of S† . [This is a very useful thing to remember.] i-th ⎛ ⎜ ⎜ ui = ⎜ ⎜ ⎜ ⎝ 0 ! 1 ! 0 ⎞ ⎟ ⎛ S† ⎟ ⎜ 1i ⎟ =⎜ ! ⎟ ⎜ S† ⎟ ⎝ Ni ⎠u mixed pure state in {u} basis ⎛ ⎞ ⎜ ⎟ † ⎟ = S1i ⎜ ⎜ ⎟ ⎠φ ⎜⎝ state in {�} basis 9 1 0 ! 0 ⎞ ⎛ ⎟ ⎜ † ⎟ + S2 i ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝ pure state in {�} basis 0 1 ! ! ⎞ ⎛ ⎟ ⎜ † ⎟ +…S Ni ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝ 0 ! ! 1 ⎞ ⎟ ⎟ ⎟ ⎟⎠ 5.73 Lecture #10 10 - 10 What kind of matrix is S? Sℓj = uℓ | φ j Sℓj* = ⎡ uℓ | φ j ⎤ * = φ j | uℓ ≡ S †jℓ ⎣ ⎦ †means take complex conjugate and interchange indices. Using the definitions of S and S† : Sℓj S †jk = uℓ φ j φ j uk † S S ∑ ℓj jk =∑ j uℓ φ j φ j uk = uℓ 1 uk = δ ℓk j = uℓ uk = δ ℓk = 1 ℓk SS† = � OR S† = S–1 “Unitary” S† is inverse of S! 10 identity matrix a very special and convenient property. 5.73 Lecture #10 10 - 11 Unitary transformations preserve both normalization and orthogonality. A φ = S †A u S SA φS† = SS†A u SS† = A u A u = SA φS† Take matrix element of both sides of equation: Aiju = ui | A | u j = ( SA φS† )ij = ∑ Sik φ k | A | φℓ S†ℓj k,ℓ ∴ u j = ∑ φℓ S†ℓj u j is j-th column of S† ℓ φ → u via S† ,S : u j is j-th column of S† 11 5.73 Lecture #10 10 - 12 Thus, ⎛ ⎜ j-th ⎜ uj = ⎜ ⎜ ⎜ ⎝ 0 ! 1 ! 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠u ⎛ S† 1j ⎜ ⎜ S2† j ⎜ =⎜ ! ⎜ ! ⎜ † ⎜ Snj ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠φ Similarly, ( A φpq = φ p | A | φq = S†Au S ) pq † = ∑ S pm u m | A | u n S nq mn ∴ φq = ∑ un Snq φq q-th column of S n φq q-th ⎛ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝ 0 ! 1 ! 0 ⎞ ⎛ S1q ⎟ ⎜ ⎟ ⎜ S2q ⎟ =⎜ ⎟ ⎜ ! ⎟ ⎜⎝ Snq ⎠φ 12 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟⎠ u 5.73 Lecture #10 10 - 13 Commutation Rules * ⎡ Â, B̂ ⎤ = ÂB̂ − B̂ ⎣ ⎦ e.g. ⎡⎣ x̂, p̂ ⎤⎦ = i! ˆ p̂x̂ ) φ = x means ( x̂p- dφ⎞ ! dφ ! ⎛ ! − ⎜ φ + x ⎟ = − φ = !iφ i dx i ⎝ dx ⎠ i = i!φ * If  and B̂ are Hermitian, is ÂB̂ Hermitian? Hermitian A and B ( AB )ij = ∑ Aik Bkj = ∑ Aki* B*jk = ∑ B*jk Aki* = ( BA )*ji k but this is not what we need to be able to show that AB is Hermitian: † † That would be: ( AB ) = ( AB )* or AB = ( AB ) ≠ ( BA ) ij ji AB is Hermitian only if ⎡⎣ A,B⎤⎦ = 0 1 However, ⎡⎣ AB + BA⎤⎦ is Hermitian if both A and B are Hermitian. 2 This is a foolproof way to construct a new Hermitian operator out of simpler Hermitian operators. This is the standard prescription for implementing the Correspondence Principle for constructing a quantum mechanical equivalent of a classical mechanical quantity. Quantities that commute in classical mechanics do not always commute in quantum mechanics. Almost everything that is not classical mechanical in quantum mechanics is derivable from [ x̂ , p̂ x ] ≠ 0! 13