10-23 ────────────────────────────────────────────────────────────────────────── graphical review of the tangent/cotangent bundle to show phases Linear Operators ───────────── rank 0 scalar ℝ,ℂ rank 1 vector basis(ê¹,ê²,ê³), components [v1, v2, v3] rank 2 𝔸:𝘃→𝘄 Linear ???rank 3 εᵢⱼᵏ - cross product, gᵢ - dot product graphically show the operation of operators graphically show the operation of rotation operators graphically distinguish active and passive transformations active - includes evolution operator U⊹ passive - matrix change of basis 𝘄 = 𝔸 𝘃. ℝ𝘄 = ℝ𝔸ℝᵀ ℝ𝘃. 𝘄′ = 𝔸′ 𝘃′. ────────────────────────────────────────────────────────────────────────── 2017-10-25 Linear transformations ⎛1 a⎞ ⎛0⎞ = ⎛a⎞ ⎝0 1⎠ ⎝1⎠ ⎝1⎠ ⎛λ₁ 0⎞ ⎛0⎞ = ⎛0 ⎞ ⎝0 λ₂⎠ ⎝1⎠ ⎝1/2⎠ M = U(ϕ)⎛λ₁ 0⎞ U⁻¹(ϕ) R(θ) ⎝0 λ₂⎠ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟ Vᵀ W Sᵀ = S, symmetric → Hermitian Rᵀ = R⁻¹ orthogonal → unitary M = U W Vᵀ singular decomposition = S R polar decomposition normal matrix analogue x ± ιn = r exp(ιϕ) Theorem ───────────── if H⁺ = H and H v⃗ᵢ = v⃗ᵢ λᵢ then λᵢ real and v⃗ᵢ⋅v⃗ⱼ = 0 if λᵢ ≠ λⱼ take ⊹: Vⱼ⊹ H = λ⃰ⱼ vⱼ⊹ λ⃰ⱼ v⃑ⱼ⊹ v⃑ᵢ = v⃑ⱼ⊹ H v⃑ᵢ = v⃑ⱼ⊹ v⃑ᵢ λᵢ I i=j, ‖vᵢ‖ ≠ 0, λᵢ⃰ = λᵢ, ℝ ∋ λ i ≠ j, λᵢ ≠ λⱼ, v⃗ⱼ⊹ v⃗ᵢ = 0 ─────────────────────────────────────────────────────────────────────── 10-30 "The importance of being Earnest" 0) Linear linear combinations separable diff. eqs. superposition basis 1) Unitary Rotations U⊹ U = I change of coordinates preserves angles and length 2) Hermitian Stretches H⊹ = H real eigenvalues → observables orthogonal eigenvectors diagonalizable H = U D U⊹ → similarity transform, rotate, stretch, rotate back H U = U D ; H⊹ = H; U⊹ U = I; U⊹ = U⁻¹ H = U D U⁻¹ = U D U⊹ U⊹ H U = D b 3) diagonal element-wise multiplication ─────────────────────────────────────────────────────────────────────── 11-01 From last time: The importance of 1) linear 2) Unitary 3) Hermitian Polar Decomposition -- A = P*U H = U D U⊹ spectral decomposition H = λ₁ P₁ + λ₂ P₂ 4) Diagonal Now: 5) Importance of Commuting tied up with diagonal matrices, because diagonal matrices commute Normal matrix: [N⊹,N] = 0. similarity transform → change of basis A = U D U⁻¹ A² = U D U⁻¹ U D U⁻¹ = U D² U⁻¹ f(A) = f₀ + f₁ A + 1/2! f₂ A² = U U⁻¹ + U f₁ D U⁻¹ + U 1/2! f₂ D² U⁻¹ = U(1 + f₁ D + 1/2! f₂ D² + ...) U⁻¹ = U(diagonal matrix) U⁻¹ Obvious that diagonal matrices commute commuting -- physical measurements represented by Hermitian operators H⊹ = H λᵢ = what you can measure 𝘃ᵢ = definte states of 𝒪. two measurements are compatible if they have the same eigenvectors, simulataneously diagonalizable every physical state is reprsented as a dot on a unit circle, brilliant expansion and projection postulate diagonal matrices are automatically diagonalizable A = U Dₐ U⁻¹, B = U Dᵦ U⁻¹ if [A,B] = 0, are A,B simulataneously diagonalizable? ex [x,p] = ιħ. Thm: if [A,B] = 0 and U⁻¹ A U = D, then U⁻¹ B U is also "block diagonal". recall ladder operators: if N|n> = n|n>, then N(a±|n>) = (n+1)|n>. let A 𝐮ᵢ = λᵢ 𝐮ᵢ A(B 𝐮ᵢ) = B(A 𝐮ᵢ)z 11-06 ───────────── Postulates - lead-up a) How does the state evolve (when not watched)? Ĥ ψ = Ê ψ. b) what happens when we measure? e.g. x i) pick random x weighted by prob │ψ(x)│² ii) collapses the wave function to δ(x-a) if measured "a" c) Observation is richer than classical mechanics due to complementarity and superposition 1) Observables are a Hermitian operator Q̂(x,-ιħ∂/∂x) on ❙ψ(x)❭ 2) Every observable has "definitey states" eigenstates of Q̂: Q̂❙ϕₙ❭ qₙ ❙ϕₙ❭ 3) Any other state is a superposition of ❙ϕₙ❭ ❙ψ❭ = ∑ cₙ ❙ϕₙ❭ cₙ = Probability Amplitude 4) Observation is an irreversible process, "collapses ❙ψ❭" 5) Measurements with same ❙ϕₙ❭ are compatible. [Q̂,R̂] = 0. Practical Application 1) Solve Ĥ ❙ψₙ❭ = Eₙ ❙ψₙ❭ stationary states 2) Find components of ❙ψ₀❭ = cₙ❙ψₙ❭ 3) Rotate State in time ❙ψ(t)❭ = exp(-ιEₙt/ħ)cₙ❙ψ₀❭ 4) Solve eigenstates of Q̂❙ϕₙ❭ = qₙ❙ϕₙ❭ 5) find components of ❙ψ(t)❭ = aₙ❙ϕₙ❭ 6) project the state ❙ψ(t)❭ ⟶ ❙ϕₙ❭ TOOLS: ❬ϕₙ❙ϕₘ❭ = δₙₘ -- orthonormality ∑ₙ ❙ϕₙ❭❬ϕₙ❙ = 𝕀 -- closure Postulates 1) superposition state ❙ψ❭ collection of probability amplitues cₙ with ∑ₙ│cₙ│² = 1. 2) expansion/projection Q̂❙ϕₙ❭ = qₙ❙ψₙ❭ ❙ψ❭ = aₙ ❙ϕₙ❭ aₙ = probability amplitude of measurement "q" after measurement qₙthe system collapses to state ❙ϕₙ❭ 3) evolution Ĥψ = Êψ. 4) Uncertainty [x,p] = ιħ. 5) wait until next semester -- exclusion principle