10-23
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graphical review of the tangent/cotangent bundle to show phases


Linear Operators
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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′.

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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
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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


 
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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

            


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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
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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