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19
2D_oscillator/formalism.motes
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19
2D_oscillator/formalism.motes
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ℋ❙φ❭ = (ℋ𝓍𝓎 + ℋ𝓏)❙φ❭ = E❙φ❭.
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ℋ𝓍𝓎 = (P²𝓍 + P²𝓎)/(2μ) + ½ μω² (X² + Y²).
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ℋ𝓏 = P²𝓏/2μ.
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❙φ❭ = ❙φ𝓏𝓎❭ ⛒ ❙φ𝓏❭.
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ℋ𝓍𝓎❙φ❭ = E𝓍𝓎❙φ❭.
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E = E𝓍𝓎 + E𝓏.
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❬z❙φ𝓏❭ = (2πħ)⁻¹ exp(ιp𝓏z/ħ).
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E𝓏 = p𝓏²/2μ where p𝓏 is an arbitrary real constant.
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ℰ = ℰ𝓍𝓎 ⛒ ℰ𝓏.
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classbfs120904402944051.pdf
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classbfs120904402944051.pdf
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griffiths/3.7
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griffiths/3.7
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3.7
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Suppose f and g are eigenfunctions of Q, with eigenvalue q. Show any linear combination of f and g are eigenfunctions of Q with eigenvalue q.
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|Qf> = q|f>
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|Qg> = w|g>
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A and B are (possibly complex) constants.
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|Q(A*f+B*g)> = A|Qf> + B|Qg> = Aq|f> + Bq|g>
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= q(A|f> + B|g>).
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QED
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Check that f(x)=exp(x) and g(x)=exp(-x) are eigenfunctions of the operator d^2/dx^2, with the same eigenvalue. construct two linear combiations of f and g that are orthogonal eigenfunctions on the interval [-1,1]
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if Q = d^2/dx^2 then we have the differential equation
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f'' = qf
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(e^x)'' = e^x, so this is an eigenfunction with q=1.
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(e^-x)'' = e^-x, so this is an eigenfunction with q=1.
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e^x + e^-x is also an eigenfunction of q, per earlier proof, and so is e^x - e^-x. These functions are just 2*sinh and 2*cosh, which are orthogonal functions because sinh is odd and cosh is even.
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21
griffiths/3.8
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griffiths/3.8
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3.8
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a)
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check that the eigenvalues of the hermitian operator in example 3.1 are real. show that the eigenfunctions are orthogonal.
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Qf = if'
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the eigenvalues are 0,+- 1, etc., which are obviously real.
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pick two arbitrary eigenfunctions:
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f = A exp(-i q phi)
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g = A exp(-i q' phi)
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<f|g> = A*A int[exp(i q phi) exp(-i q' phi)] dphi[0,2pi]
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= A*A int[exp(i (q - q') phi)] dphi[0,2pi]
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= A*A [i(q-q')]^-1 [exp(i(q-q') phi)]|[0,2pi]
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hw/12/H12.pdf
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hw/12/H12.pdf
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hw/12/HW12.motes
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hw/12/HW12.motes
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𝓗𝓍𝓎 = ½ (-∇² + ρ²).
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ρ² = x² + y² and ħ = m = ω = 1.
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Griffith's Eq. 2.71:
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Hₙ₊₁(ξ) = 2ξ Hₙ(ξ) - 2n Hₙ₋₁(ξ)
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In 2D cartesian coordinates, the del operator is defined
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∇f = [∂/∂x f, ∂/∂y f].
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∇²f = ∇⋅∇f = ∇⋅[∂/∂x f, ∂/∂y f]
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= ∂²/∂x² f + ∂²/∂y² f.
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∴ ∇² = ∂²/∂²x + ∂²/∂²y.
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Then,
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𝓗𝓍𝓎 = ½ (-(∂²/∂²x + ∂²/∂²y) + (x² + y²)).
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𝓗𝓍𝓎 = ½ (-(∂²/∂²x + ∂²/∂²y) + (x² + y²))
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= ½ (x² - ∂²/∂²x + y² - ∂²/∂²y)
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= ½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y).
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𝓗𝓍 + 𝓗𝓎 = ½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y).
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∎
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The Schrodinger Equation then reads,
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[½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y)] Ψ = (E𝓍 + E𝓎) Ψ.
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Assuming a separable solution Ψ(x,y) = X(x) Y(y), with E = E𝓍 + E𝓎.
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[½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y)] X(x) Y(y) = (E𝓍 + E𝓎) X(x) Y(y).
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½(x² - ∂²/∂²x) X(x) Y(y) + ½(y² - ∂²/∂²y) X(x) Y(y)
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= (E𝓍 + E𝓎) X(x) Y(y).
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[1/X(x)] [½(x² - ∂²/∂²x) X(x)] + [1/Y(y)] [½(y² - ∂²/∂²y) Y(y)]
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= (E𝓍 + E𝓎).
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So, I have two differential equations,
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½(x² - ∂²/∂²x) X(x) = E𝓍 X(x), and
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½(y² - ∂²/∂²y) Y(y) = E𝓎 Y(y).
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The solutions to these differential equations are the same as for the 1D harmonic oscillator. They have eigenvalues (n + 1/2), where ħ = ω = 1, with n = 0,1,2,... .
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∴ the eigenvalues for the combined operator are n𝓍 + n𝓎 + 1.
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The degeneracy is pretty obvious, just from counting the possibilities: there is n+1 degeneracy for each value of n = n𝓍 + n𝓎.
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So,
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n degeneracy
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─────────────────
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0 1
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1 2
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2 3
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3 4
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4 5
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5 6
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The Hermite polynomials help to generate the eigenstates of this:
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Hₙ(x) = (-1)ⁿ exp(x²) d/dxⁿ exp(-x²/2) = (2x - d/dx)ⁿ * 1.
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The first six polynomials are
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H₀(x) = 1.
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H₁(x) = 2x.
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H₂(x) = 4x² - 2.
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H₃(x) = 8x³ - 12x.
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H₄(x) = 16x⁴ - 48x² + 12.
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H₅(x) = 32x⁵ - 160x³ + 120x.
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H₆(x) = 64x⁶ - 480x⁴ + 720x² - 120.
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The wave functions involving these polynomials, with the unitizations given in the intro, are
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Ψₙₘ(x) = π^(-1/4) 1/√(2ⁿ n!) Hₙ(x) exp(-x²/2)
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π^(-1/4) 1/√(2ᵐ m!) Hₘ(y) exp(-y²/2).
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Ψₙₘ(x) = 1/√π 1/√(2ⁿ n! 2ᵐ m!) Hₙ(x) Hₘ(y) exp(-(x²/2 + y²/2)).
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For n = {1,...,∞}, the lowest possible energy is 3, and this level has no degeneracy.
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For additional levels,
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a𝓍 = 1/√2(x + ιp𝓍)
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a𝓎 = 1/√2(y + ιp𝓎)
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=======
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The lowering operators are
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a𝓍 = 1/√2 (x + ιp𝓍) and
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a𝓎 = 1/√2 (y + ιp𝓎).
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They are not hermitian, but x,y and p𝓍,p𝓎 are, so the raising operators are
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a𝓍᛭ = 1/√2 (x - ιp𝓍) and
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a𝓎᛭ = 1/√2 (y - ιp𝓎).
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Applying these to the ground state ❙00❭, I can find the first six states, with normalization:
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a𝓍᛭❙00❭ = 1/√2 (x - ιp𝓍)❙00❭
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= 1/√2 (x❙00❭ - ιp𝓍❙00❭)
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b) & c)
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I'm still working out the algebra, here. I will try to finish it as soon as I can, but I know I also have new work to do.
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I finished much of this assignment, but need to get done faster in the future.
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3883
hw/12/HW12.motes.ps
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hw/12/HW12.motes.ps
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hw/12/bookprobs_othoulrich_hw12.pdf
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hw/12/othoulrich_hw12.pdf
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hw/hw14.pdf
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hw/hw14.pdf
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presentation/.ipynb_checkpoints/Entanglement-checkpoint.ipynb
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presentation/.ipynb_checkpoints/Entanglement-checkpoint.ipynb
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presentation/Document.docx
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presentation/Entanglement.ipynb
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presentation/Entanglement.ipynb
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presentation/entanglement.cpp
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presentation/entanglement.cpp
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#include <vector>
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struct spin {bool up; bool down;}
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struct entangled_spin {bool up_1; bool up_2; bool down_1; bool down_2; }
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struct state { double prob_up; double prob_down; }
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struct entangled_state {}
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spin measure (particle measured) {
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}
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int main(int argc, char const *argv[])
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{
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std::vector<spin> v;
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spin electron_spin;
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spin positron_spin;
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\\ uncorrelated measurements
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for (int trial=1; trial<=1000; trial++) {
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particle positron;
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particle electron;
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measure_spin(positron)
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}
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\\ entangled measurements
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for (int trial=1; trial<=1000; trial++) {
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entangled_pair pair = new entangled_pair(
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particle position,
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particle electron);
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measure_spin(positron)
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}
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return 0;
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//=============================================================================
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presentation/notebook.tex
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% Default to the notebook output style
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% Inherit from the specified cell style.
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\documentclass[11pt]{article}
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\usepackage[T1]{fontenc}
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% Nicer default font (+ math font) than Computer Modern for most use cases
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\usepackage{mathpazo}
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% Basic figure setup, for now with no caption control since it's done
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% automatically by Pandoc (which extracts  syntax from Markdown).
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\usepackage{graphicx}
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% We will generate all images so they have a width \maxwidth. This means
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% that they will get their normal width if they fit onto the page, but
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% are scaled down if they would overflow the margins.
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\makeatletter
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\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth
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\else\Gin@nat@width\fi}
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\makeatother
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\let\Oldincludegraphics\includegraphics
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% Set max figure width to be 80% of text width, for now hardcoded.
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\renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}}
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% Ensure that by default, figures have no caption (until we provide a
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% proper Figure object with a Caption API and a way to capture that
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% in the conversion process - todo).
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\usepackage{caption}
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\DeclareCaptionLabelFormat{nolabel}{}
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\captionsetup{labelformat=nolabel}
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\usepackage{adjustbox} % Used to constrain images to a maximum size
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\usepackage{xcolor} % Allow colors to be defined
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\usepackage{enumerate} % Needed for markdown enumerations to work
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\usepackage{geometry} % Used to adjust the document margins
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\usepackage{amsmath} % Equations
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\usepackage{amssymb} % Equations
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\usepackage{textcomp} % defines textquotesingle
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% Hack from http://tex.stackexchange.com/a/47451/13684:
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\AtBeginDocument{%
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\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
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}
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\usepackage{upquote} % Upright quotes for verbatim code
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\usepackage{eurosym} % defines \euro
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\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
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\usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document
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\usepackage{fancyvrb} % verbatim replacement that allows latex
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\usepackage{grffile} % extends the file name processing of package graphics
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% to support a larger range
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% The hyperref package gives us a pdf with properly built
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% internal navigation ('pdf bookmarks' for the table of contents,
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% internal cross-reference links, web links for URLs, etc.)
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\usepackage{hyperref}
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\usepackage{longtable} % longtable support required by pandoc >1.10
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\usepackage{booktabs} % table support for pandoc > 1.12.2
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\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
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\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
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% normalem makes italics be italics, not underlines
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% Colors for the hyperref package
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\definecolor{urlcolor}{rgb}{0,.145,.698}
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\definecolor{linkcolor}{rgb}{.71,0.21,0.01}
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\definecolor{citecolor}{rgb}{.12,.54,.11}
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% ANSI colors
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\definecolor{ansi-black}{HTML}{3E424D}
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\definecolor{ansi-red}{HTML}{E75C58}
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\definecolor{ansi-red-intense}{HTML}{B22B31}
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\definecolor{ansi-green}{HTML}{00A250}
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\definecolor{ansi-green-intense}{HTML}{007427}
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\definecolor{ansi-yellow}{HTML}{DDB62B}
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\definecolor{ansi-yellow-intense}{HTML}{B27D12}
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\definecolor{ansi-blue}{HTML}{208FFB}
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\definecolor{ansi-blue-intense}{HTML}{0065CA}
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\definecolor{ansi-magenta}{HTML}{D160C4}
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\definecolor{ansi-magenta-intense}{HTML}{A03196}
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\definecolor{ansi-cyan}{HTML}{60C6C8}
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\definecolor{ansi-cyan-intense}{HTML}{258F8F}
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\definecolor{ansi-white}{HTML}{C5C1B4}
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\definecolor{ansi-white-intense}{HTML}{A1A6B2}
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% commands and environments needed by pandoc snippets
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\providecommand{\tightlist}{%
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\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
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\newenvironment{Shaded}{}{}
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\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
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% Additional commands for more recent versions of Pandoc
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|
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|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
|
||||
\maketitle
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{incolor}In [{\color{incolor}1}]:} \PY{c+c1}{\PYZsh{} EPR Paradox Example}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} A neutral pi meson (pi0) decays into an electron/positron (e\PYZhy{}/e+) }
|
||||
\PY{c+c1}{\PYZsh{} pair.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} pi0 \PYZhy{}\PYZhy{}\PYZgt{} e+ + e\PYZhy{} (electron\PYZhy{}positron pair)}
|
||||
\PY{c+c1}{\PYZsh{} pi0 has angular momentum l = s = 0}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} Align electron and positron detectors in opposite directions.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} | pi0 |}
|
||||
\PY{c+c1}{\PYZsh{} |e\PYZhy{} \PYZlt{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{} \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZgt{} e+|}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} Where hbar = 1, the measurement of the spin in some direction is }
|
||||
\PY{c+c1}{\PYZsh{} +/\PYZhy{}1 with spin state [1 0] (up) or [0 1] (down).}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} The Pauli exclusion principle with conserved angular momentum 0}
|
||||
\PY{c+c1}{\PYZsh{} dictates this system must be in the singlet state }
|
||||
\PY{c+c1}{\PYZsh{} chi = [1/sqrt(2) (|up+\PYZgt{}|down\PYZhy{}\PYZgt{} \PYZhy{} |down+\PYZgt{}|up\PYZhy{}\PYZgt{})].}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} In this state, if the positron is measured to have spin [1 0], the }
|
||||
\PY{c+c1}{\PYZsh{} electron must have spin [0 1], or vice versa. There is an equal }
|
||||
\PY{c+c1}{\PYZsh{} probability to find either state during the first measurement.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} This view is consistent with the realist view. The realist view could }
|
||||
\PY{c+c1}{\PYZsh{} hold that the electron and position had those angular momenta }
|
||||
\PY{c+c1}{\PYZsh{} from creation.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} EPR assumes influences cannot propagate faster than the speed of }
|
||||
\PY{c+c1}{\PYZsh{} light. \PYZdq{}Wave function collapse\PYZdq{} is apparently instantaneous, however.}
|
||||
|
||||
|
||||
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
|
||||
\PY{k+kn}{import} \PY{n+nn}{matplotlib}
|
||||
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt}
|
||||
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{patches} \PY{k}{as} \PY{n+nn}{mpatches}
|
||||
\PY{o}{\PYZpc{}}\PY{k}{matplotlib} inline
|
||||
\end{Verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{incolor}In [{\color{incolor}2}]:} \PY{c+c1}{\PYZsh{} If information about the measurement of the wave function propagated}
|
||||
\PY{c+c1}{\PYZsh{} at a finite speed, the particles could conceivably be measured such}
|
||||
\PY{c+c1}{\PYZsh{} that both are equally likely to hold either spin up or spin down.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} What would happen if the measurements were uncorrelated?}
|
||||
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
|
||||
\PY{n}{elepatch} \PY{o}{=} \PY{n}{mpatches}\PY{o}{.}\PY{n}{Patch}\PY{p}{(}\PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{blue}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{e\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
|
||||
\PY{n}{pospatch} \PY{o}{=} \PY{n}{mpatches}\PY{o}{.}\PY{n}{Patch}\PY{p}{(}\PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{orange}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{e+}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
|
||||
\PY{c+c1}{\PYZsh{}plt.legend(handles=[elepatch,pospatch])}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{text}\PY{p}{(}\PY{l+m+mf}{0.5}\PY{p}{,}\PY{l+m+mi}{565}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{e\PYZhy{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{size}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{text}\PY{p}{(}\PY{l+m+mf}{3.5}\PY{p}{,}\PY{l+m+mi}{565}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{e+}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{size}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
|
||||
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Uncorrelated Spins}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{600}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{,}\PY{l+m+mf}{3.125}\PY{p}{,}\PY{l+m+mf}{3.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{down}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{up}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{down}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{up}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
|
||||
\end{Verbatim}
|
||||
|
||||
|
||||
\begin{center}
|
||||
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_1_0.png}
|
||||
\end{center}
|
||||
{ \hspace*{\fill} \\}
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{incolor}In [{\color{incolor}3}]:} \PY{c+c1}{\PYZsh{} How many violations of angular momentum would be measured?}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} From running several simulations, it\PYZsq{}s evident a violation in the }
|
||||
\PY{c+c1}{\PYZsh{} conservation of angular momentum would be measured half of the time.}
|
||||
\PY{c+c1}{\PYZsh{} We can conclude that the information that the entangled particles are in}
|
||||
\PY{c+c1}{\PYZsh{} orthogonal spin states is instantaneously agreed once a measurement}
|
||||
\PY{c+c1}{\PYZsh{} is made.}
|
||||
|
||||
\PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0}
|
||||
|
||||
\PY{k}{for} \PY{n}{trial} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{:}
|
||||
\PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}
|
||||
\PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}
|
||||
\PY{k}{if} \PY{n}{elespin} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:}
|
||||
\PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
|
||||
\PY{k}{else}\PY{p}{:}
|
||||
\PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
|
||||
\PY{k}{if} \PY{n}{posspin} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:}
|
||||
\PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
|
||||
\PY{k}{else}\PY{p}{:}
|
||||
\PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
|
||||
|
||||
\PY{n}{elespin}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}
|
||||
\PY{n}{posspin}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}
|
||||
\PY{n}{chi\PYZus{}squared} \PY{o}{=} \PY{n}{elespin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1; 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{o}{*}\PY{n}{posspin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0; 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{posspin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1; 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{o}{*}\PY{n}{elespin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0; 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
|
||||
|
||||
\PY{k}{if} \PY{n}{chi\PYZus{}squared} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:}
|
||||
\PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1}
|
||||
|
||||
\PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
|
||||
\PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{l+m+mi}{1000}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
|
||||
\PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)}
|
||||
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{600}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Conservation Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
|
||||
\end{Verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{outcolor}Out[{\color{outcolor}3}]:} <matplotlib.figure.Figure at 0x7f22020419e8>
|
||||
\end{Verbatim}
|
||||
|
||||
\begin{center}
|
||||
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_2_1.png}
|
||||
\end{center}
|
||||
{ \hspace*{\fill} \\}
|
||||
|
||||
|
||||
\begin{verbatim}
|
||||
<matplotlib.figure.Figure at 0x7f22020419e8>
|
||||
\end{verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{incolor}In [{\color{incolor}4}]:} \PY{c+c1}{\PYZsh{} Bell\PYZsq{}s Experiment took this a step further, to rule out locality }
|
||||
\PY{c+c1}{\PYZsh{} completely. Establish the detectors to \PYZdq{}float\PYZdq{} such that they }
|
||||
\PY{c+c1}{\PYZsh{} measure the components of the spins of the electron and positron }
|
||||
\PY{c+c1}{\PYZsh{} along a unit vector a or b, with angles phi\PYZus{}a and phi\PYZus{}b, }
|
||||
\PY{c+c1}{\PYZsh{} respectively. Compute a product P of the spins in units of hbar/2. }
|
||||
\PY{c+c1}{\PYZsh{} This will give +/\PYZhy{}1. }
|
||||
|
||||
\PY{c+c1}{\PYZsh{} / pi0 \PYZbs{}}
|
||||
\PY{c+c1}{\PYZsh{} /e\PYZhy{} \PYZlt{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{} \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZgt{} e+\PYZbs{}}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} QM predicts P(a,b) = \PYZhy{}a dot b, the expectation value of the product }
|
||||
\PY{c+c1}{\PYZsh{} of the spins.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} In 1964, Bell derived the Bell inequality for a local hidden variable}
|
||||
\PY{c+c1}{\PYZsh{} theory: abs(P(a,b) \PYZhy{} P(a,c)) \PYZlt{}= 1 + P(b,c)}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} For any local hidden variable theory, the Bell inequality must hold. }
|
||||
\PY{c+c1}{\PYZsh{} It introduces a third unit vector c, which is any other unit vector }
|
||||
\PY{c+c1}{\PYZsh{} than a or b.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} Does the quantum mechanical prediction violate the Bell inequality?}
|
||||
\PY{c+c1}{\PYZsh{} Testing several detector configurations in a plane, systematically }
|
||||
\PY{c+c1}{\PYZsh{} from 0 to pi, we determine whether the QM prediction is consistent }
|
||||
\PY{c+c1}{\PYZsh{} with a local hidden variable theory.}
|
||||
|
||||
\PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0}
|
||||
\PY{n}{trials} \PY{o}{=} \PY{l+m+mi}{0}
|
||||
\PY{k}{for} \PY{n}{step\PYZus{}a} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{:}
|
||||
\PY{k}{for} \PY{n}{step\PYZus{}b} \PY{o+ow}{in} \PY{n+nb}{range} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{:}
|
||||
\PY{n}{phi\PYZus{}a} \PY{o}{=} \PY{n}{step\PYZus{}a}\PY{o}{/}\PY{l+m+mi}{6}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
|
||||
\PY{n}{phi\PYZus{}b} \PY{o}{=} \PY{n}{step\PYZus{}b}\PY{o}{/}\PY{l+m+mi}{6}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
|
||||
\PY{n}{phi\PYZus{}c} \PY{o}{=} \PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{phi\PYZus{}b}
|
||||
\PY{n}{P\PYZus{}ab} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}b}\PY{p}{)}
|
||||
\PY{n}{P\PYZus{}ac} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
|
||||
\PY{n}{P\PYZus{}bc} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}b} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
|
||||
|
||||
\PY{n}{bell\PYZus{}lhs} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{P\PYZus{}ab} \PY{o}{\PYZhy{}} \PY{n}{P\PYZus{}ac}\PY{p}{)}
|
||||
\PY{n}{bell\PYZus{}rhs} \PY{o}{=} \PY{l+m+mi}{1} \PY{o}{+} \PY{n}{P\PYZus{}bc}
|
||||
|
||||
\PY{k}{if} \PY{n}{bell\PYZus{}lhs} \PY{o}{\PYZgt{}} \PY{n}{bell\PYZus{}rhs}\PY{p}{:}
|
||||
\PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1}
|
||||
|
||||
\PY{n}{trials} \PY{o}{=} \PY{n}{trials} \PY{o}{+} \PY{l+m+mi}{1}
|
||||
|
||||
\PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
|
||||
\PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{n}{trials}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
|
||||
\PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)}
|
||||
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{l+m+mi}{10}\PY{p}{,}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mi}{10}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Hidden Locality Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
|
||||
\end{Verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{outcolor}Out[{\color{outcolor}4}]:} <matplotlib.figure.Figure at 0x7f22020737f0>
|
||||
\end{Verbatim}
|
||||
|
||||
\begin{center}
|
||||
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_3_1.png}
|
||||
\end{center}
|
||||
{ \hspace*{\fill} \\}
|
||||
|
||||
|
||||
\begin{verbatim}
|
||||
<matplotlib.figure.Figure at 0x7f22020737f0>
|
||||
\end{verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{incolor}In [{\color{incolor}5}]:} \PY{c+c1}{\PYZsh{} It is seen that the QM prediction disagrees with a local hidden }
|
||||
\PY{c+c1}{\PYZsh{} variable theory in a significant number of configurations.}
|
||||
|
||||
\PY{c+c1}{\PYZsh{} On average, for random orientations between 0 and pi, how often? }
|
||||
\PY{c+c1}{\PYZsh{} After running several trials, it appears to be about half of the, }
|
||||
\PY{c+c1}{\PYZsh{} which is what one would expect from the quantum mechanical}
|
||||
\PY{c+c1}{\PYZsh{} prediction.}
|
||||
|
||||
\PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0}
|
||||
\PY{n}{trials} \PY{o}{=} \PY{l+m+mi}{0}
|
||||
\PY{k}{for} \PY{n}{rand\PYZus{}a} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)}\PY{p}{:}
|
||||
\PY{k}{for} \PY{n}{rand\PYZus{}b} \PY{o+ow}{in} \PY{n+nb}{range} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)}\PY{p}{:}
|
||||
\PY{n}{phi\PYZus{}a} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{rand}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
|
||||
\PY{n}{phi\PYZus{}b} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{rand}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
|
||||
\PY{n}{phi\PYZus{}c} \PY{o}{=} \PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{phi\PYZus{}b}
|
||||
\PY{n}{P\PYZus{}ab} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}b}\PY{p}{)}
|
||||
\PY{n}{P\PYZus{}ac} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
|
||||
\PY{n}{P\PYZus{}bc} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}b} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
|
||||
|
||||
\PY{n}{bell\PYZus{}lhs} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{P\PYZus{}ab} \PY{o}{\PYZhy{}} \PY{n}{P\PYZus{}ac}\PY{p}{)}
|
||||
\PY{n}{bell\PYZus{}rhs} \PY{o}{=} \PY{l+m+mi}{1} \PY{o}{+} \PY{n}{P\PYZus{}bc}
|
||||
|
||||
\PY{k}{if} \PY{n}{bell\PYZus{}lhs} \PY{o}{\PYZgt{}} \PY{n}{bell\PYZus{}rhs}\PY{p}{:}
|
||||
\PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1}
|
||||
|
||||
\PY{n}{trials} \PY{o}{=} \PY{n}{trials} \PY{o}{+} \PY{l+m+mi}{1}
|
||||
|
||||
\PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
|
||||
\PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{n}{trials}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
|
||||
\PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)}
|
||||
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{l+m+mi}{10}\PY{p}{,}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mi}{10}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Hidden Locality Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
|
||||
\PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
|
||||
\end{Verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{outcolor}Out[{\color{outcolor}5}]:} <matplotlib.figure.Figure at 0x7f220202b080>
|
||||
\end{Verbatim}
|
||||
|
||||
\begin{center}
|
||||
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_4_1.png}
|
||||
\end{center}
|
||||
{ \hspace*{\fill} \\}
|
||||
|
||||
|
||||
\begin{verbatim}
|
||||
<matplotlib.figure.Figure at 0x7f220202b080>
|
||||
\end{verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{incolor}In [{\color{incolor}6}]:} \PY{c+c1}{\PYZsh{} The prediction (Bell\PYZsq{}s inequality) made by assuming a local hidden}
|
||||
\PY{c+c1}{\PYZsh{} variable is violated by a significant number of the possibile }
|
||||
\PY{c+c1}{\PYZsh{} bborientations. This simulation cannot determine which theory is}
|
||||
\PY{c+c1}{\PYZsh{} correct, but the QM prediction has been confirmed through experiment.}
|
||||
\PY{c+c1}{\PYZsh{} No hidden local variable holds actionable information about the }
|
||||
\PY{c+c1}{\PYZsh{} state. Entangled states retain their entanglement in a non\PYZhy{}local }
|
||||
\PY{c+c1}{\PYZsh{} nature.}
|
||||
\end{Verbatim}
|
||||
|
||||
|
||||
\begin{Verbatim}[commandchars=\\\{\}]
|
||||
{\color{incolor}In [{\color{incolor}7}]:} \PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
|
||||
\end{Verbatim}
|
||||
|
||||
|
||||
|
||||
% Add a bibliography block to the postdoc
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
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|
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70
presentation/sim_notes.motes
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70
presentation/sim_notes.motes
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@ -0,0 +1,70 @@
|
||||
Three viewpoints: realist, orthodox, agnostic
|
||||
|
||||
|
||||
|
||||
// |phi_1> |phi_2>
|
||||
|
||||
|
||||
|
||||
For parallel detectors:
|
||||
|
||||
P(a,b) = -1
|
||||
|
||||
|
||||
arbitrary orientation:
|
||||
|
||||
P(a,b) = -a⋅b
|
||||
|
||||
Our understanding of entanglement is consistent with the idea that modern local variables
|
||||
|
||||
|
||||
|
||||
A(a,λ) = ±1.
|
||||
B(b,λ) = ±1.
|
||||
|
||||
If detectors are aligned:
|
||||
|
||||
A(a,λ) = -B(b,λ).
|
||||
|
||||
Average of product of measurements
|
||||
|
||||
P(a,b) = ∫ ρ(λ) A(a,λ) B(b,λ) dλ
|
||||
|
||||
but since A(a,λ) = -B(b,λ),
|
||||
|
||||
P(a,b) = - ∫ ρ(λ) A(a,λ) A(b,λ) dλ
|
||||
|
||||
c is any other unit vector...
|
||||
|
||||
P(a,b) - P(a,c) = - ∫ ρ(λ) [ A(a,λ) A(b,λ) - A(a,λ) A(c,λ) ] dλ
|
||||
|
||||
= - ∫ ρ(λ) [ 1 - A(a,λ) A(c,λ) ] A(a,λ) A(b,λ) dλ
|
||||
|
||||
Because A(a,λ) = ±1 and B(b,λ) = ±1,
|
||||
|
||||
-1 ≤ [A(a,λ) A(b,λ)] ≤ +1.
|
||||
|
||||
ρ(λ) [1 - A(b,λ) A(c,λ)] ≥ 0, so
|
||||
|
||||
│P(a,b) - P(a,c)│ ≤ ∫ ρ(λ) [1 - A(B,λ) A(c,λ)] dλ.
|
||||
|
||||
|
||||
|
||||
|
||||
│P(a,b) - P(a,c)│ ≤ 1 + P(b,c)
|
||||
|
||||
|
||||
|
||||
simulation:
|
||||
|
||||
|
||||
pion decays, leaving two particles
|
||||
|
||||
each particle has a spin state
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
207
presentation/simulation.py
Normal file
207
presentation/simulation.py
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@ -0,0 +1,207 @@
|
||||
# EPR Paradox Example
|
||||
|
||||
# A pion (pi0) decays into an electron/positron (e-/e+) pair.
|
||||
|
||||
# pi0 --> e+ + e- (electron-positron pair)
|
||||
# pi0 has angular momentum l = 0
|
||||
|
||||
# Align electron and positron detectors in opposite directions.
|
||||
|
||||
# | pi0 |
|
||||
# |e- <------ ------> e+|
|
||||
|
||||
# Where hbar = 1, the measurement of the spin in some direction is either
|
||||
# [1 0] or [0 1].
|
||||
|
||||
# Pauli exclusion principle with conserved angular momentum l=0 says this
|
||||
# system must be in the singlet state
|
||||
# chi = [1/sqrt(2) (|up+>|down-> - |down+>|up->)].
|
||||
|
||||
# In this state, if the positron is measured to have spin up, the electron
|
||||
# must have spin down, or vice versa. There is an equal probability to
|
||||
# measure either spin by the first measurement.
|
||||
|
||||
# This view is consistent with the realist view. The realist view could hold
|
||||
# that the electron and position had those angular momenta from creation.
|
||||
|
||||
# EPR assumes influences cannot propagate faster than the speed of light.
|
||||
# "Wave function collapse" is instantaneous.
|
||||
|
||||
|
||||
import numpy as np
|
||||
import matplotlib
|
||||
import matplotlib.pyplot as plt
|
||||
import matplotlib.patches as mpatches
|
||||
|
||||
|
||||
|
||||
# If information about the measurement of the wave function propagated
|
||||
# at a finite speed, the particles could conceivably be measured such
|
||||
# that both are equally likely to hold either spin up or spin down.
|
||||
|
||||
# What would happen if the measurements were uncorrelated?
|
||||
|
||||
plt.ylim([400,600])
|
||||
plt.xlim([-1,5])
|
||||
plt.xticks([0.125,0.85,3.125,3.85],["down","up","down","up"])
|
||||
plt.tick_params(axis='both',labelsize=15)
|
||||
plt.hist(np.random.randint(0,2,1000),bins=4)
|
||||
plt.hist(np.random.randint(3,5,1000),bins=4)
|
||||
elepatch = mpatches.Patch(color='blue', label='e-')
|
||||
pospatch = mpatches.Patch(color='orange', label='e+')
|
||||
plt.legend(handles=[elepatch,pospatch])
|
||||
plt.figure()
|
||||
|
||||
|
||||
|
||||
# On average, how many violations of angular momentum would be measured?
|
||||
|
||||
# Clearly, a violation in angular momentum would be measured half of the time.
|
||||
# We can conclude that the information that the entangled particles are in
|
||||
# the opposite spin states of eachother is instantaneously known once
|
||||
# a measurement is made.
|
||||
|
||||
violations = 0
|
||||
|
||||
for trial in range(0,1000):
|
||||
elespin = np.random.randint(0,2)
|
||||
posspin = np.random.randint(0,2)
|
||||
if elespin == 0:
|
||||
elespin = np.matrix('0 1')
|
||||
else:
|
||||
elespin = np.matrix('1 0')
|
||||
if posspin == 0:
|
||||
posspin = np.matrix('0 1')
|
||||
else:
|
||||
posspin = np.matrix('1 0')
|
||||
|
||||
elespin.transpose()
|
||||
posspin.transpose()
|
||||
chi_squared = elespin*np.matrix('1; 0')*posspin*np.matrix('0; 1') - posspin*np.matrix('1; 0')*elespin*np.matrix('0; 1')
|
||||
|
||||
if chi_squared == 0:
|
||||
violations = violations + 1
|
||||
|
||||
zeroes = np.zeros(violations,dtype=int)
|
||||
ones = np.full((1000-violations),1,dtype=int)
|
||||
result = np.concatenate((ones,zeroes))
|
||||
|
||||
plt.ylim([400,600])
|
||||
plt.xlim([-1,2])
|
||||
plt.xticks([0.125,0.85],["violation","adherence"])
|
||||
plt.tick_params(axis='both',labelsize=15)
|
||||
plt.suptitle("Conservation Violations",fontsize=20)
|
||||
plt.hist([result],bins=4)
|
||||
plt.figure()
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
# Bell's Experiment took this a step further, to rule out locality completely.
|
||||
# Establish the detectors to "float" such that they measure the components of
|
||||
# the spins of the electron and positron along a unit vector a or b, with
|
||||
# angles phi_a and phi_b, respectively. Compute a product P of the spins in
|
||||
# units of hbar/2. This will give +/-1.
|
||||
|
||||
# / pi0 \
|
||||
# /e- <------ ------> e+\
|
||||
|
||||
# QM predicts P(a,b) = -a dot b, the expectation value of the product of
|
||||
# the spins.
|
||||
|
||||
# In 1964, Bell derived the Bell inequality for a local hidden variable
|
||||
# theory: abs(P(a,b) - P(a,c)) <= 1 + P(b,c)
|
||||
|
||||
# For any local hidden variable theory, the Bell inequality must hold. It
|
||||
# introduces a third unit vector c, which is any other unit vector than
|
||||
# a or b.
|
||||
|
||||
# Does the quantum mechanical prediction violate the Bell inequality? Testing
|
||||
# several detector configurations in a plane, systematically from 0 to pi,
|
||||
# we determine whether the QM prediction is consistent with a local hidden
|
||||
# variable theory.
|
||||
|
||||
violations = 0
|
||||
trials = 0
|
||||
for step_a in range(0,6):
|
||||
for step_b in range (0,6):
|
||||
phi_a = step_a/6*np.pi
|
||||
phi_b = step_b/6*np.pi
|
||||
phi_c = phi_a - 0.5*phi_b
|
||||
P_ab = -1 * np.cos(phi_a - phi_b)
|
||||
P_ac = -1 * np.cos(phi_a - phi_c)
|
||||
P_bc = -1 * np.cos(phi_b - phi_c)
|
||||
|
||||
bell_lhs = np.abs(P_ab - P_ac)
|
||||
bell_rhs = 1 + P_bc
|
||||
|
||||
if bell_lhs > bell_rhs:
|
||||
violations = violations + 1
|
||||
|
||||
trials = trials + 1
|
||||
|
||||
zeroes = np.zeros(violations,dtype=int)
|
||||
ones = np.full((trials-violations),1,dtype=int)
|
||||
result = np.concatenate((ones,zeroes))
|
||||
|
||||
plt.ylim([trials/2-10,trials/2+10])
|
||||
plt.xlim([-1,2])
|
||||
plt.xticks([0.125,0.85],["violation","adherence"])
|
||||
plt.tick_params(axis='both',labelsize=15)
|
||||
plt.suptitle("Hidden Locality Violations",fontsize=20)
|
||||
plt.hist([result],bins=4)
|
||||
plt.figure()
|
||||
|
||||
|
||||
|
||||
|
||||
# It is seen that the QM prediction disagrees with a local hidden variable
|
||||
# theory in a significant number of configurations.
|
||||
|
||||
# On average, for random orientations between 0 and pi, how often? After running several trials,
|
||||
# it appears to be about half of the time.
|
||||
|
||||
violations = 0
|
||||
trials = 0
|
||||
for rand_a in range(0,10):
|
||||
for rand_b in range (0,10):
|
||||
phi_a = np.random.rand(1)*np.pi
|
||||
phi_b = np.random.rand(1)*np.pi
|
||||
phi_c = phi_a - 0.5*phi_b
|
||||
P_ab = -1 * np.cos(phi_a - phi_b)
|
||||
P_ac = -1 * np.cos(phi_a - phi_c)
|
||||
P_bc = -1 * np.cos(phi_b - phi_c)
|
||||
|
||||
bell_lhs = np.abs(P_ab - P_ac)
|
||||
bell_rhs = 1 + P_bc
|
||||
|
||||
if bell_lhs > bell_rhs:
|
||||
violations = violations + 1
|
||||
|
||||
trials = trials + 1
|
||||
|
||||
zeroes = np.zeros(violations,dtype=int)
|
||||
ones = np.full((trials-violations),1,dtype=int)
|
||||
result = np.concatenate((ones,zeroes))
|
||||
|
||||
plt.ylim([trials/2-10,trials/2+10])
|
||||
plt.xlim([-1,2])
|
||||
plt.xticks([0.125,0.85],["violation","adherence"])
|
||||
plt.tick_params(axis='both',labelsize=15)
|
||||
plt.suptitle("Hidden Locality violations",fontsize=20)
|
||||
plt.hist([result],bins=4)
|
||||
plt.figure()
|
||||
|
||||
|
||||
|
||||
# The prediction (Bell's inequality) made by assuming a local hidden
|
||||
# variable is violated by a significant number of the possibile orientations.
|
||||
# This simulation cannot determine which theory is correct, but the QM
|
||||
# prediction has been confirmed through experiment. No hidden local variable
|
||||
# holds actionable information about the state. Entangled states retain
|
||||
# their entanglement in a non-local nature.
|
||||
|
||||
|
||||
|
||||
|
||||
BIN
recursion.pdf
Normal file
BIN
recursion.pdf
Normal file
Binary file not shown.
Loading…
Reference in New Issue
Block a user