Legendary Physicists

Learn directly from the masters who shaped our understanding of quantum mechanics and modern physics

Learn from the Legends

These video lectures feature legendary physicists and mathematicians explaining quantum mechanics and fundamental physics in their own words. There's no better way to understand physics than learning directly from the geniuses who discovered it.

Featured: Richard Feynman (Nobel 1965) • Emmy Noether (Noether's Theorem) • Paul Dirac (Nobel 1933) • Werner Heisenberg (Nobel 1932) • Erwin Schrödinger (Nobel 1933) • Alain Aspect (Nobel 2022)

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Richard Feynman

Nobel Prize in Physics 1965 • "The Great Explainer"

Richard Feynman won the Nobel Prize for his work on Quantum Electrodynamics (QED). Known as "The Great Explainer," Feynman had an extraordinary ability to make complex physics accessible and exciting. Bill Gates called him "the best teacher I never had."

Nobel Prize:1965 (QED)

Institution:Caltech

Contributions:QED, Feynman diagrams

⚛️

QED: The Strange Theory of Light and Matter

Douglas Robb Memorial Lectures • Auckland 1979 • 4 lectures

In 1979, Feynman traveled to the University of Auckland, New Zealand, to deliver four lectures on Quantum Electrodynamics—the theory that earned him the Nobel Prize. These lectures are legendary for making the most advanced quantum physics accessible to general audiences.

Lecture 1: Photons - Corpuscles of Light

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Video Lecture

Feynman QED Lecture 1 - Photons

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 2: Fits of Reflection and Transmission - Quantum Behaviour

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Video Lecture

Feynman QED Lecture 2 - Quantum Behaviour

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 3: Electrons and Their Interactions

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Video Lecture

Feynman QED Lecture 3 - Electrons

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 4: New Queries

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Video Lecture

Feynman QED Lecture 4 - New Queries

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Book: These lectures became the basis for "QED: The Strange Theory of Light and Matter" (1985), one of the most popular physics books ever written.

📚

The Character of Physical Law

Messenger Lectures • Cornell University 1964 • 7 lectures

Feynman's famous Cornell Messenger Lectures from 1964, filmed by the BBC. These lectures explore the fundamental nature of physical law through examples from gravitation, conservation principles, symmetry, and quantum mechanics.

Lecture 1: The Law of Gravitation

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Video Lecture

Feynman - The Law of Gravitation

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 2: The Relation of Mathematics and Physics

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Video Lecture

Feynman - Mathematics and Physics

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 3: The Great Conservation Principles

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Video Lecture

Feynman - Conservation Principles

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 4: Symmetry in Physical Law

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Video Lecture

Feynman - Symmetry

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 5: The Distinction of Past and Future

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Video Lecture

Feynman - Past and Future

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 6: Probability and Uncertainty - The Quantum Mechanical View

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Video Lecture

Feynman - Quantum Uncertainty

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Lecture 7: Seeking New Laws

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Video Lecture

Feynman - Seeking New Laws

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Official Source: High-quality versions with interactive transcripts are available at feynmanlectures.caltech.edu

🎬

Fun to Imagine

BBC Television Series • 1983 • 6 episodes

In this beloved BBC series, Feynman sits in his home and explains everyday physics phenomena with infectious enthusiasm. Why are rubber bands stretchy? How do magnets work? What is fire, really? Feynman's joy of discovery is contagious.

Complete Series: Fun to Imagine (All Episodes)

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Video Lecture

Feynman - Fun to Imagine Complete

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Episodes included:

• Jiggling Atoms
• Fire
• Rubber Bands
• Magnets
• Electricity
• Mirror Reflections
• Train Wheels & Refrigerators
• Seeing Things
• Big Numbers

Why Watch: Even if you've studied physics for years, Feynman's explanations will give you new insights. His famous "magnets" explanation is particularly brilliant.

∞

Emmy Noether

"The most significant creative mathematical genius" — Albert Einstein

Emmy Noether (1882–1935) proved what is arguably the most important theorem in theoretical physics: Noether's Theorem, which reveals the deep connection between symmetries and conservation laws. Every symmetry in nature corresponds to a conserved quantity—time symmetry gives energy conservation, space symmetry gives momentum conservation, rotational symmetry gives angular momentum.

Famous For:Noether's Theorem

Field:Mathematics & Physics

Impact:Foundation of modern physics

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Emmy Noether Lectures

Symmetry, Conservation Laws & Modern Physics

These lectures explore Noether's theorem and its profound implications for physics. Understanding why conservation laws exist (rather than just accepting them) is essential for deep comprehension of quantum mechanics and all of modern physics.

Complete Lecture Playlist

Open in YouTube →

Why This Matters: Noether's theorem is the reason physicists search for symmetries. Every fundamental force in the Standard Model comes from a gauge symmetry. Without Noether, there would be no modern particle physics.

Noether's Theorem at a Glance

Time Translation Symmetry

Laws of physics don't change with time → Energy Conservation

Space Translation Symmetry

Laws are the same everywhere → Momentum Conservation

Rotational Symmetry

Laws don't depend on direction → Angular Momentum Conservation

Gauge Symmetry

U(1), SU(2), SU(3) symmetries → Charge Conservation, Forces

🏆

Paul Dirac

Nobel Prize in Physics 1933 • "The Strangest Man"

Paul Adrien Maurice Dirac (1902–1963) was one of the most brilliant theoretical physicists of the 20th century. He formulated the Dirac equation, which unified quantum mechanics with special relativity and predicted the existence of antimatter before the positron was experimentally discovered by Carl Anderson in 1932. Dirac also made foundational contributions to quantum electrodynamics (QED), introduced the delta function and bra-ket notation, and laid the mathematical groundwork for modern quantum field theory.

Nobel Prize:1933 (with Schrödinger)

Institution:Cambridge University

Contributions:Dirac equation, antimatter, QED

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The Dirac Equation & Key Concepts

Relativistic quantum mechanics • Antimatter prediction • Spin from first principles

The Dirac Equation

The Dirac equation is the relativistic wave equation for spin-1/2 particles. It naturally produces electron spin, the correct magnetic moment, and predicts antimatter:

$$(i\gamma^\mu \partial_\mu - m)\psi = 0$$

where $\gamma^\mu$ are the 4×4 gamma matrices satisfying the Clifford algebra:

$$\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}I_4$$

Dirac Sea & Positron

The Dirac equation yields both positive and negative energy solutions:

$$E = \pm\sqrt{p^2c^2 + m^2c^4}$$

Dirac proposed the "sea" of filled negative-energy states. A hole in this sea appears as a particle with positive charge—the positron, discovered in 1932.

Bra-Ket Notation & Delta Function

Dirac introduced the elegant bra-ket notation used throughout quantum mechanics:

$$\langle \phi | \psi \rangle = \int \phi^*(x)\psi(x)\,dx$$

He also formalized the Dirac delta function $\delta(x)$, essential for completeness relations: $\int |x\rangle\langle x|\,dx = \hat{I}$.

Historical Note: When Dirac first derived negative-energy solutions, he initially tried to identify the "holes" as protons. It was only after Oppenheimer showed this was inconsistent that Dirac boldly predicted a new particle—the positron—with the electron's mass but positive charge.

Computational Explorations

Dirac Equation Energy Spectrum

Python

Visualize the positive and negative energy branches E = +/-sqrt(p^2c^2 + m^2c^4) and the Dirac sea interpretation

script.py83 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Dirac Equation Energy Spectrum
# E = +/- sqrt(p^2 c^2 + m^2 c^4)
# ──────────────────────────────────────────────

c = 3e8          # speed of light (m/s)
m_e = 9.109e-31  # electron mass (kg)
mc2 = m_e * c**2 # rest energy (J)
mc2_MeV = 0.511  # rest energy (MeV)

p = np.linspace(-5 * m_e * c, 5 * m_e * c, 500)
p_norm = p / (m_e * c)  # in units of mc

E_pos = np.sqrt(p**2 * c**2 + m_e**2 * c**4) / (m_e * c**2)
E_neg = -np.sqrt(p**2 * c**2 + m_e**2 * c**4) / (m_e * c**2)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

# Left plot: energy spectrum
ax1.plot(p_norm, E_pos, 'cyan', linewidth=2, label='Positive energy (electron)')
ax1.plot(p_norm, E_neg, 'magenta', linewidth=2, label='Negative energy (positron/Dirac sea)')
ax1.axhline(y=1, color='gray', linestyle='--', alpha=0.5, label='$E = +mc^2$')
ax1.axhline(y=-1, color='gray', linestyle='--', alpha=0.5, label='$E = -mc^2$')
ax1.axhspan(-1, 1, alpha=0.1, color='yellow', label='Forbidden gap (2$mc^2$)')
ax1.set_xlabel('Momentum p / (mc)', fontsize=12, color='white')
ax1.set_ylabel('Energy E / ($mc^2$)', fontsize=12, color='white')
ax1.set_title('Dirac Equation Energy Spectrum', fontsize=14, color='cyan', fontweight='bold')
ax1.legend(fontsize=9, loc='upper left', facecolor='#1a1a2e', edgecolor='cyan', labelcolor='white')
ax1.set_facecolor('#0a0a1a')
ax1.tick_params(colors='white')
ax1.set_ylim(-5, 5)
ax1.grid(True, alpha=0.2)
for spine in ax1.spines.values():
    spine.set_color('white')

# Right plot: Dirac sea visualization
ax2.set_xlim(-3, 3)
ax2.set_ylim(-5, 5)
ax2.axhspan(-5, -1, alpha=0.3, color='magenta', label='Dirac sea (filled)')
ax2.axhspan(-1, 1, alpha=0.15, color='yellow', label='Energy gap $2mc^2$')
ax2.axhspan(1, 5, alpha=0.1, color='cyan', label='Continuum states')

for y_val in np.linspace(-4.5, -1.2, 8):
    ax2.plot(0, y_val, 'o', color='magenta', markersize=8, alpha=0.7)

ax2.annotate('Hole = positron', xy=(0, -2), xytext=(1.5, -2),
            arrowprops=dict(arrowstyle='->', color='yellow'),
            fontsize=10, color='yellow')
ax2.plot(0, -2, 'o', color='black', markersize=10, markeredgecolor='yellow', markeredgewidth=2)
ax2.plot(0, 2.5, '*', color='cyan', markersize=15, label='Excited electron')

ax2.set_xlabel('State index', fontsize=12, color='white')
ax2.set_ylabel('Energy / ($mc^2$)', fontsize=12, color='white')
ax2.set_title("Dirac Sea Interpretation", fontsize=14, color='magenta', fontweight='bold')
ax2.legend(fontsize=9, loc='upper left', facecolor='#1a1a2e', edgecolor='magenta', labelcolor='white')
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.2)
for spine in ax2.spines.values():
    spine.set_color('white')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()

print("Dirac Equation Energy Spectrum")
print("=" * 50)
print(f"Electron rest energy: {mc2_MeV:.3f} MeV")
print(f"Energy gap: {2*mc2_MeV:.3f} MeV")
print(f"\nPositive branch: E = +sqrt(p²c² + m²c⁴)  →  electrons")
print(f"Negative branch: E = -sqrt(p²c² + m²c⁴)  →  positrons (Dirac sea holes)")
print(f"\nKey predictions of the Dirac equation:")
print(f"  1. Antimatter (positron) - confirmed by Anderson (1932)")
print(f"  2. Electron spin emerges naturally (spin-1/2)")
print(f"  3. Correct magnetic moment g ≈ 2")
print(f"  4. Fine structure of hydrogen spectrum")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Dirac Spinor Components

Fortran

Compute the four-component Dirac spinor for a spin-up electron at various momenta

dirac_spinor.f9049 lines

program dirac_spinor
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: c = 3.0d8
  real(dp), parameter :: m_e = 9.109d-31
  real(dp), parameter :: hbar = 1.0546d-34
  real(dp), parameter :: MeV = 1.602d-13

real(dp) :: p, E, mc, mc2
  real(dp) :: norm, u1, u2, u3, u4
  real(dp) :: p_values(7)
  integer :: i

mc = m_e * c
  mc2 = m_e * c * c

write(*,'(A)') '=== Dirac Spinor Components ==='
  write(*,'(A)') 'For spin-up electron along z-axis'
  write(*,'(A)') ''
  write(*,'(A,A12,A14,A14,A14,A14)') ' p/(mc)  ', 'E/(mc²)', &
        '  u1  ', '  u2  ', '  u3  ', '  u4  '
  write(*,'(A)') repeat('-', 72)

p_values = (/ -2.0_dp, -1.0_dp, -0.5_dp, 0.0_dp, 0.5_dp, 1.0_dp, 2.0_dp /)

do i = 1, 7
    p = p_values(i) * mc
    E = sqrt(p**2 * c**2 + mc2**2)

! Positive-energy spin-up Dirac spinor (standard representation)
    ! u = N * (1, 0, p*c/(E+mc2), 0)^T for spin-up along z
    norm = sqrt((E + mc2) / (2.0_dp * mc2))
    u1 = norm * 1.0_dp
    u2 = norm * 0.0_dp
    u3 = norm * (p * c) / (E + mc2)
    u4 = norm * 0.0_dp

write(*,'(F7.2,F12.4,4F14.6)') p_values(i), E/mc2, u1, u2, u3, u4
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Notes:'
  write(*,'(A)') '  - u1,u2 are large components (particle)'
  write(*,'(A)') '  - u3,u4 are small components (antiparticle admixture)'
  write(*,'(A)') '  - At rest (p=0): only u1 is nonzero (pure particle)'
  write(*,'(A)') '  - At high p: u3 grows (relativistic mixing)'
  write(*,'(A)') '  - Spinor normalized: |u1|²+|u2|²+|u3|²+|u4|² ≈ E/(mc²)'

end program dirac_spinor

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

🎬

Video Lectures on Dirac

Learn about the Dirac equation, antimatter, and relativistic quantum mechanics

The Dirac Equation — A Visual Explanation

▶️

Video Lecture

The Dirac Equation Explained

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Paul Dirac and the Prediction of Antimatter

▶️

Video Lecture

Dirac and Antimatter

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Dirac's Legacy: From QED to the Standard Model

▶️

Video Lecture

Dirac's Legacy

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Book: Dirac's own textbook "The Principles of Quantum Mechanics" (1930) remains one of the most influential physics texts ever written and introduced the bra-ket notation used worldwide.

🏆

Werner Heisenberg

Nobel Prize in Physics 1932 • "Father of Matrix Mechanics"

Werner Karl Heisenberg (1901–1976) revolutionized physics by developing matrix mechanics, the first mathematically consistent formulation of quantum mechanics, in 1925. He is best known for the uncertainty principle, which establishes fundamental limits on how precisely complementary variables (like position and momentum) can be simultaneously known. This is not a limitation of measurement technology but a fundamental property of nature. Heisenberg also made major contributions to nuclear physics, turbulence theory, and the S-matrix formulation of particle physics.

Nobel Prize:1932 (Quantum Mechanics)

Institution:Leipzig, Göttingen, Munich

Contributions:Uncertainty principle, matrix mechanics

📐

Uncertainty Principle & Matrix Mechanics

Fundamental limits of measurement • Observable algebra • Heisenberg picture

The Uncertainty Relations

Heisenberg's uncertainty principle states that certain pairs of physical properties cannot both be known to arbitrary precision simultaneously:

$$\Delta x \, \Delta p \geq \frac{\hbar}{2}$$

$$\Delta E \, \Delta t \geq \frac{\hbar}{2}$$

The general form for any two observables $\hat{A}$ and $\hat{B}$ is:$$\Delta A \, \Delta B \geq \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle|$$

Matrix Mechanics (1925)

Heisenberg replaced classical trajectories with matrices of transition amplitudes. Physical observables become non-commuting operators:

$$[\hat{x}, \hat{p}] = i\hbar$$

This non-commutativity is the mathematical root of the uncertainty principle. Energy levels are found by diagonalizing the Hamiltonian matrix.

Heisenberg vs Schrödinger Picture

In the Heisenberg picture, operators evolve in time while states remain fixed:

$$\frac{d\hat{A}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}] + \frac{\partial \hat{A}}{\partial t}$$

In the Schrödinger picture, states evolve while operators are fixed. Both pictures give identical physical predictions.

Historical Note: Heisenberg developed matrix mechanics during a retreat to Helgoland island in 1925, recovering from hay fever. He later recalled being so excited by his results that he stayed up all night and climbed a rock to watch the sunrise.

Computational Explorations

Uncertainty Principle Visualization

Python

Gaussian wave packets demonstrating the position-momentum trade-off and minimum uncertainty states

script.py126 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Heisenberg Uncertainty Principle Visualization
# Gaussian wave packets: Δx·Δp = ℏ/2
# ──────────────────────────────────────────────

hbar = 1.0  # natural units

def gaussian_wavepacket(x, x0, sigma):
    """Position-space Gaussian wave packet"""
    psi = (1.0/(2*np.pi*sigma**2))**0.25 * np.exp(-(x - x0)**2 / (4*sigma**2))
    return psi

def momentum_wavepacket(p, p0, sigma_x):
    """Momentum-space Gaussian (FT of position Gaussian)"""
    sigma_p = hbar / (2 * sigma_x)
    phi = (1.0/(2*np.pi*sigma_p**2))**0.25 * np.exp(-(p - p0)**2 / (4*sigma_p**2))
    return phi

x = np.linspace(-15, 15, 1000)
p = np.linspace(-5, 5, 1000)

sigmas = [0.5, 1.0, 2.0, 4.0]
colors = ['#ff6b6b', '#ffd93d', '#6bcb77', '#4d96ff']

fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Top-left: Position-space wave packets
ax1 = axes[0, 0]
for sigma, color in zip(sigmas, colors):
    psi = gaussian_wavepacket(x, 0, sigma)
    prob = np.abs(psi)**2
    ax1.plot(x, prob, color=color, linewidth=2,
             label=f'\u0394x = {sigma:.1f}')
    ax1.fill_between(x, prob, alpha=0.15, color=color)
ax1.set_xlabel('Position x', fontsize=12, color='white')
ax1.set_ylabel('|\u03c8(x)|\u00b2', fontsize=12, color='white')
ax1.set_title('Position Space', fontsize=14, color='cyan', fontweight='bold')
ax1.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='cyan', labelcolor='white')
ax1.set_facecolor('#0a0a1a')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.2)
for spine in ax1.spines.values():
    spine.set_color('white')

# Top-right: Momentum-space wave packets
ax2 = axes[0, 1]
for sigma, color in zip(sigmas, colors):
    sigma_p = hbar / (2 * sigma)
    phi = momentum_wavepacket(p, 0, sigma)
    prob_p = np.abs(phi)**2
    ax2.plot(p, prob_p, color=color, linewidth=2,
             label=f'\u0394p = {sigma_p:.2f}')
    ax2.fill_between(p, prob_p, alpha=0.15, color=color)
ax2.set_xlabel('Momentum p', fontsize=12, color='white')
ax2.set_ylabel('|\u03c6(p)|\u00b2', fontsize=12, color='white')
ax2.set_title('Momentum Space', fontsize=14, color='magenta', fontweight='bold')
ax2.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='magenta', labelcolor='white')
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.2)
for spine in ax2.spines.values():
    spine.set_color('white')

# Bottom-left: Uncertainty product
ax3 = axes[1, 0]
sigma_range = np.linspace(0.2, 5.0, 100)
delta_x = sigma_range
delta_p = hbar / (2 * sigma_range)
product = delta_x * delta_p

ax3.plot(delta_x, product, 'yellow', linewidth=3, label='\u0394x \u00b7 \u0394p (Gaussian)')
ax3.axhline(y=hbar/2, color='red', linestyle='--', linewidth=2,
            label=f'\u0127/2 = {hbar/2:.2f} (minimum)')
ax3.fill_between(delta_x, hbar/2, product, alpha=0.1, color='yellow')
ax3.set_xlabel('\u0394x', fontsize=12, color='white')
ax3.set_ylabel('\u0394x \u00b7 \u0394p', fontsize=12, color='white')
ax3.set_title('Uncertainty Product', fontsize=14, color='yellow', fontweight='bold')
ax3.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='yellow', labelcolor='white')
ax3.set_facecolor('#0a0a1a')
ax3.tick_params(colors='white')
ax3.set_ylim(0, 2)
ax3.grid(True, alpha=0.2)
for spine in ax3.spines.values():
    spine.set_color('white')

# Bottom-right: Δx vs Δp trade-off
ax4 = axes[1, 1]
ax4.loglog(delta_x, delta_p, 'cyan', linewidth=3, label='\u0394p = \u0127/(2\u0394x)')
for sigma, color in zip(sigmas, colors):
    sp = hbar / (2 * sigma)
    ax4.plot(sigma, sp, 'o', color=color, markersize=12, markeredgecolor='white',
             markeredgewidth=2, zorder=5)
    ax4.annotate(f'({sigma:.1f}, {sp:.2f})', (sigma, sp),
                textcoords="offset points", xytext=(10, 10),
                fontsize=9, color=color)

ax4.set_xlabel('\u0394x', fontsize=12, color='white')
ax4.set_ylabel('\u0394p', fontsize=12, color='white')
ax4.set_title('Position-Momentum Trade-off', fontsize=14, color='cyan', fontweight='bold')
ax4.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='cyan', labelcolor='white')
ax4.set_facecolor('#0a0a1a')
ax4.tick_params(colors='white')
ax4.grid(True, alpha=0.2, which='both')
for spine in ax4.spines.values():
    spine.set_color('white')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()

print("Heisenberg Uncertainty Principle Demonstration")
print("=" * 50)
print(f"For Gaussian wave packets (minimum uncertainty states):")
print(f"  \u0394x \u00b7 \u0394p = \u0127/2 = {hbar/2:.4f}")
print(f"\nKey insight: Gaussians SATURATE the inequality (equality holds)")
print(f"\nWave packet widths tested:")
for sigma in sigmas:
    sp = hbar / (2 * sigma)
    print(f"  \u0394x = {sigma:.1f}  →  \u0394p = {sp:.3f}  →  product = {sigma*sp:.4f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Quantum Hamiltonian Diagonalization

Fortran

Matrix diagonalization of a quantum harmonic oscillator Hamiltonian with anharmonic perturbation

hamiltonian_diag.f9077 lines

program hamiltonian_diag
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  integer, parameter :: N = 6
  real(dp) :: H(N,N), eigenvalues(N)
  real(dp) :: work(3*N)
  integer :: info, i, j
  real(dp), parameter :: hbar = 1.0_dp
  real(dp), parameter :: m = 1.0_dp
  real(dp), parameter :: omega = 1.0_dp
  real(dp) :: dx, expected

write(*,'(A)') '=== Quantum Hamiltonian Matrix Diagonalization ==='
  write(*,'(A)') 'System: 1D Quantum Harmonic Oscillator'
  write(*,'(A,I3,A)') 'Truncated to ', N, ' basis states'
  write(*,'(A)') ''

! Build Hamiltonian in number-state basis |n>
  ! H = hbar*omega*(n + 1/2)
  ! Off-diagonal from perturbation: <n|x^2|m> terms
  H = 0.0_dp

! Diagonal: E_n = hbar*omega*(n + 0.5)
  do i = 1, N
    H(i,i) = hbar * omega * (real(i-1, dp) + 0.5_dp)
  end do

! Add anharmonic perturbation lambda*x^4
  ! x = sqrt(hbar/(2*m*omega)) * (a + a†)
  ! <n|x^2|m> = (hbar/(2*m*omega)) * coupling terms
  ! For small perturbation demonstration
  dx = 0.05_dp  ! perturbation strength
  do i = 1, N
    do j = 1, N
      if (j == i+2 .and. i+2 <= N) then
        H(i,j) = H(i,j) + dx * sqrt(real((i)*(i+1), dp))
        H(j,i) = H(i,j)
      end if
      if (j == i-2 .and. i-2 >= 1) then
        H(i,j) = H(i,j) + dx * sqrt(real((i-1)*(i-2), dp))
        H(j,i) = H(i,j)
      end if
    end do
  end do

write(*,'(A)') 'Hamiltonian Matrix H (with anharmonic perturbation):'
  do i = 1, N
    write(*,'(6F10.4)') (H(i,j), j=1,N)
  end do
  write(*,'(A)') ''

! Diagonalize using LAPACK dsyev
  call dsyev('V', 'U', N, H, N, eigenvalues, work, 3*N, info)

if (info == 0) then
    write(*,'(A)') 'Eigenvalues (energy levels in units of hbar*omega):'
    write(*,'(A)') '  n    E_numerical   E_harmonic    Shift'
    write(*,'(A)') ' ---   -----------   ----------   ------'
    do i = 1, N
      expected = hbar * omega * (real(i-1, dp) + 0.5_dp)
      write(*,'(I4,F13.6,F13.6,F10.6)') i-1, eigenvalues(i), expected, &
            eigenvalues(i) - expected
    end do
    write(*,'(A)') ''
    write(*,'(A)') 'Eigenvectors (columns = states):'
    do i = 1, N
      write(*,'(6F10.4)') (H(i,j), j=1,N)
    end do
  else
    write(*,'(A,I4)') 'Diagonalization failed with info = ', info
  end if

write(*,'(A)') ''
  write(*,'(A)') 'Note: Matrix mechanics (Heisenberg 1925) represents'
  write(*,'(A)') 'observables as matrices. Diagonalizing H gives energy levels.'

end program hamiltonian_diag

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

🎬

Video Lectures on Heisenberg

The uncertainty principle, matrix mechanics, and quantum foundations

The Uncertainty Principle Explained

▶️

Video Lecture

Heisenberg Uncertainty Principle

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Matrix Mechanics — How Heisenberg Built Quantum Theory

▶️

Video Lecture

Matrix Mechanics Explained

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Heisenberg and the Quantum Revolution

▶️

Video Lecture

Heisenberg and the Quantum Revolution

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Book: Heisenberg's "Physics and Philosophy" (1958) remains a classic exploration of the philosophical implications of quantum mechanics, written for general audiences.

🏆

Erwin Schrödinger

Nobel Prize in Physics 1933 • "The Cat Man of Quantum Mechanics"

Erwin Rudolf Josef Alexander Schrödinger (1887–1961) developed wave mechanics, an equivalent but more intuitive formulation of quantum mechanics compared to Heisenberg's matrix mechanics. His famous Schrödinger equation describes how quantum states evolve in time and remains the central equation of non-relativistic quantum mechanics. He is also known for the Schrödinger's cat thought experiment, which highlights the measurement problem and the paradox of quantum superposition at macroscopic scales.

Nobel Prize:1933 (with Dirac)

Institution:Vienna, Zürich, Dublin

Contributions:Wave equation, cat paradox

📐

The Schrödinger Equation & Wave Mechanics

Wave functions • Born interpretation • Measurement problem

The Schrödinger Equation

The time-dependent Schrödinger equation governs the evolution of all non-relativistic quantum systems:

$$i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi$$

For a particle in a potential $V(x)$, the Hamiltonian gives:

$$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi$$

The time-independent form $\hat{H}\psi_n = E_n\psi_n$ gives the stationary states and energy eigenvalues.

Wave-Particle Duality & Born Rule

Schrödinger's wave function $\psi(x,t)$ encodes all information about a quantum system. Max Born interpreted its squared modulus as a probability density:

$$P(x,t) = |\psi(x,t)|^2$$

This probabilistic interpretation was initially resisted by Schrödinger himself, who preferred a wave-only picture.

Schrödinger's Cat

In 1935, Schrödinger proposed his famous thought experiment: a cat in a sealed box is entangled with a radioactive atom, placing it in a superposition of alive and dead:

$$|\text{cat}\rangle = \frac{1}{\sqrt{2}}(|\text{alive}\rangle + |\text{dead}\rangle)$$

This paradox highlights the measurement problem: when and how does superposition collapse to a definite outcome?

Historical Note: Schrödinger derived his wave equation during a Christmas holiday in the Swiss Alps in December 1925. He later said it came from the de Broglie hypothesis that particles have wave-like properties, combined with the Hamilton-Jacobi formulation of classical mechanics.

Computational Explorations

Wave Packet Evolution in a Potential Well

Python

Time evolution of a Gaussian wave packet in an infinite square well showing collapse, revival, and probability density

script.py102 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Quantum Wave Packet in an Infinite Potential Well
# Time evolution via eigenstate expansion
# ──────────────────────────────────────────────

L = 1.0          # well width
hbar = 1.0       # natural units
m = 1.0          # particle mass
N_basis = 50     # number of eigenstates
N_x = 500        # spatial points

x = np.linspace(0, L, N_x)

# Energy eigenvalues: E_n = n²π²ℏ²/(2mL²)
def E_n(n):
    return (n * np.pi * hbar)**2 / (2 * m * L**2)

# Eigenstates: ψ_n(x) = sqrt(2/L) sin(nπx/L)
def psi_n(n, x):
    return np.sqrt(2.0/L) * np.sin(n * np.pi * x / L)

# Initial Gaussian wave packet centered at L/3
x0 = L / 3.0
sigma = L / 15.0
p0 = 20.0 * np.pi / L  # initial momentum

psi0 = np.exp(-(x - x0)**2 / (4*sigma**2)) * np.exp(1j * p0 * x)
# Normalize
norm = np.trapezoid(np.abs(psi0)**2, x)
psi0 = psi0 / np.sqrt(norm)

# Expand in eigenstates: c_n = <ψ_n|ψ_0>
c_n = np.zeros(N_basis, dtype=complex)
for n in range(1, N_basis + 1):
    c_n[n-1] = np.trapezoid(psi_n(n, x) * psi0, x)

# Time evolution: ψ(x,t) = Σ c_n ψ_n(x) exp(-iE_n t/ℏ)
def psi_t(t):
    psi = np.zeros_like(x, dtype=complex)
    for n in range(1, N_basis + 1):
        psi += c_n[n-1] * psi_n(n, x) * np.exp(-1j * E_n(n) * t / hbar)
    return psi

# Revival time: T_rev = 4mL²/(πℏ)
T_rev = 4 * m * L**2 / (np.pi * hbar)

times = [0, T_rev/20, T_rev/8, T_rev/4, T_rev/2, T_rev]
time_labels = ['t = 0', 'T/20', 'T/8', 'T/4', 'T/2', 'T (revival)']
colors = ['#ff6b6b', '#ffa502', '#ffd93d', '#6bcb77', '#4d96ff', '#cc6bff']

fig, axes = plt.subplots(2, 3, figsize=(16, 10))
axes_flat = axes.flatten()

for idx, (t, label, color) in enumerate(zip(times, time_labels, colors)):
    ax = axes_flat[idx]
    psi = psi_t(t)
    prob = np.abs(psi)**2

ax.fill_between(x, prob, alpha=0.3, color=color)
    ax.plot(x, prob, color=color, linewidth=2)
    ax.plot(x, psi.real, '--', color='cyan', linewidth=1, alpha=0.6, label='Re(\u03c8)')
    ax.plot(x, psi.imag, '--', color='magenta', linewidth=1, alpha=0.6, label='Im(\u03c8)')

ax.set_title(label, fontsize=13, color=color, fontweight='bold')
    ax.set_xlabel('x / L', fontsize=10, color='white')
    ax.set_ylabel('|\u03c8|\u00b2', fontsize=10, color='white')
    ax.set_xlim(0, L)
    ax.set_facecolor('#0a0a1a')
    ax.tick_params(colors='white')
    ax.grid(True, alpha=0.2)
    if idx == 0:
        ax.legend(fontsize=8, facecolor='#1a1a2e', edgecolor='cyan', labelcolor='white')
    for spine in ax.spines.values():
        spine.set_color('white')

fig.suptitle('Quantum Wave Packet in Infinite Well — Time Evolution',
             fontsize=16, color='cyan', fontweight='bold', y=1.02)
fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()

print("Wave Packet Evolution in Infinite Potential Well")
print("=" * 50)
print(f"Well width L = {L:.1f}")
print(f"Initial position: x0 = L/{L/x0:.0f}")
print(f"Wave packet width: \u03c3 = L/{L/sigma:.0f}")
print(f"Initial momentum: p0 = {p0:.1f}\u03c0/L")
print(f"Revival time: T_rev = {T_rev:.4f}")
print(f"Number of basis states: {N_basis}")
print(f"\nExpansion coefficients |c_n|² (top 5):")
probs = np.abs(c_n)**2
top_idx = np.argsort(probs)[::-1][:5]
for idx in top_idx:
    print(f"  n={idx+1:3d}: |c_n|² = {probs[idx]:.4f}")
print(f"\nTotal probability captured: {np.sum(probs):.6f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

1D Schrödinger Equation — Shooting Method

Fortran

Numerically solve the time-independent Schrödinger equation for the infinite square well using the shooting method with bisection

schrodinger_shooting.f9087 lines

program schrodinger_shooting
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  integer, parameter :: N = 1000
  real(dp), parameter :: L = 1.0_dp
  real(dp), parameter :: hbar = 1.0_dp
  real(dp), parameter :: m = 1.0_dp
  real(dp), parameter :: pi = 3.14159265358979323846_dp

real(dp) :: dx, x(0:N), psi(0:N), V(0:N)
  real(dp) :: E, E_low, E_high, E_mid
  real(dp) :: k2, psi_end
  integer :: i, level, iter
  real(dp) :: exact_E, error_pct
  real(dp) :: norm

dx = L / real(N, dp)

! Set up spatial grid and potential
  do i = 0, N
    x(i) = real(i, dp) * dx
    V(i) = 0.0_dp  ! infinite well (V=0 inside)
  end do

write(*,'(A)') '=== 1D Schrodinger Equation — Shooting Method ==='
  write(*,'(A)') 'System: Infinite square well, width L = 1.0'
  write(*,'(A)') 'Units: hbar = m = 1'
  write(*,'(A)') ''
  write(*,'(A,A7,A16,A16,A12)') '  ', 'Level', 'E_numerical', 'E_exact', 'Error(%)'
  write(*,'(A)') '  ' // repeat('-', 55)

do level = 1, 8
    ! Exact energy for infinite well
    exact_E = (real(level,dp) * pi * hbar)**2 / (2.0_dp * m * L**2)

! Bisection search for energy eigenvalue
    E_low = 0.0_dp
    E_high = exact_E * 2.5_dp

do iter = 1, 200
      E_mid = (E_low + E_high) / 2.0_dp

! Shooting: integrate from x=0 with psi(0)=0, psi(1)=small
      psi(0) = 0.0_dp
      psi(1) = 1.0d-5

do i = 1, N-1
        k2 = 2.0_dp * m * (E_mid - V(i)) / hbar**2
        psi(i+1) = 2.0_dp * psi(i) - psi(i-1) + dx**2 * (-k2) * psi(i)
      end do

psi_end = psi(N)

! Check boundary condition psi(L) = 0
      ! Determine sign to bisect correctly
      if (mod(level, 2) == 1) then
        ! Odd levels: psi_end should be positive for E too low
        if (psi_end > 0.0_dp) then
          E_low = E_mid
        else
          E_high = E_mid
        end if
      else
        ! Even levels: opposite sign convention
        if (psi_end < 0.0_dp) then
          E_low = E_mid
        else
          E_high = E_mid
        end if
      end if

if (abs(E_high - E_low) < 1.0d-12) exit
    end do

error_pct = abs(E_mid - exact_E) / exact_E * 100.0_dp
    write(*,'(A,I5,F16.8,F16.8,F10.4,A)') '  ', level, E_mid, exact_E, error_pct, '%'
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Method: Numerov shooting with bisection on energy'
  write(*,'(A,I6,A)') 'Grid points: ', N, ' (dx = 0.001)'
  write(*,'(A)') 'Boundary conditions: psi(0) = psi(L) = 0'
  write(*,'(A)') ''
  write(*,'(A)') 'The shooting method integrates the ODE from one boundary'
  write(*,'(A)') 'and adjusts E until the other boundary condition is satisfied.'

end program schrodinger_shooting

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

🎬

Video Lectures on Schrödinger

Wave mechanics, the Schrödinger equation, and the measurement problem

The Schrödinger Equation — What Does It Really Mean?

▶️

Video Lecture

Schrödinger Equation Explained

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Schrödinger's Cat — The Quantum Measurement Problem

▶️

Video Lecture

Schrödinger's Cat

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Wave Mechanics — From de Broglie to Schrödinger

▶️

Video Lecture

Wave Mechanics History

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Book: Schrödinger's "What is Life?" (1944) profoundly influenced molecular biology. Crick and Watson credited it as inspiration for their discovery of DNA's structure.

🏆

Alain Aspect

Nobel Prize in Physics 2022 • "The Man Who Proved Entanglement"

Alain Aspect (born 1947) performed the pioneering experiments that definitively tested Bell's inequalities and confirmed the reality of quantum entanglement. His 1982 experiments at the Institut d'Optique in Paris used rapid switching of measurement settings to close the "locality loophole," demonstrating that no local hidden variable theory can reproduce quantum mechanical predictions. He shared the 2022 Nobel Prize with John Clauser and Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science."

Nobel Prize:2022 (with Clauser, Zeilinger)

Institution:Institut d'Optique, Paris

Contributions:Bell tests, entanglement verification

📐

Bell Inequalities & Quantum Entanglement

EPR paradox • CHSH inequality • Loophole-free Bell tests

Bell's Inequality & the CHSH Form

John Bell showed in 1964 that any local hidden variable theory must satisfy an inequality. The most common experimental form is the CHSH inequality:

$$|S| = |E(a,b) - E(a,b') + E(a',b) + E(a',b')| \leq 2$$

Quantum mechanics predicts a maximum violation with the Tsirelson bound:

$$|S|_{\text{QM}} \leq 2\sqrt{2} \approx 2.828$$

Aspect's experiments measured $S \approx 2.70$, violating the classical limit by over 40 standard deviations.

The EPR Paradox

Einstein, Podolsky, and Rosen (1935) argued that quantum mechanics was incomplete because entangled particles seem to communicate instantaneously. They proposed "hidden variables" to restore local realism. For a singlet state:

$$|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)$$

Measuring one particle instantly determines the other, regardless of distance. Bell showed this cannot be explained classically.

Loophole-Free Bell Tests

Aspect's key innovation was rapid switching of measurement settings during photon flight, closing the locality loophole. Later experiments addressed:

• Detection loophole: closed by Giustina et al. (2013)
• Freedom-of-choice loophole: addressed by BIG Bell test (2018)
• All loopholes simultaneously: Hensen et al. (2015)

Why It Matters: Aspect's experiments settled a 50-year debate between Einstein and Bohr about the nature of reality. Nature is fundamentally non-local: entangled particles share correlations that cannot be explained by any classical mechanism. This is the foundation of quantum computing, quantum cryptography, and quantum teleportation.

Computational Explorations

CHSH Bell Inequality Violation

Python

Simulate quantum vs classical correlations and visualize the CHSH inequality violation with historic experiment results

script.py166 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# CHSH Bell Inequality Violation Simulation
# Classical limit: |S| ≤ 2
# Quantum maximum: |S| = 2√2 ≈ 2.828
# ──────────────────────────────────────────────

N_pairs = 10000  # entangled pairs per trial
N_trials = 50    # number of trials for statistics
np.random.seed(42)

def quantum_correlation(theta):
    """Quantum prediction: E(a,b) = -cos(a-b) for singlet state"""
    return -np.cos(theta)

def classical_correlation(theta):
    """Classical (local hidden variable) prediction"""
    return -1.0 + 2.0 * abs(theta) / np.pi

# CHSH measurement angles
# Optimal quantum settings: a1=0, a2=π/4, b1=π/8, b2=3π/8
a1, a2 = 0, np.pi/4
b1, b2 = np.pi/8, 3*np.pi/8

def simulate_quantum_CHSH(N):
    """Simulate CHSH with quantum mechanics predictions"""
    E11 = quantum_correlation(a1 - b1)
    E12 = quantum_correlation(a1 - b2)
    E21 = quantum_correlation(a2 - b1)
    E22 = quantum_correlation(a2 - b2)
    S = E11 - E12 + E21 + E22
    return S, E11, E12, E21, E22

def simulate_classical_CHSH(N):
    """Simulate CHSH with local hidden variables"""
    # Hidden variable: uniform random angle lambda
    lam = np.random.uniform(0, 2*np.pi, N)

def measure(angle, lam_vals):
        return np.sign(np.cos(angle - lam_vals))

A1 = measure(a1, lam)
    A2 = measure(a2, lam)
    B1 = measure(b1, lam)
    B2 = measure(b2, lam)

E11 = np.mean(A1 * B1)
    E12 = np.mean(A1 * B2)
    E21 = np.mean(A2 * B1)
    E22 = np.mean(A2 * B2)
    S = E11 - E12 + E21 + E22
    return S, E11, E12, E21, E22

# Run multiple trials
S_quantum_vals = []
S_classical_vals = []
for _ in range(N_trials):
    Sq, *_ = simulate_quantum_CHSH(N_pairs)
    Sc, *_ = simulate_classical_CHSH(N_pairs)
    S_quantum_vals.append(Sq)
    S_classical_vals.append(Sc)

S_quantum_vals = np.array(S_quantum_vals)
S_classical_vals = np.array(S_classical_vals)

fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Top-left: Correlation function comparison
ax1 = axes[0, 0]
theta = np.linspace(-np.pi, np.pi, 500)
E_qm = quantum_correlation(theta)
E_cl = np.array([-1 + 2*abs(t)/np.pi for t in theta])

ax1.plot(theta*180/np.pi, E_qm, 'cyan', linewidth=2.5, label='Quantum (singlet)')
ax1.plot(theta*180/np.pi, E_cl, 'red', linewidth=2.5, linestyle='--', label='Classical (LHV)')
ax1.axhline(y=0, color='gray', linestyle='-', alpha=0.3)
ax1.axvline(x=0, color='gray', linestyle='-', alpha=0.3)
ax1.set_xlabel('Angle difference (degrees)', fontsize=11, color='white')
ax1.set_ylabel('E(a,b) correlation', fontsize=11, color='white')
ax1.set_title('Quantum vs Classical Correlations', fontsize=13, color='cyan', fontweight='bold')
ax1.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='cyan', labelcolor='white')
ax1.set_facecolor('#0a0a1a')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.2)
for spine in ax1.spines.values():
    spine.set_color('white')

# Top-right: CHSH S-value distribution
ax2 = axes[0, 1]
ax2.hist(S_classical_vals, bins=15, alpha=0.6, color='red', edgecolor='white', label='Classical LHV')
ax2.axvline(x=2, color='yellow', linewidth=2.5, linestyle='--', label='Bell limit |S|=2')
ax2.axvline(x=2*np.sqrt(2), color='cyan', linewidth=2.5, linestyle='-', label=f'QM prediction S={2*np.sqrt(2):.3f}')
ax2.axvline(x=np.mean(S_quantum_vals), color='lime', linewidth=2, linestyle=':',
            label=f'QM simulated S={np.mean(S_quantum_vals):.3f}')
ax2.set_xlabel('CHSH S value', fontsize=11, color='white')
ax2.set_ylabel('Count', fontsize=11, color='white')
ax2.set_title('CHSH S-Value Distribution', fontsize=13, color='yellow', fontweight='bold')
ax2.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='yellow', labelcolor='white')
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.2)
for spine in ax2.spines.values():
    spine.set_color('white')

# Bottom-left: Violation significance
ax3 = axes[1, 0]
cumulative_S = np.cumsum(S_classical_vals) / np.arange(1, N_trials+1)
ax3.plot(range(1, N_trials+1), cumulative_S, 'red', linewidth=2, label='Classical running mean')
ax3.axhline(y=2, color='yellow', linewidth=2, linestyle='--', label='Bell limit |S|=2')
ax3.axhline(y=2*np.sqrt(2), color='cyan', linewidth=2, linestyle='--', label='QM prediction')
ax3.fill_between(range(1, N_trials+1), 2, 2*np.sqrt(2), alpha=0.1, color='cyan',
                 label='Quantum advantage region')
ax3.set_xlabel('Number of trials', fontsize=11, color='white')
ax3.set_ylabel('Running mean S', fontsize=11, color='white')
ax3.set_title('Convergence of S Value', fontsize=13, color='lime', fontweight='bold')
ax3.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='lime', labelcolor='white')
ax3.set_facecolor('#0a0a1a')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.2)
for spine in ax3.spines.values():
    spine.set_color('white')

# Bottom-right: Aspect experiment summary
ax4 = axes[1, 1]
experiments = ['Aspect\n1982', 'Aspect\n(switching)', 'Weihs\n1998', 'Hensen\n2015', 'BIG Bell\n2018']
S_values = [2.70, 2.73, 2.73, 2.42, 2.63]
colors_bar = ['#ff6b6b', '#ffa502', '#ffd93d', '#6bcb77', '#4d96ff']

bars = ax4.bar(experiments, S_values, color=colors_bar, edgecolor='white', linewidth=0.5, alpha=0.8)
ax4.axhline(y=2, color='yellow', linewidth=2.5, linestyle='--', label='Classical limit |S|=2')
ax4.axhline(y=2*np.sqrt(2), color='cyan', linewidth=2, linestyle=':', label='QM maximum 2\u221a2')

for bar, val in zip(bars, S_values):
    ax4.text(bar.get_x() + bar.get_width()/2., bar.get_height() + 0.03,
            f'{val:.2f}', ha='center', va='bottom', fontsize=10, color='white', fontweight='bold')

ax4.set_ylabel('CHSH S value', fontsize=11, color='white')
ax4.set_title('Historic Bell Test Experiments', fontsize=13, color='#ff6b6b', fontweight='bold')
ax4.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#ff6b6b', labelcolor='white')
ax4.set_facecolor('#0a0a1a')
ax4.tick_params(colors='white')
ax4.set_ylim(0, 3.2)
ax4.grid(True, alpha=0.2, axis='y')
for spine in ax4.spines.values():
    spine.set_color('white')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()

print("CHSH Bell Inequality Simulation")
print("=" * 50)
print(f"Classical limit:     |S| \u2264 2")
print(f"Quantum maximum:     |S| = 2\u221a2 \u2248 {2*np.sqrt(2):.4f}")
print(f"\nQuantum prediction:  S = {np.mean(S_quantum_vals):.4f}")
print(f"Classical simulation: S = {np.mean(S_classical_vals):.4f} \u00b1 {np.std(S_classical_vals):.4f}")
print(f"\nViolation: QM exceeds classical limit by {np.mean(S_quantum_vals)-2:.4f}")
print(f"\nOptimal CHSH angles:")
print(f"  Alice:  a1 = 0\u00b0, a2 = 45\u00b0")
print(f"  Bob:    b1 = 22.5\u00b0, b2 = 67.5\u00b0")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Bell Correlation Function Computation

Fortran

Compute the quantum and classical correlation functions E(theta) and evaluate the CHSH S-parameter

bell_correlation.f90102 lines

program bell_correlation
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.14159265358979323846_dp
  integer, parameter :: N_angles = 37
  integer, parameter :: N_pairs = 100000

real(dp) :: theta, d_theta
  real(dp) :: E_quantum, E_classical
  real(dp) :: a_angle, b_angle
  real(dp) :: lambda_val, A_meas, B_meas
  real(dp) :: sum_AB
  integer :: i, j
  real(dp) :: S_qm, S_cl
  real(dp) :: a1, a2, b1, b2
  real(dp) :: E11, E12, E21, E22
  integer :: seed_size
  integer, allocatable :: seed(:)

! Initialize random seed
  call random_seed(size=seed_size)
  allocate(seed(seed_size))
  seed = 12345
  call random_seed(put=seed)

write(*,'(A)') '=== Bell Correlation Function ==='
  write(*,'(A)') ''
  write(*,'(A)') 'E(theta) for entangled singlet state vs classical LHV'
  write(*,'(A)') ''
  write(*,'(A8,A16,A16,A16)') 'Angle', 'E_quantum', 'E_classical', 'Difference'
  write(*,'(A)') repeat('-', 58)

d_theta = 180.0_dp / real(N_angles - 1, dp)

do i = 1, N_angles
    theta = real(i - 1, dp) * d_theta * pi / 180.0_dp

! Quantum prediction: E = -cos(theta)
    E_quantum = -cos(theta)

! Classical (LHV) simulation
    sum_AB = 0.0_dp
    do j = 1, N_pairs
      call random_number(lambda_val)
      lambda_val = lambda_val * 2.0_dp * pi

! Deterministic measurement based on hidden variable
      if (cos(0.0_dp - lambda_val) >= 0.0_dp) then
        A_meas = 1.0_dp
      else
        A_meas = -1.0_dp
      end if

if (cos(theta - lambda_val) >= 0.0_dp) then
        B_meas = 1.0_dp
      else
        B_meas = -1.0_dp
      end if

sum_AB = sum_AB + A_meas * B_meas
    end do
    E_classical = sum_AB / real(N_pairs, dp)

if (mod(i-1, 3) == 0) then
      write(*,'(F7.1,A,F14.6,F16.6,F16.6)') &
        real(i-1,dp)*d_theta, '°', E_quantum, E_classical, &
        E_quantum - E_classical
    end if
  end do

! CHSH calculation
  write(*,'(A)') ''
  write(*,'(A)') '=== CHSH Inequality Test ==='
  a1 = 0.0_dp
  a2 = pi / 4.0_dp
  b1 = pi / 8.0_dp
  b2 = 3.0_dp * pi / 8.0_dp

! Quantum CHSH
  E11 = -cos(a1 - b1)
  E12 = -cos(a1 - b2)
  E21 = -cos(a2 - b1)
  E22 = -cos(a2 - b2)
  S_qm = E11 - E12 + E21 + E22

write(*,'(A,F10.6)') '  Quantum S  = ', S_qm
  write(*,'(A,F10.6)') '  2*sqrt(2)  = ', 2.0_dp * sqrt(2.0_dp)
  write(*,'(A,F10.6)') '  Classical limit = ', 2.0_dp
  write(*,'(A)') ''

if (abs(S_qm) > 2.0_dp) then
    write(*,'(A)') '  *** BELL INEQUALITY VIOLATED ***'
    write(*,'(A,F8.4)') '  Violation amount: ', abs(S_qm) - 2.0_dp
  end if

write(*,'(A)') ''
  write(*,'(A)') 'Conclusion: Quantum mechanics predicts correlations'
  write(*,'(A)') 'that no local hidden variable theory can reproduce.'
  write(*,'(A)') 'Aspect (1982) confirmed this experimentally.'

deallocate(seed)
end program bell_correlation

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