Scrolling bug in JupyterLab 4.3.5 - simple notebook that triggers the bug for me, to help reproducing the problem

scrolling-bug.ipynb

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    "# The Double-Slit Experiment"
   ]
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    ":::{admonition} Model thoughts\n",
    ":class: dropdown\n",
    "\n",
    "Okay, so I'm trying to understand the double-slit experiment in quantum mechanics. From what I remember, it's this experiment where particles like electrons or photons are sent towards a barrier with two slits, and then they go through each slit separately before interfering with each other on a detection screen. This interference is supposed to show that particles behave both like waves (interfering) and particles (passing through the slits).\n",
    ":::"
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    ":::{admonition} Model thought process\n",
    ":class: dropdown\n",
    "\n",
    "Okay, so I'm trying to understand the double-slit experiment in quantum mechanics. From what I remember, it's this experiment where particles like electrons or photons are sent towards a barrier with two slits, and then they go through each slit separately before interfering with each other on a detection screen. This interference is supposed to show that particles behave both like waves (interfering) and particles (passing through the slits).\n",
    "\n",
    "But wait, I'm not entirely sure about all the details. Let me think this through step by step.\n",
    "\n",
    "First, when these particles come out of the barrier, they should be in a state where their wavefunction is spread out over the entire screen because the two slits are so close together. This spreading out is what's called interference or superposition. So each particle isn't actually going through one slit or the other; instead, it exists in both states at once until it's observed.\n",
    "\n",
    "I remember something about electrons having wave-like properties and photons having particle-like properties. But I'm not sure how that plays into the double-slit experiment specifically. Maybe the electrons aren't behaving like particles when they're sent through the slits because their behavior is determined by the waves passing through both slits simultaneously?\n",
    "\n",
    "Also, I think there's something about the probability distribution of where the particles hit the screen. If it were just a wave, we'd see an interference pattern similar to what we observe in double-slit experiments. But if it were just particles going through one slit, we wouldn't get that pattern because they would interfere with themselves.\n",
    "\n",
    "Wait, but I also recall reading somewhere that even when you send a single particle through the slits, you can still see the wave-like behavior by detecting it on both slits at once. So maybe the experiment is designed to test whether particles behave like waves or particles based on how their detection is done.\n",
    "\n",
    "Another thing I'm confused about is the concept of quantum interference. How does that lead to the pattern observed? Is it because the wavefunction splits into two parts, and then they interfere when detected? If that's the case, wouldn't each particle just pass through one slit without interfering with itself?\n",
    "\n",
    "I also wonder about the practical aspects. Can we really detect particles individually in this experiment, or do they behave too much like waves to be observed as individual entities? Because if you can't see individual particles, then interference doesn't happen because there's no basis for observing them.\n",
    "\n",
    "Maybe it's about quantum measurement and how sensitive the detectors are. If the detectors are so precise that they can detect both particles passing through slit A and B simultaneously, then the wave-like behavior is seen as an interference pattern. But if the detectors are too crude, maybe you just see which slit each particle goes through before interfering.\n",
    "\n",
    "I'm also thinking about the theory behind this. The double-slit experiment is a foundational part of quantum mechanics. It was proposed by Young in the 19th century and demonstrated the wave-particle duality of light and matter. However, it wasn't until much later that experiments like Schrödinger's cat (where particles are both alive and dead) showed even more about this principle.\n",
    "\n",
    "So, putting it all together, the double-slit experiment tests whether particles exhibit wave-like behavior when detected individually or if they behave like particles by passing through one slit at a time. The interference pattern observed suggests that quantum mechanics is the correct framework for describing these particles' behavior.\n",
    ":::\n",
    "\n",
    "The double-slit experiment is a foundational demonstration in quantum mechanics that illustrates the wave-particle duality of matter and energy. Here's a concise summary of its key principles:\n",
    "\n",
    "1. **Wave Function Splitting**: When particles (e.g., electrons or photons) are sent through a barrier with two slits, their wave function splits into two parts simultaneously. This means both paths are active until the particle is detected.\n",
    "\n",
    "2. **Interference Pattern**: The interference pattern observed on a detection screen arises from the overlapping wave functions that pass through both slits. This phenomenon suggests that particles exhibit wave-like behavior, even when detected individually.\n",
    "\n",
    "3. **Particle-Indivisibility Detection**: The experiment relies on detecting particles with sufficient precision to observe their individual paths and interactions. If detectors are too sensitive, they might only reveal which slit each particle passes through without interference.\n",
    "\n",
    "4. **Measurement Sensitivity**: The effectiveness of the interference pattern depends on the detectors' sensitivity. More precise detection allows for wave-like behavior to be observed, while less sensitive detectors may show particle-like behavior by revealing path choices.\n",
    "\n",
    "5. **Theoretical Foundation**: This experiment is pivotal in quantum mechanics, confirming that particles such as electrons and photons exhibit both wave-like and particle-like properties. It underscores the principles of quantum superposition and interference, which are central to understanding modern physics.\n",
    "\n",
    "In essence, the double-slit experiment serves as a test for quantum mechanics' ability to describe particles' behavior through both waves and particles, supported by observable patterns when detected with sufficient sensitivity."
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    ":::{admonition} Model thought process\n",
    ":class: dropdown\n",
    "\n",
    "Okay, so I need to figure out what Schrödinger's cat thought experiment is. Hmm, I remember hearing about it in some basic physics class or maybe a documentary. Let me try to piece this together.\n",
    "\n",
    "First off, Erwin Schrödinger was a physicist, right? He worked on quantum mechanics, which is all about the tiny particles like electrons and photons. But what's the cat thing about?\n",
    "\n",
    "I think it has something to do with states of being. Like, whether something is alive or dead, but in a quantum context. Maybe it's about how particles can be in multiple states at once until observed. Oh yeah, that's superposition! So before you look, things are kind of all possibilities.\n",
    "\n",
    "So Schrödinger probably created this hypothetical scenario to explain or challenge the Copenhagen interpretation of quantum mechanics. Wait, what's that? I think that's when particles don't have definite properties until they're measured. So like, an electron doesn't have a position until you measure it.\n",
    "\n",
    "But how does the cat fit in? Okay, imagine putting a cat in a box with some radioactive material, a Geiger counter, and a vial of poison. If the Geiger counter detects decay, it triggers the poison, killing the cat. According to quantum theory, before you open the box, the radioactive atom is both decayed and not decayed at the same time.\n",
    "\n",
    "So if that's the case, does that mean the cat is both alive and dead simultaneously? That seems pretty strange, which is probably why Schrödinger came up with this example. He was trying to show how ridiculous it sounds when you apply quantum superposition to everyday objects like a cat instead of just particles.\n",
    "\n",
    "But then, isn't there something about decoherence or the observer effect? Maybe that's where people talk about consciousness affecting reality by observing. But I think most physicists don't take it literally; they use it as a tool for understanding, not a real explanation.\n",
    "\n",
    "So in summary, Schrödinger's cat is a thought experiment illustrating how quantum superposition could lead to macroscopic objects being in multiple states at once until observed. It's meant to highlight the weirdness of quantum mechanics and perhaps point out limitations or interpretations that need refining.\n",
    ":::\n",
    "\n",
    "Schrödinger's cat is a famous thought experiment in quantum physics, conceptualized by physicist Erwin Schrödinger in 1935. The experiment was designed to illustrate the paradoxical nature of quantum superposition when applied to macroscopic objects.\n",
    "\n",
    "**The Setup:**\n",
    "Imagine a scenario where a cat is placed inside a sealed box along with a radioactive atom, a Geiger counter, and a vial of poison. The apparatus is set up such that if the Geiger counter detects radiation (indicating decay of the radioactive atom), it triggers the release of poison, killing the cat.\n",
    "\n",
    "**Quantum Superposition:**\n",
    "According to quantum mechanics, until an observation or measurement is made, particles can exist in multiple states simultaneously—a phenomenon known as superposition. In this case, the radioactive atom exists in a superposition of both decayed and not decayed states before being observed.\n",
    "\n",
    "**The Paradox:**\n",
    "Extending this principle, Schrödinger's thought experiment suggests that until the box is opened and an observation is made, the cat would theoretically exist in a superposition of states—both alive and dead simultaneously. This illustrates how quantum principles applied to macroscopic objects lead to paradoxical conclusions.\n",
    "\n",
    "**Purpose and Interpretation:**\n",
    "Schrödinger proposed this paradox to highlight the peculiarities of applying quantum mechanics to everyday objects and to challenge the Copenhagen interpretation, which posits that particles have no definite state until measured. The experiment underscores the need for a more comprehensive understanding or alternative interpretations of quantum theory, such as decoherence or many-worlds interpretations.\n",
    "\n",
    "In essence, Schrödinger's cat serves as a tool to explore and discuss the foundational questions in quantum mechanics, emphasizing the enigmatic nature of reality at the quantum level."
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    "# Kramers-Krönig relations\n",
    "\n",
    "Let's play with some electricity and magnetism today...\n",
    "\n",
    "If $\\epsilon(\\omega)$ is the dielectric constant in an isotropic medium, we know from fundamental electromagnetic theory that its real and imaginary parts ($Re \\epsilon(\\omega)$ and , $Im \\epsilon(\\omega)$ respectively) are connected by the Kramers-Kronig dispersion relations, which state that\n",
    "\n",
    "$$\n",
    "Re \\epsilon(\\omega_0) = 1 + \\frac{2}{\\pi} P \\int_{0}^{\\infty} \\frac{\\omega Im \\epsilon(\\omega)}{\\omega^2 - \\omega_0^2} dw \n",
    "$$\n",
    "\n",
    "where $P$ denotes the Principal Part of the integral. "
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    ":::{admonition} Model thought process\n",
    ":class: dropdown\n",
    "\n",
    "Okay, so I'm trying to understand this Kramers-Kronig relation that was mentioned. The user wrote out the equation for the real part of the dielectric constant, Re ε(ω), in terms of the imaginary part Im ε(w). They also included some context about isotropic media and electromagnetic theory.\n",
    "\n",
    "First, I need to recall what the Kramers-Kronig relations are. From what I remember, these are a pair of integral equations that relate the real and imaginary parts of a material's dielectric constant or susceptibility. They're important because they ensure causality in the response of the medium to an external electromagnetic field.\n",
    "\n",
    "Looking at the equation provided:\n",
    "\n",
    "Re ε(ω) = 1 + (2/π) P ∫₀^∞ [w Im ε(w)] / (w² - ω²) dw\n",
    "\n",
    "Where P denotes the Cauchy principal value. I think this is one of the two relations, and there's another similar equation that relates the imaginary part to the real part.\n",
    "\n",
    "I should check if this equation makes sense dimensionally and mathematically. The integral involves a convolution with 1/(w² - ω²), which can be tricky because of the singularity when w = ω. That's why the principal value is necessary here—it tells us how to handle that point to make the integral converge.\n",
    "\n",
    "I also recall that the Kramers-Kronig relations come from the theory of analytic functions, particularly the fact that ε(ω) must be an analytic function in the upper half of the complex plane. This analyticity imposes constraints on its real and imaginary parts, leading to these dispersion relations.\n",
    "\n",
    "Another thought: how does this relate to causality? I think it's because if a material responds instantaneously to an external field (non-causal), then certain singularities would appear that aren't allowed by the Kramers-Kronig relations. So these relations ensure that the response is causal, meaning effects can't precede their causes.\n",
    "\n",
    "Let me also consider some examples. For instance, in a simple Drude model for metals, ε(ω) has a specific form involving plasma frequency and damping. Applying the Kramers-Kronig relation should allow us to compute one part from the other.\n",
    "\n",
    "Wait, I'm not sure about the exact derivation of this integral. Maybe I can sketch it out mentally. Starting with the fact that ε(ω) is related to the susceptibility χ(ω), which is analytic in the upper half-plane. Using Hilbert transforms, we can express Re χ in terms of Im χ and vice versa.\n",
    "\n",
    "The Hilbert transform involves integrating over all frequencies, weighted by 1/(w - ω), but since we're dealing with real functions, there's a symmetry that leads to the specific form given here with w² - ω² in the denominator.\n",
    "\n",
    "I should also remember that these relations apply to causal response functions, so they are fundamental in not just electromagnetism, but also in other areas like linear response theory in statistical mechanics.\n",
    "\n",
    "Another point: The factor of 2/π comes from normalization when performing the integral. I think it's related to the residue theorem or Fourier transforms involved in deriving these relations.\n",
    "\n",
    "I wonder about the applications. For example, if you have experimental data for the imaginary part of ε(ω), you can use this relation to compute the real part without needing an independent measurement. This is useful in spectroscopy and material characterization.\n",
    "\n",
    "But I also need to be cautious about approximations. In reality, materials might not perfectly satisfy these relations due to noise or finite bandwidths in measurements. However, they still provide a valuable framework for analyzing data.\n",
    "\n",
    "Let me think if there are any boundary conditions or specific cases where this relation simplifies. For example, at ω approaching zero, what does Re ε(ω) look like? The integral might involve low-frequency behavior of Im ε(w), which could relate to DC conductivity.\n",
    "\n",
    "Conversely, as ω becomes very large, the upper limit of the integral would dominate, and one might approximate the behavior for high frequencies. This could give insights into the material's response at different regions of the spectrum.\n",
    "\n",
    "Also, considering that this is a principal value integral, numerical integration can be tricky because you have to handle the singularity carefully. Maybe using symmetric limits around ω or applying specific quadrature rules designed for such integrals.\n",
    "\n",
    "I think I've covered most aspects. To summarize my thoughts:\n",
    "\n",
    "- Kramers-Kronig relations connect Re and Im parts of ε(ω).\n",
    "- They ensure causality and come from analyticity requirements.\n",
    "- The given equation is one such relation, involving a principal value integral over Im ε(w).\n",
    "- Applications include determining real part from imaginary part in experiments.\n",
    "- There are specific examples like the Drude model where these relations can be demonstrated.\n",
    "\n",
    "I might have missed some mathematical nuances, but this gives me a solid foundation to discuss Kramers-Kronig dispersion relations with someone else or to explore them further on my own.\n",
    ":::\n",
    "\n",
    "The Kramers-Kronig dispersion relations are fundamental in electromagnetic theory, connecting the real and imaginary parts of a material's dielectric constant, ε(ω). These relations ensure causality and arise from the analyticity of ε(ω) in the upper half of the complex plane. The provided equation:\n",
    "\n",
    "$$\n",
    "Re \\, \\varepsilon(\\omega) = 1 + \\frac{2}{\\pi} P \\int_{0}^{\\infty} \\frac{\\omega \\, Im \\, \\varepsilon(\\omega)}{\\omega^2 - \\omega'^2} d\\omega'\n",
    "$$\n",
    "\n",
    "demonstrates how the real part of ε(ω) is related to its imaginary part through a principal value integral. This relationship is crucial for understanding material responses to electromagnetic fields and has applications in spectroscopy and material characterization.\n",
    "\n",
    "**Key Points:**\n",
    "\n",
    "1. **Causality and Analyticity:** The relations ensure that physical systems respond causally, meaning effects cannot precede their causes. This stems from ε(ω) being an analytic function in the upper half-plane.\n",
    "\n",
    "2. **Integral Structure:** The principal value integral handles a singularity at ω' = ω, ensuring convergence. It uses the Hilbert transform framework, reflecting the Fourier relationship between real and imaginary parts.\n",
    "\n",
    "3. **Applications:** These relations allow determination of one part (real or imaginary) from the other, useful in experiments where direct measurement of both might be challenging.\n",
    "\n",
    "4. **Examples and Simplifications:** In specific models like the Drude model, these relations can illustrate material behavior at different frequency limits, providing insights into low- and high-frequency responses.\n",
    "\n",
    "In summary, the Kramers-Kronig relations are essential for analyzing electromagnetic responses in materials, ensuring physical consistency and enabling practical applications in various scientific domains."
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