Understanding Physical Laws in Quantum Systems

Quest - ION Everything — Think Quantum — State of Being — Here’s your previous explanation with all the asterisks (bold markers) removed: The Heisenberg Uncertainty Principle is one of the most famous (and most misunderstood) ideas in all of physics. It lies at the very heart of why the quantum world behaves so differently from the classical one we experience every day. What it actually says In its modern, precise form (Kennard 1927 / Weyl 1928), the principle states that for any quantum system, there is a fundamental lower limit to how precisely we can simultaneously know certain pairs of properties — called canonically conjugate variables. The most famous version is for position (x) and momentum (p): Δx ⋅ Δp ≥ ℏ/2 where: - Δx is the standard deviation (spread/uncertainty) in position - Δp is the standard deviation in momentum - ℏ = h/(2π) is the reduced Planck's constant ≈ 1.0545718 × 10⁻³⁴ J⋅s This means the product of the uncertainties can never be smaller than ℏ/2 — it's an intrinsic property of nature, not a limitation of our measuring instruments. There's also the very important energy-time version (a bit more subtle because time isn't an operator): ΔE ⋅ Δt ≥ ℏ/2 This shows up in things like the lifetime of unstable particles, linewidths in spectroscopy, and virtual particles in quantum field theory. Why it exists — the wave connection Particles aren't little bullets; they're described by wavefunctions (thanks to Schrödinger). A very sharp, localized position (small Δx) requires a wave packet made of many different wavelengths → many different momenta → large Δp A very precise momentum (small Δp) means a nearly pure wavelength → the wave is very spread out → large Δx It's the same mathematical trade-off you see in Fourier analysis: you can't have both a sharply peaked function in space and in frequency (momentum) at the same time. Common misconceptions (very important!) Many popular explanations get this wrong — even some textbooks from decades ago: 1. It's not just about disturbing the particle The old "measurement disturbs the system" picture (e.g., Heisenberg's gamma-ray microscope thought experiment) is an illustration, not the fundamental reason. Modern experiments (like weak measurements and Ozawa's work) show you can sometimes disturb less than the old picture suggested — but the intrinsic uncertainty in the state itself still obeys the limit.

I updated my Schrödinger equation visuals. This time I included the unbounded inner product Gaussian in the first 2 animations, and used the more familiar localized inner product on the last. To review: The Schrödinger equation is one of the cornerstones of quantum mechanics, describing how the quantum state of a physical system changes over time. Here's a detailed explanation without using any equations: ### **Core Idea:** The Schrödinger equation governs the behavior of quantum systems, much like Newton's laws govern classical mechanics. Instead of predicting exact positions and velocities of particles, it tells us how the *probability amplitude* (a complex-valued function related to the likelihood of finding a particle in a certain state) evolves over time. ### **Key Concepts:** 1. **Wavefunction (ψ):** - In quantum mechanics, particles don’t have definite positions or paths. Instead, their state is described by a *wavefunction*, which contains all the probabilistic information about the system. - The wavefunction doesn’t tell us where a particle *is* but rather where it *might be* and with what probability. 2. **Time Evolution:** - The Schrödinger equation explains how the wavefunction changes with time. It doesn’t determine a single outcome but describes a smooth, deterministic evolution of probabilities. - If you know the wavefunction at one moment, the equation tells you how it will look in the next instant. 3. **Energy and Hamiltonian:** - The equation depends on the *Hamiltonian*, which represents the total energy of the system (kinetic + potential energy). - Different potentials (e.g., an electron in an atom vs. a free particle) lead to different wavefunction behaviors. 4. **Superposition & Quantization:** - The equation naturally leads to *superposition*—where a quantum system can exist in multiple states at once until measured. - For bound systems (like electrons in atoms), it predicts *quantized* energy levels, explaining why electrons occupy discrete orbitals. 5. **Uncertainty & Probabilities:** - The wavefunction’s square magnitude gives the probability density of finding a particle in a certain state. - Unlike classical physics, quantum mechanics is inherently probabilistic, and the Schrödinger equation encodes this randomness. ### **Analogy (Rough but Helpful):** Imagine a ripple spreading on a pond. The shape and motion of the ripple depend on the water’s properties (like depth and obstacles). Similarly, the Schrödinger equation describes how the "quantum ripple" (the wavefunction) evolves based on the system’s energy landscape. ### **Interpretations:** - The equation itself doesn’t explain *why* the wavefunction behaves this way or what it "really" is—that’s the realm of quantum interpretations (e.g., Copenhagen, Many-Worlds). #quantum #quantumphysics #quantummechanics #physics #math #engineering #programming #Schrödinger #science

Electron is everywhere until observed/measured is the worst myth in quantum mechanics, spread by so-called science YouTubers who oversimplify everything for clicks. In Quantum Physics, observed means interaction. Interaction with another particle like a photon, electron, or anything that exchanges energy or information. Come back to our main topic. The electron is not everywhere until observed or measured. First we need to know, what actually is electron? Electron is nothing but localised excitation in electron field at some location of space time fabric. Electron is not a tiny ball. Now you might think, what is the electron field? Electron field is energy configuration at every location of space-time fabric (x, y, z, t). x, y, z are Spatial dimensions t is the Time dimension. There are other fields also: - Electromagnetic (EM) field - Higgs field - Many more Photon is excitation in the EM field. Mass arises due to excitation in the Higgs field. The electron field itself is everywhere, but the excitation, that ripple which represents one electron, is not spread across the entire space-time fabric. When we talk about where the electron might be, we don't talk about its location. We talk about its wave function. Wave function is not something physical. It is a mathematical function that tells about the probability amplitude of finding the electron at each position if you were to check. For example: You visit three stores: - Grocery store - Medical store - Electronics store You come back home and realize you left your wallet in one of those stores but don't know which one. You assign probability of finding the wallet to grocery, medical, and electronic stores. We all know probability formula: P = number of favorable outcomes / total outcomes So P = 1/3 (33.3%) You go to the electronics store and check the CCTV and find out this is not where you left your wallet. Now probability of finding the wallet at the electronics store becomes 0. And for the medical and grocery stores, it becomes 1/2 (50%) because the number of total outcomes decreased from 3 to 2. Now you go to the medical store and find your wallet there. That means probability of finding your wallet at the medical store becomes 1 (100%) and for the grocery store it becomes 0, because there is no way you can find your wallet in two places at once. That’s exactly how wavefunction collapse works. Before measurement, the electron’s position is uncertain, it’s described by probabilities. Once you measure it (meaning once it interacts with something), the probability at that point becomes 1, and everywhere else becomes 0. You didn’t summon the electron into existence; You just forced the field excitation to reveal its position through interaction. Electrons aren’t 'everywhere until observed', they’re localized excitations in a field. We just don’t know where until they interact. No consciousness. No magic. Just physics.

Everybody’s asking about the 𝗸𝗶𝗹𝗹𝗲𝗿 𝗮𝗽𝗽 𝗳𝗼𝗿 𝗾𝘂𝗮𝗻𝘁𝘂𝗺 𝗰𝗼𝗺𝗽𝘂𝘁𝗲𝗿𝘀. But when a team actually uses one to explore 𝗳𝘂𝗻𝗱𝗮𝗺𝗲𝗻𝘁𝗮𝗹 𝗽𝗵𝘆𝘀𝗶𝗰𝘀 in a way we couldn't before, the 𝘀𝗶𝗹𝗲𝗻𝗰𝗲 from the broader community is deafening. Really? I’ve talked about using quantum computers for exploring physics before. I get it - 𝗶𝘁'𝘀 𝗻𝗼𝘁 𝘁𝗵𝗲 𝗶𝗺𝗺𝗲𝗱𝗶𝗮𝘁𝗲, 𝗱𝗶𝘀𝗿𝘂𝗽𝘁𝗶𝘃𝗲 𝗮𝗽𝗽𝗹𝗶𝗰𝗮𝘁𝗶𝗼𝗻 𝘁𝗵𝗮𝘁 𝗩𝗖𝘀 𝗮𝗻𝗱 𝗺𝗮𝗿𝗸𝗲𝘁 𝗮𝗻𝗮𝗹𝘆𝘀𝘁𝘀 𝘄𝗮𝗻𝘁 𝘁𝗼 𝗵𝗲𝗮𝗿 𝗮𝗯𝗼𝘂𝘁. 𝗕𝘂𝘁 𝗜 𝗳𝗶𝗻𝗱 𝗶𝘁 𝗮𝗯𝘀𝗼𝗹𝘂𝘁𝗲𝗹𝘆 𝗮𝗺𝗮𝘇𝗶𝗻𝗴 𝘁𝗵𝗮𝘁 𝘄𝗲'𝗿𝗲 𝗳𝗶𝗻𝗮𝗹𝗹𝘆 𝗯𝘂𝗶𝗹𝗱𝗶𝗻𝗴 𝘁𝗼𝗼𝗹𝘀 𝘁𝗵𝗮𝘁 𝗮𝗹𝗹𝗼𝘄 𝘂𝘀 𝘁𝗼 𝘂𝗻𝗱𝗲𝗿𝘀𝘁𝗮𝗻𝗱 𝗼𝘂𝗿 𝘄𝗼𝗿𝗹𝗱 𝗼𝗻𝗲 𝗹𝗮𝘆𝗲𝗿 𝗱𝗲𝗲𝗽𝗲𝗿. A new paper from Google 𝗤𝘂𝗮𝗻𝘁𝘂𝗺 𝗔𝗜 & 𝗰𝗼𝗹𝗹𝗮𝗯𝗼𝗿𝗮𝘁𝗼𝗿𝘀, is a perfect case in point. The team tackled a monster of a problem in condensed matter physics: 𝗵𝗼𝘄 𝘁𝗼 𝘀𝗶𝗺𝘂𝗹𝗮𝘁𝗲 𝘀𝘆𝘀𝘁𝗲𝗺𝘀 𝘄𝗶𝘁𝗵 𝗱𝗶𝘀𝗼𝗿𝗱𝗲𝗿. Classically, this is a brute-force nightmare: You have to simulate thousands or even millions of different disorder configurations one by one, which can take an exponential amount of time. 𝗜𝗻𝘀𝘁𝗲𝗮𝗱 𝗼𝗳 𝘀𝗶𝗺𝘂𝗹𝗮𝘁𝗶𝗻𝗴 𝗼𝗻𝗲 𝗰𝗼𝗻𝗳𝗶𝗴𝘂𝗿𝗮𝘁𝗶𝗼𝗻 𝗮𝘁 𝗮 𝘁𝗶𝗺𝗲, 𝗚𝗼𝗼𝗴𝗹𝗲 𝘂𝘀𝗲𝗱 𝘁𝗵𝗲𝗶𝗿 𝟴𝟭-𝗾𝘂𝗯𝗶𝘁 𝗾𝘂𝗮𝗻𝘁𝘂𝗺 𝗽𝗿𝗼𝗰𝗲𝘀𝘀𝗼𝗿 𝘁𝗼 𝗽𝗿𝗲𝗽𝗮𝗿𝗲 𝗮 𝘀𝘁𝗮𝘁𝗲 𝘁𝗵𝗮𝘁 𝗶𝘀 𝗮 𝘀𝘂𝗽𝗲𝗿𝗽𝗼𝘀𝗶𝘁𝗶𝗼𝗻 𝗼𝗳 𝗮𝗹𝗹 𝗽𝗼𝘀𝘀𝗶𝗯𝗹𝗲 𝗱𝗶𝘀𝗼𝗿𝗱𝗲𝗿 𝗰𝗼𝗻𝗳𝗶𝗴𝘂𝗿𝗮𝘁𝗶𝗼𝗻𝘀. Then they gave it a tiny kick of energy in one spot, and watched what happened. The result? The energy stayed put. It refused to spread. This is a phenomenon called 𝗗𝗶𝘀𝗼𝗿𝗱𝗲𝗿-𝗙𝗿𝗲𝗲 𝗟𝗼𝗰𝗮𝗹𝗶𝘇𝗮𝘁𝗶𝗼𝗻 (𝗗𝗙𝗟). Even though the system's evolution and the initial state were perfectly uniform and disorder-free, the underlying superposition over different "backgrounds" caused the system to localize. 𝗜𝘁’𝘀 𝗮 𝘀𝘁𝘂𝗻𝗻𝗶𝗻𝗴 𝗱𝗲𝗺𝗼𝗻𝘀𝘁𝗿𝗮𝘁𝗶𝗼𝗻 𝗼𝗳 𝗾𝘂𝗮𝗻𝘁𝘂𝗺 𝗺𝗲𝗰𝗵𝗮𝗻𝗶𝗰𝘀 𝗮𝘁 𝘄𝗼𝗿𝗸 𝗼𝗻 𝗮 𝘀𝗰𝗮𝗹𝗲 𝘁𝗵𝗮𝘁’𝘀 𝗶𝗻𝗰𝗿𝗲𝗱𝗶𝗯𝗹𝘆 𝗱𝗶𝗳𝗳𝗶𝗰𝘂𝗹𝘁 𝗳𝗼𝗿 𝗰𝗹𝗮𝘀𝘀𝗶𝗰𝗮𝗹 𝗰𝗼𝗺𝗽𝘂𝘁𝗲𝗿𝘀 𝘁𝗼 𝗵𝗮𝗻𝗱𝗹𝗲, 𝗲𝘀𝗽𝗲𝗰𝗶𝗮𝗹𝗹𝘆 𝗶𝗻 𝟮𝗗. But this isn't just a cool physics experiment. This work carves out a concrete path to quantum advantage. The team proposed an 𝗮𝗹𝗴𝗼𝗿𝗶𝘁𝗵𝗺 based on this technique that offers a 𝗽𝗼𝗹𝘆𝗻𝗼𝗺𝗶𝗮𝗹 𝘀𝗽𝗲𝗲𝗱𝘂𝗽 𝗳𝗼𝗿 𝘀𝗮𝗺𝗽𝗹𝗶𝗻𝗴 𝗱𝗶𝘀𝗼𝗿𝗱𝗲𝗿𝗲𝗱 𝘀𝘆𝘀𝘁𝗲𝗺𝘀. So yes, let's keep working toward fault-tolerant machines that can break RSA and optimize your portfolio. But let's not ignore the incredible science happening right now. 📸 Credits: Google 𝗤𝘂𝗮𝗻𝘁𝘂𝗺 𝗔𝗜 & 𝗖𝗼𝗹𝗹𝗮𝗯𝗼𝗿𝗮𝘁𝗼𝗿𝘀 (arXiv:2410.06557) Pedram Roushan

A research team at TU Wien has uncovered something astonishing: quantum entanglement the mysterious bond connecting particles across space doesn’t form instantly. Instead, it takes about 232 attoseconds (a quintillionth of a second) to fully emerge. Using advanced computer simulations of atoms hit by laser pulses, scientists observed that entanglement develops gradually as one electron escapes and another shifts energy levels, slowly weaving their quantum link through time. This finding challenges decades of assumptions that entanglement happens outside of time itself. It reveals that even the universe’s fastest phenomena have measurable stages a kind of “quantum heartbeat.” Researchers now aim to confirm the results experimentally, a daunting task at speeds where light barely crosses a human hair’s width. Cracking these fleeting moments could reshape quantum computing, encryption, and communication, showing that even instant mysteries unfold with rhythm and order. Sources: NASA, Scientific American, Physical Review Letters

In the well-known double-slit experiment, electrons exhibit wave-like behavior when not being measured, producing an interference pattern on the detection screen. But when we attempt to determine which slit an electron goes through, that pattern disappears, and the electrons behave like particles. This shift is not due to electrons “knowing” they’re being watched. Instead, it’s a fundamental consequence of quantum measurement. According to quantum mechanics—specifically the Copenhagen interpretation and the uncertainty principle—observing a quantum particle requires interaction. To detect an electron’s path, we use photons, which carry energy. Since electrons are extremely small, even a single photon can significantly disturb their motion or momentum, effectively collapsing their wave function into a definite state. This collapse destroys the superposition—the state where an electron exists in multiple possible paths—and eliminates the interference pattern. The act of measurement turns a probability wave into a single, classical outcome. This isn't mysticism or magic. It's a well-documented quantum phenomenon with decades of experimental support. Measurement affects quantum systems—not because of observation in the human sense, but because of unavoidable physical interaction. It's not magic. It's quantum physics.

The principle of least action takes on a deeper meaning in quantum mechanics. In classical mechanics, it gives a single trajectory, the path that extremizes the action. In quantum mechanics, there is no single path. Instead, a system explores all possible trajectories, each contributing a probability amplitude. Each path carries a phase determined by its action. Near the optimal trajectory, these phases vary slowly and interfere constructively, leading to a strong contribution. Far from it, phases fluctuate rapidly and cancel out through destructive interference. The classical path emerges not because other paths vanish, but because it dominates the interference pattern. The observable probability is obtained from the squared magnitude of the total amplitude.

QUANTUM SYSTEM AT THE EDGE OF CHAOS: A PATH TOWARD STABLE QUANTUM COMPUTATION Quantum physics rarely offers moments where theory, engineering, and the raw behavior of many‑body systems collide to reveal a new dynamical regime. Yet that is exactly what the 78‑qubit Chuang‑tzu 2.0 processor has uncovered: a quantum system pushed to the brink of chaos can be held in a long‑lived, tunable prethermal state—an island of order suspended inside non‑equilibrium turbulence. This discovery goes far beyond Floquet physics. Periodic driving has already given us time crystals and engineered topological phases, but non‑periodic driving—especially with structured randomness—has long been synonymous with rapid heating and the loss of quantum information. Instead, this experiment shows that temporal randomness can be engineered to suppress heating, stabilize dynamics, and preserve coherence far longer than expected. Random multipolar driving, neither periodic nor chaotic, acts as a hidden temporal scaffold that shapes how energy flows through the system. Applied to a two‑dimensional Bose–Hubbard model across 78 qubits and 137 couplers, this protocol prevents the system from collapsing into chaos. Instead, it enters a robust prethermal plateau where imbalance decays slowly, entanglement grows in a controlled way, and the heating rate becomes tunable—matching universal algebraic scaling predicted for multipolar drives. This is not a subtle correction; it is a macroscopic reshaping of the system’s dynamical landscape. The geometry of entanglement is equally striking. Different subsystems show distinct behaviors—some oscillate coherently, others settle into plateaus—revealing a highly non‑uniform spread of correlations across the lattice. It is the first time such fine‑grained entanglement dynamics have been observed in a large, non‑periodically driven quantum simulator. Classical tensor‑network methods like GMPS and PEPS cannot keep pace once heating accelerates, confirming that these dynamics lie firmly beyond classical reach. Quantum systems at the brink of chaos are not doomed to disorder. With the right temporal geometry, they can be shaped, stabilized, and made computationally powerful. This work demonstrates that the boundary between coherence and chaos is not a hard limit but a navigable frontier—and that the future of quantum computation may lie precisely in mastering this edge. # https://lnkd.in/eJBkGts5

In quantum physics, the concept of a wave is crucial to understanding the behavior of particles at the atomic and subatomic levels. This wave behavior is encapsulated in the wave-particle duality, which states that every particle or quantum entity exhibits both wave-like and particle-like properties. Key Concepts: 1. Wave Function (Ψ): The wave function is a mathematical function that describes the quantum state of a particle or system of particles. It contains all the information about a system and allows for the calculation of probabilities of finding a particle in a certain state or position. The square of the wave function's absolute value () gives the probability density of the particle's position. 2. Schrödinger Equation: The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function evolves over time. It can be written in time-dependent and time-independent forms and is used to solve for the wave function of a quantum system. 3. Superposition: Quantum particles can exist in a superposition of states, meaning they can be in multiple states at once until measured. This principle is illustrated in phenomena such as the double-slit experiment, where particles display interference patterns characteristic of waves when not observed. 4. Quantum Entanglement: Entangled particles remain correlated regardless of the distance separating them. Measurement of one particle instantly affects the state of the other. This phenomenon showcases the non-local behavior of quantum systems and challenges classical intuitions about locality and separation. 5. Uncertainty Principle: Formulated by Werner Heisenberg, this principle states that certain pairs of physical properties (like position and momentum) cannot be simultaneously known with arbitrary precision. This intrinsic uncertainty is a fundamental aspect of quantum mechanics and arises from the wave-like nature of particles.

I’m happy to share an extensive notebook I’ve written on the physics of single qubits. It starts from the very basics and, step by step, moves toward more advanced concepts, a trajectory from complex numbers all the way to tensors, SU(2) and POVMs: https://wolfr.am/QIS-Book I’ve tried to explain things with a computation-first narrative: if I don’t compute, I don’t learn. Many concepts and ideas are introduced and then followed by plenty of examples and computations, before we get to the formal (and often boring) definitions. Of course, I believe Wolfram Mathematica is a great tool for this purpose. In writing this notebook (which is as long as a short book), I had myself in mind, and my students in the quantum courses I’ve taught over the years, both inside and outside academia. I actually had to W-drop the very first quantum course I took about 21 years ago, taught by a famous, top-notch quantum researcher in Iran, simply because I was completely confused and the professor didn’t have the patience to explain and answer my questions. Ironically, I had to resign one of my US postdocs too, because my PI (a famous quantum researcher) was too harsh and impatient with me trying to understand the foundations of the quantum systems we were studying and wanted me only to simulate. These experiences have shaped how I teach and how I wrote this notebook: I’ve tried to collect and present all the essential ideas that I believe someone should know when learning about qubits and doing computation. Of course, there are still missing concepts (e.g., stochastic dynamical equations, Schrödinger and master equations) that I plan to add over time. Take a look, and let me know what you think, I’d really appreciate your feedback. #quantum #wolfram #mathematica #teaching #learing