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Today we introduce Stochastic Differential Equations (SDEs), and the main thing to watch for is this: We’ll use Brownian motion as the basic noise source, then see how well-known SDEs drop out of it naturally, without guessing. I still think the best way into these concepts is through an...

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Mathelirium

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The Probability Density of a Brownian Particle Solves the. Heat Equation. Einstein’s 1905 paper on Brownian motion doesn’t get the same spotlight as General Relativity, but it’s still an astonishing piece of work. It takes a messy, noisy phenomenon and extracts three hard facts from it: First, the randomness of the particles is physical. The motion is molecular impacts made visible. Second, one path looks like chaos, but the cloud has a rigid scaling law...mean-square displacement grows like t, so the width grows like √t. Third, the cloud evolves deterministically. Its density solves the heat equation PDE! So Brownian probability spreads the same way heat spreads!
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The Probability Density of a Brownian Particle Solves the. Heat Equation. Einstein’s 1905 paper on Brownian motion doesn’t get the same spotlight as General Relativity, but it’s still an astonishing piece of work. It takes a messy, noisy phenomenon and extracts three hard facts from it: First, the randomness of the particles is physical. The motion is molecular impacts made visible. Second, one path looks like chaos, but the cloud has a rigid scaling law...mean-square displacement grows like t, so the width grows like √t. Third, the cloud evolves deterministically. Its density solves the heat equation PDE! So Brownian probability spreads the same way heat spreads!

Mathelirium

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Once you’ve accepted that Brownian motion doesn’t have a classical dX/dt but does have this rigid quadratic variation, you’re ready for the first real Itô vs chain rule punch: take X_t = B_t and look at f(x) = x². In ordinary calculus you’d write d(B_t²) = 2 B_t dB_t and move on, because the only thing that survives is f′(x)·dx. Itô’s formula says that’s not enough for Brownian motion: the infinitely many tiny (dB_t)² terms don’t wash out, they add up to dt, so the correct identity is d(B_t²) = 2 B_t dB_t + dt (see insert formula). In the animation we fix one Brownian path, draw B_t² as the pale truth, then build the naive path that accumulates only the 2 B_t dB_t term and watch it drift away from B_t², while the corrected path that accumulates 2 B_t dB_t plus an extra linear drift t hugs B_t² almost perfectly. The whole point is visible on screen: the missing +t isn’t a technicality, it’s the deterministic shadow of quadratic variation correcting the chain rule so calculus survives contact with noise. #ItosFormula #BrownianMotion #StochasticCalculus #QuadraticVariation #RandomPaths #ProbabilityTheory
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Once you’ve accepted that Brownian motion doesn’t have a classical dX/dt but does have this rigid quadratic variation, you’re ready for the first real Itô vs chain rule punch: take X_t = B_t and look at f(x) = x². In ordinary calculus you’d write d(B_t²) = 2 B_t dB_t and move on, because the only thing that survives is f′(x)·dx. Itô’s formula says that’s not enough for Brownian motion: the infinitely many tiny (dB_t)² terms don’t wash out, they add up to dt, so the correct identity is d(B_t²) = 2 B_t dB_t + dt (see insert formula). In the animation we fix one Brownian path, draw B_t² as the pale truth, then build the naive path that accumulates only the 2 B_t dB_t term and watch it drift away from B_t², while the corrected path that accumulates 2 B_t dB_t plus an extra linear drift t hugs B_t² almost perfectly. The whole point is visible on screen: the missing +t isn’t a technicality, it’s the deterministic shadow of quadratic variation correcting the chain rule so calculus survives contact with noise. #ItosFormula #BrownianMotion #StochasticCalculus #QuadraticVariation #RandomPaths #ProbabilityTheory

Mathelirium

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The Curve That Forced Mathematicians To Invent A New Calculus There is a strange curve called Brownian path. It traces a continuous path without ever jumping, but it never settles into a well-defined direction at any point, so it’s continuous everywhere but differentiable nowhere. We’re used to the idea that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation
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The Curve That Forced Mathematicians To Invent A New Calculus There is a strange curve called Brownian path. It traces a continuous path without ever jumping, but it never settles into a well-defined direction at any point, so it’s continuous everywhere but differentiable nowhere. We’re used to the idea that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation

Mathelirium

31,688 Aufrufe • vor 6 Monaten

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.
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Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Mathelirium

54,730 Aufrufe • vor 4 Monaten

Why Does Quantum Mechanics Use a Complex Wavefunction? Schrödinger’s equation doesn’t start from mystery. It starts from a very specific bet. The state of a particle is a complex field ψ(x,t), and whatever time-evolution rule we choose has to move ψ forward while preserving total probability. So the basic question is simple. What equation should ψ satisfy so that |ψ|² behaves like a conserved density, the way mass density does in fluid flow? What is ψ? Think of ψ(x,t) as an amplitude attached to the statement the particle is at position x at time t. It’s not a probability. It’s the thing you add first, and only at the end do you square it: p(x,t) = |ψ(x,t)|² Because ψ is complex, it has magnitude and phase. Write it as ψ(x,t) = r(x,t) exp(i θ(x,t)) Then r² = |ψ|² is the density, and the phase θ ends up controlling the flow through the probability current. Where does Schrödinger’s equation come from? Start with two empirical inputs that tie waves to particles: E = ħ ω p = ħ k Here ħ is Planck’s constant divided by 2π. It’s the conversion factor between frequency and energy, and between wavenumber and momentum. A plane wave with angular frequency ω and wavevector k is ψ(x,t) = A exp(i(k·x − ωt)) Now watch what derivatives do to this wave: ∂ψ/∂t = −i ω ψ ∇ψ = i k ψ ∇²ψ = −|k|² ψ Multiply by ħ and you get: i ħ ∂ψ/∂t = ħ ω ψ = E ψ −i ħ ∇ψ = ħ k ψ = p ψ −ħ² ∇²ψ = ħ² |k|² ψ = p² ψ So for plane waves, the operators Ê = i ħ ∂/∂t p̂ = −i ħ ∇ act like energy and momentum. Now bring in the classical, nonrelativistic energy bookkeeping: E = p²/(2m) + V(x) Kinetic plus potential. That’s it. Turn it into an equation for ψ by replacing E and p with the operators above: Ê ψ = (p̂²/(2m) + V) ψ Since p̂² = (−i ħ ∇)·(−i ħ ∇) = −ħ² ∇², this becomes i ħ ∂ψ/∂t = ( −ħ²/(2m) ∇² + V(x) ) ψ That’s the time-dependent Schrödinger equation. This derivation is a controlled heuristic. Match the plane-wave identities to the measured relations E = ħω and p = ħk, then impose the same energy bookkeeping you trust in classical mechanics. Why this is the right kind of rule If ψ is the state, we need a rule that preserves total probability: ∫ |ψ(x,t)|² dx = 1 Schrödinger evolution does, and you can see it by deriving a continuity equation. Let ρ(x,t) = |ψ|² = ψ*ψ. Differentiate: ∂ρ/∂t = ψ* ∂ψ/∂t + ψ ∂ψ*/∂t Use Schrödinger and its complex conjugate. The potential terms cancel, and what’s left can be rearranged into ∂ρ/∂t + ∇·j = 0 with probability current j = (ħ/(2mi)) ( ψ* ∇ψ − ψ ∇ψ* ) That’s the cleanest way to say what ψ is. |ψ|² behaves like a conserved density, the phase drives a current, and the time evolution is fixed, up to V, by combining wave relations with energy bookkeeping: i ħ ∂ψ/∂t = ( −ħ²/(2m) ∇² + V ) ψ #QuantumMechanics #SchrodingerEquation #WaveFunction #BornRule #Physics #MathematicalPhysics
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Why Does Quantum Mechanics Use a Complex Wavefunction? Schrödinger’s equation doesn’t start from mystery. It starts from a very specific bet. The state of a particle is a complex field ψ(x,t), and whatever time-evolution rule we choose has to move ψ forward while preserving total probability. So the basic question is simple. What equation should ψ satisfy so that |ψ|² behaves like a conserved density, the way mass density does in fluid flow? What is ψ? Think of ψ(x,t) as an amplitude attached to the statement the particle is at position x at time t. It’s not a probability. It’s the thing you add first, and only at the end do you square it: p(x,t) = |ψ(x,t)|² Because ψ is complex, it has magnitude and phase. Write it as ψ(x,t) = r(x,t) exp(i θ(x,t)) Then r² = |ψ|² is the density, and the phase θ ends up controlling the flow through the probability current. Where does Schrödinger’s equation come from? Start with two empirical inputs that tie waves to particles: E = ħ ω p = ħ k Here ħ is Planck’s constant divided by 2π. It’s the conversion factor between frequency and energy, and between wavenumber and momentum. A plane wave with angular frequency ω and wavevector k is ψ(x,t) = A exp(i(k·x − ωt)) Now watch what derivatives do to this wave: ∂ψ/∂t = −i ω ψ ∇ψ = i k ψ ∇²ψ = −|k|² ψ Multiply by ħ and you get: i ħ ∂ψ/∂t = ħ ω ψ = E ψ −i ħ ∇ψ = ħ k ψ = p ψ −ħ² ∇²ψ = ħ² |k|² ψ = p² ψ So for plane waves, the operators Ê = i ħ ∂/∂t p̂ = −i ħ ∇ act like energy and momentum. Now bring in the classical, nonrelativistic energy bookkeeping: E = p²/(2m) + V(x) Kinetic plus potential. That’s it. Turn it into an equation for ψ by replacing E and p with the operators above: Ê ψ = (p̂²/(2m) + V) ψ Since p̂² = (−i ħ ∇)·(−i ħ ∇) = −ħ² ∇², this becomes i ħ ∂ψ/∂t = ( −ħ²/(2m) ∇² + V(x) ) ψ That’s the time-dependent Schrödinger equation. This derivation is a controlled heuristic. Match the plane-wave identities to the measured relations E = ħω and p = ħk, then impose the same energy bookkeeping you trust in classical mechanics. Why this is the right kind of rule If ψ is the state, we need a rule that preserves total probability: ∫ |ψ(x,t)|² dx = 1 Schrödinger evolution does, and you can see it by deriving a continuity equation. Let ρ(x,t) = |ψ|² = ψ*ψ. Differentiate: ∂ρ/∂t = ψ* ∂ψ/∂t + ψ ∂ψ*/∂t Use Schrödinger and its complex conjugate. The potential terms cancel, and what’s left can be rearranged into ∂ρ/∂t + ∇·j = 0 with probability current j = (ħ/(2mi)) ( ψ* ∇ψ − ψ ∇ψ* ) That’s the cleanest way to say what ψ is. |ψ|² behaves like a conserved density, the phase drives a current, and the time evolution is fixed, up to V, by combining wave relations with energy bookkeeping: i ħ ∂ψ/∂t = ( −ħ²/(2m) ∇² + V ) ψ #QuantumMechanics #SchrodingerEquation #WaveFunction #BornRule #Physics #MathematicalPhysics

Mathelirium

20,781 Aufrufe • vor 6 Monaten

Lecture 2 on our Quantum Mechanics Series Schrödinger’s equation doesn’t start from mystery. It starts from a very specific bet…the state of a particle is a complex field ψ(x,t), and whatever dynamics we write down must move ψ forward in time in a way that preserves total probability. We ask a basic question…what equation should ψ satisfy so that |ψ|² behaves like a conserved density, the way mass density does in fluid flow? What is ψ? Think of ψ(x,t) as the amplitude assigned to “the particle is at position x at time t”. It’s not a probability. It’s the object you add first, and only at the end do you square p(x,t) = |ψ(x,t)|² Because ψ is complex, it has magnitude and phase. Write it in polar form ψ(x,t) = r(x,t) exp(i θ(x,t)) Then r² = |ψ|² is the density, and θ will end up controlling flow (the probability current). Where does Schrödinger’s equation come from? Start with two empirical inputs about waves and particles: E = ħ ω p = ħ k Here ħ (“h-bar”) is Planck’s constant divided by 2π. It’s the unit conversion factor between the wave description (frequency ω, wavevector k) and the particle description (energy E, momentum p). In units, ħ has units of joule-seconds, so multiplying ω (1/seconds) gives energy (joules), and multiplying k (1/meters) gives momentum (kg·m/s). It’s the number that tells you how much energy or momentum you get per unit frequency or wavenumber. A plane wave with angular frequency ω and wavevector k is ψ(x,t) = A exp(i(k·x − ω t)) Now notice what derivatives do to this wave: ∂ψ/∂t = −i ω ψ ∇ψ = i k ψ ∇²ψ = −|k|² ψ Multiply those identities by ħ: i ħ ∂ψ/∂t = ħ ω ψ = E ψ −i ħ ∇ψ = ħ k ψ = p ψ −ħ² ∇²ψ = ħ² |k|² ψ = p² ψ So for plane waves, the operators Ê = i ħ ∂/∂t p̂ = −i ħ ∇ act like energy and momentum! Now use the classical nonrelativistic energy relation: E = p²/(2m) + V(x) This is bookkeeping for a particle moving slow enough that relativity can be ignored. The term p²/(2m) is kinetic energy. If p = mv, then p²/(2m) = (m²v²)/(2m) = (1/2)mv². The term V(x) is potential energy. It depends on position because forces come from spatially varying energy. A slope in V pushes the particle. Examples: for a charged particle in an electric potential φ(x), V(x) = q φ(x). Near Earth, V(z) = mgz. The point is total energy equals kinetic plus potential. Turn that into an equation for ψ by replacing E and p with the operators above: Ê ψ = (p̂²/(2m) + V) ψ Compute p̂² = (−i ħ ∇)·(−i ħ ∇) = −ħ² ∇², so we get i ħ ∂ψ/∂t = ( −ħ²/(2m) ∇² + V(x) ) ψ That is the time-dependent Schrödinger equation. The derivation here is a controlled heuristic: we matched the plane-wave identities to the measured relations E = ħω and p = ħk, then imposed the same energy bookkeeping as classical mechanics. Why this equation is the right kind of rule If ψ is the state, we need a rule that preserves total probability: ∫ |ψ(x,t)|² dx = 1 Schrödinger evolution does. You can see it by deriving a continuity equation. Let ρ(x,t) = |ψ|² = ψ* ψ. Take a time derivative: ∂ρ/∂t = ψ* ∂ψ/∂t + ψ ∂ψ*/∂t Use Schrödinger and its complex conjugate: ∂ψ/∂t = (1/(i ħ)) ( −ħ²/(2m) ∇²ψ + Vψ ) ∂ψ*/∂t = (−1/(i ħ)) ( −ħ²/(2m) ∇²ψ* + Vψ* ) Plug in. The V terms cancel exactly, and what remains can be rearranged into a divergence: ∂ρ/∂t + ∇·j = 0 where the probability current is j = (ħ/(2mi)) ( ψ* ∇ψ − ψ ∇ψ* ) This is the best way to explain ehat ψ is: |ψ|² behaves like a conserved density, and the phase of ψ is what drives the current j. So in this series, ψ isn’t a slogan. It’s the object whose modulus squared is the density, whose phase generates flow, and whose time evolution is fixed (up to V) by matching wave relations to energy bookkeeping: i ħ ∂ψ/∂t = ( -ħ²/(2m) ∇² + V ) ψ #QuantumMechanics #SchrodingerEquation #WaveFunction #BornRule #Physics #MathematicalPhysics
0:40

Lecture 2 on our Quantum Mechanics Series Schrödinger’s equation doesn’t start from mystery. It starts from a very specific bet…the state of a particle is a complex field ψ(x,t), and whatever dynamics we write down must move ψ forward in time in a way that preserves total probability. We ask a basic question…what equation should ψ satisfy so that |ψ|² behaves like a conserved density, the way mass density does in fluid flow? What is ψ? Think of ψ(x,t) as the amplitude assigned to “the particle is at position x at time t”. It’s not a probability. It’s the object you add first, and only at the end do you square p(x,t) = |ψ(x,t)|² Because ψ is complex, it has magnitude and phase. Write it in polar form ψ(x,t) = r(x,t) exp(i θ(x,t)) Then r² = |ψ|² is the density, and θ will end up controlling flow (the probability current). Where does Schrödinger’s equation come from? Start with two empirical inputs about waves and particles: E = ħ ω p = ħ k Here ħ (“h-bar”) is Planck’s constant divided by 2π. It’s the unit conversion factor between the wave description (frequency ω, wavevector k) and the particle description (energy E, momentum p). In units, ħ has units of joule-seconds, so multiplying ω (1/seconds) gives energy (joules), and multiplying k (1/meters) gives momentum (kg·m/s). It’s the number that tells you how much energy or momentum you get per unit frequency or wavenumber. A plane wave with angular frequency ω and wavevector k is ψ(x,t) = A exp(i(k·x − ω t)) Now notice what derivatives do to this wave: ∂ψ/∂t = −i ω ψ ∇ψ = i k ψ ∇²ψ = −|k|² ψ Multiply those identities by ħ: i ħ ∂ψ/∂t = ħ ω ψ = E ψ −i ħ ∇ψ = ħ k ψ = p ψ −ħ² ∇²ψ = ħ² |k|² ψ = p² ψ So for plane waves, the operators Ê = i ħ ∂/∂t p̂ = −i ħ ∇ act like energy and momentum! Now use the classical nonrelativistic energy relation: E = p²/(2m) + V(x) This is bookkeeping for a particle moving slow enough that relativity can be ignored. The term p²/(2m) is kinetic energy. If p = mv, then p²/(2m) = (m²v²)/(2m) = (1/2)mv². The term V(x) is potential energy. It depends on position because forces come from spatially varying energy. A slope in V pushes the particle. Examples: for a charged particle in an electric potential φ(x), V(x) = q φ(x). Near Earth, V(z) = mgz. The point is total energy equals kinetic plus potential. Turn that into an equation for ψ by replacing E and p with the operators above: Ê ψ = (p̂²/(2m) + V) ψ Compute p̂² = (−i ħ ∇)·(−i ħ ∇) = −ħ² ∇², so we get i ħ ∂ψ/∂t = ( −ħ²/(2m) ∇² + V(x) ) ψ That is the time-dependent Schrödinger equation. The derivation here is a controlled heuristic: we matched the plane-wave identities to the measured relations E = ħω and p = ħk, then imposed the same energy bookkeeping as classical mechanics. Why this equation is the right kind of rule If ψ is the state, we need a rule that preserves total probability: ∫ |ψ(x,t)|² dx = 1 Schrödinger evolution does. You can see it by deriving a continuity equation. Let ρ(x,t) = |ψ|² = ψ* ψ. Take a time derivative: ∂ρ/∂t = ψ* ∂ψ/∂t + ψ ∂ψ*/∂t Use Schrödinger and its complex conjugate: ∂ψ/∂t = (1/(i ħ)) ( −ħ²/(2m) ∇²ψ + Vψ ) ∂ψ*/∂t = (−1/(i ħ)) ( −ħ²/(2m) ∇²ψ* + Vψ* ) Plug in. The V terms cancel exactly, and what remains can be rearranged into a divergence: ∂ρ/∂t + ∇·j = 0 where the probability current is j = (ħ/(2mi)) ( ψ* ∇ψ − ψ ∇ψ* ) This is the best way to explain ehat ψ is: |ψ|² behaves like a conserved density, and the phase of ψ is what drives the current j. So in this series, ψ isn’t a slogan. It’s the object whose modulus squared is the density, whose phase generates flow, and whose time evolution is fixed (up to V) by matching wave relations to energy bookkeeping: i ħ ∂ψ/∂t = ( -ħ²/(2m) ∇² + V ) ψ #QuantumMechanics #SchrodingerEquation #WaveFunction #BornRule #Physics #MathematicalPhysics

Mathelirium

40,835 Aufrufe • vor 7 Monaten

There’s a random curve that is perfectly continuous in time, never jumps, models smooth noise in physics and finance… and yet with probability 1 it has no derivative at any point. If you zoom in anywhere, it never straightens out. Brownian motion started as a physics headache long before it became a clean mathematical object. In 1827, Robert Brown watched pollen grains jitter under a microscope and had no microscopic theory to blame; almost a century later, Einstein and Smoluchowski explained that jitter as the visible effect of countless molecular collisions, and then Wiener turned that physical picture into a precise random function of time that now underpins much of modern probability, physics, and quantitative finance. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #MathematicalPhysics #QuantFinance
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There’s a random curve that is perfectly continuous in time, never jumps, models smooth noise in physics and finance… and yet with probability 1 it has no derivative at any point. If you zoom in anywhere, it never straightens out. Brownian motion started as a physics headache long before it became a clean mathematical object. In 1827, Robert Brown watched pollen grains jitter under a microscope and had no microscopic theory to blame; almost a century later, Einstein and Smoluchowski explained that jitter as the visible effect of countless molecular collisions, and then Wiener turned that physical picture into a precise random function of time that now underpins much of modern probability, physics, and quantitative finance. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #MathematicalPhysics #QuantFinance

Mathelirium

33,072 Aufrufe • vor 7 Monaten

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics
0:46

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,892 Aufrufe • vor 7 Monaten

In 1996, James Sethian showed a remarkably clean idea. You can get shortest routes through a messy world by letting a wave expand once. No trial paths. No scanning beams. Just one growing front. You compute an arrival time field T(x,y). At each point, T means how long the front needs to reach that point. The rule is ‖∇T‖ = 1/F. If the medium is fast, meaning F is large, the front moves quickly and T grows slowly. If the medium is slow, meaning F is small, the front moves slowly and T grows quickly. Obstacles act like F is near zero, so the front cannot pass through and instead wraps around. Then you get the path without searching. Once T exists, you pick any start point and follow xdot proportional to minus grad T. You slide downhill on the time landscape and trace a fastest route back to the source. This is the Hamilton Jacobi and optimal control view that Tsitsiklis (1995) made precise. Compute the value or arrival time function first, and the optimal trajectories follow from it. #FastMarching #EikonalEquation #HamiltonJacobi #OptimalControl #ShortestPath #ComputationalGeometry
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In 1996, James Sethian showed a remarkably clean idea. You can get shortest routes through a messy world by letting a wave expand once. No trial paths. No scanning beams. Just one growing front. You compute an arrival time field T(x,y). At each point, T means how long the front needs to reach that point. The rule is ‖∇T‖ = 1/F. If the medium is fast, meaning F is large, the front moves quickly and T grows slowly. If the medium is slow, meaning F is small, the front moves slowly and T grows quickly. Obstacles act like F is near zero, so the front cannot pass through and instead wraps around. Then you get the path without searching. Once T exists, you pick any start point and follow xdot proportional to minus grad T. You slide downhill on the time landscape and trace a fastest route back to the source. This is the Hamilton Jacobi and optimal control view that Tsitsiklis (1995) made precise. Compute the value or arrival time function first, and the optimal trajectories follow from it. #FastMarching #EikonalEquation #HamiltonJacobi #OptimalControl #ShortestPath #ComputationalGeometry

Mathelirium

203,228 Aufrufe • vor 6 Monaten

In 1996, James Sethian showed something almost unfair...you can find shortest routes through a messy world by letting a wave expand once...no trial paths, no search beams, just one growing front. Here’s how: we solve for an arrival-time field T(x,y) so that T literally means how long the wave needs to reach this point. The rule is ||∇ T|| = 1/F, where the medium is fast (F large) the front sprints, where it’s slow it trudges, and obstacles are speed ≈ 0, so the front wraps around them because that’s the only way forward. Then comes the satisfying part: once T exists, a path doesn’t need to search at all...drop a bead anywhere and let it follow ẋ ∝ -∇ T, it slides downhill on the time landscape and traces a globally fastest route back to the source. This “wave = optimal control” viewpoint is exactly what Tsitsiklis (1995) made precise from the Hamilton-Jacobi side...compute the value/arrival-time function and the optimal trajectories fall out from it. #FastMarching #EikonalEquation
0:31

In 1996, James Sethian showed something almost unfair...you can find shortest routes through a messy world by letting a wave expand once...no trial paths, no search beams, just one growing front. Here’s how: we solve for an arrival-time field T(x,y) so that T literally means how long the wave needs to reach this point. The rule is ||∇ T|| = 1/F, where the medium is fast (F large) the front sprints, where it’s slow it trudges, and obstacles are speed ≈ 0, so the front wraps around them because that’s the only way forward. Then comes the satisfying part: once T exists, a path doesn’t need to search at all...drop a bead anywhere and let it follow ẋ ∝ -∇ T, it slides downhill on the time landscape and traces a globally fastest route back to the source. This “wave = optimal control” viewpoint is exactly what Tsitsiklis (1995) made precise from the Hamilton-Jacobi side...compute the value/arrival-time function and the optimal trajectories fall out from it. #FastMarching #EikonalEquation

Mathelirium

766,230 Aufrufe • vor 8 Monaten

In this incredible demonstration from the Maritime Technical and Safety Institute in Japan, a large wave tank uses many synchronized wave-generating devices to create visible symbols and patterns on the surface of the water. Each paddle produces a small wave, but when many waves are emitted at specific timings, amplitudes, and phases, they begin to overlap. Where the wave crests reinforce one another, the water rises. Where a crest meets a trough, the motion cancels out. This is called interference, and it is one of the most fundamental principles of wave physics. By controlling the phase relationship between many individual wave sources, researchers can shape the surface of the water into temporary patterns, symbols, and directional flows. In other words, the visible image is not “drawn” onto the water. It emerges from the mathematics of waves interacting with one another. This is the same underlying principle behind acoustics, cymatics, ocean modeling, signal processing, holography, and even the way complex field patterns arise throughout nature. A single wave carries motion, but many waves, when organized in relationship, create structure. This is why water is such a powerful medium for understanding resonance. It makes invisible dynamics visible and shows us that form can emerge from rhythm, timing, frequency, and relational coherence. At a deeper level, this demonstration reveals something profound about reality itself… Patterns appear when many small movements become synchronized enough to behave as one field. The ocean, sound, light, and even the nervous system all do this, because life is vibrational in nature… Did this expand your perception?
0:59

In this incredible demonstration from the Maritime Technical and Safety Institute in Japan, a large wave tank uses many synchronized wave-generating devices to create visible symbols and patterns on the surface of the water. Each paddle produces a small wave, but when many waves are emitted at specific timings, amplitudes, and phases, they begin to overlap. Where the wave crests reinforce one another, the water rises. Where a crest meets a trough, the motion cancels out. This is called interference, and it is one of the most fundamental principles of wave physics. By controlling the phase relationship between many individual wave sources, researchers can shape the surface of the water into temporary patterns, symbols, and directional flows. In other words, the visible image is not “drawn” onto the water. It emerges from the mathematics of waves interacting with one another. This is the same underlying principle behind acoustics, cymatics, ocean modeling, signal processing, holography, and even the way complex field patterns arise throughout nature. A single wave carries motion, but many waves, when organized in relationship, create structure. This is why water is such a powerful medium for understanding resonance. It makes invisible dynamics visible and shows us that form can emerge from rhythm, timing, frequency, and relational coherence. At a deeper level, this demonstration reveals something profound about reality itself… Patterns appear when many small movements become synchronized enough to behave as one field. The ocean, sound, light, and even the nervous system all do this, because life is vibrational in nature… Did this expand your perception?

🧬Maxpein🧬

31,782 Aufrufe • vor 3 Monaten

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics
0:46

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 Aufrufe • vor 6 Monaten

Here's how they did it. If you look closely in the slow motion video, you can see that the silicone mask he's wearing inflates slightly. That's when the pressure of the pump is activated and the hole in the neck of the mask is opened and the fake blood pumps out. The piece shooting away from his t-shirt is possibly a magnetic microphone. What happens is that the tubing (which appears to be running through the front of his t-shirt is pressurized, causing the t-shirt to jump up slightly and the magnetic microphone to pop off his shirt, but at the same time you can see the shirt jump up slightly in front of the opening in the mask (at the level where the hole in the neck should be). It's all a logical consequence of pressurizing the hose with the fake blood. All that of course remote controlled triggered the moment the blank was shot. That was the reason he was sitting on a chair. He had to be hooked up to a hose and no kink in that hose was allowed.
1:02

Here's how they did it. If you look closely in the slow motion video, you can see that the silicone mask he's wearing inflates slightly. That's when the pressure of the pump is activated and the hole in the neck of the mask is opened and the fake blood pumps out. The piece shooting away from his t-shirt is possibly a magnetic microphone. What happens is that the tubing (which appears to be running through the front of his t-shirt is pressurized, causing the t-shirt to jump up slightly and the magnetic microphone to pop off his shirt, but at the same time you can see the shirt jump up slightly in front of the opening in the mask (at the level where the hole in the neck should be). It's all a logical consequence of pressurizing the hose with the fake blood. All that of course remote controlled triggered the moment the blank was shot. That was the reason he was sitting on a chair. He had to be hooked up to a hose and no kink in that hose was allowed.

Martin Vrijland

187,541 Aufrufe • vor 11 Monaten

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics
0:40

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics

Mathelirium

35,799 Aufrufe • vor 2 Monaten

Yann LeCun (Yann LeCun) just revealed why he left Meta, Why are LLMs an extremely narrow field, and how "World model" is the way to build really meaningful agentic future. 🎯 Beautiful and simple in 1.5 mints. "I can’t imagine we can build agentic systems without those systems having the ability to predict, in advance, what the consequences of their actions are going to be. The way we act in the world is that we can predict the consequences of our actions, and that’s what allows us to plan. So what is a world model? Given the state of an environment, a system you want to control at time t, and given an action or intervention you imagine taking, can you predict the state of the world (or the system) at time t + 1? If you can, that’s a world model. You don’t do this at a pixel level, if it’s video. You do this in an abstract representation space, and that’s a crucial key insight." --- From 'AI House Davos" YT channel (full link in comment)
1:40

Yann LeCun (Yann LeCun) just revealed why he left Meta, Why are LLMs an extremely narrow field, and how "World model" is the way to build really meaningful agentic future. 🎯 Beautiful and simple in 1.5 mints. "I can’t imagine we can build agentic systems without those systems having the ability to predict, in advance, what the consequences of their actions are going to be. The way we act in the world is that we can predict the consequences of our actions, and that’s what allows us to plan. So what is a world model? Given the state of an environment, a system you want to control at time t, and given an action or intervention you imagine taking, can you predict the state of the world (or the system) at time t + 1? If you can, that’s a world model. You don’t do this at a pixel level, if it’s video. You do this in an abstract representation space, and that’s a crucial key insight." --- From 'AI House Davos" YT channel (full link in comment)

Rohan Paul

383,490 Aufrufe • vor 6 Monaten

What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:25

What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

17,285 Aufrufe • vor 2 Monaten

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

47,308 Aufrufe • vor 7 Monaten

What are Physics-Informed Neural Networks (PINNs) Physics-Informed Neural Networks (PINNs) are neural nets trained to satisfy a differential equation. The trick is simple. You bake the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work. Meshes. Stencils. Stability tuning. And once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain. Most people first meet PINNs in the least helpful way. A pretty solution surface, almost no clarity on what was enforced to make it appear. In this series we keep the enforcement visible. We pick a PDE, represent the unknown solution as a flexible function, measure how badly that function violates the equation across the domain, and train it to reduce that mismatch at the points we sample. A normal neural net learns from labels. You give it inputs and target outputs. A PINN learns from an equation. You give it inputs (x,t), and it gets penalized whenever its output fails the PDE. Smaller mismatch means smaller loss. Bigger mismatch means bigger loss. That’s all “punish” and “reward” mean here. The network isn’t replacing physics. It’s just a flexible function that we force to obey the same calculus you’d demand from any candidate solution. The math breakdown: We start with a PDE on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we’d have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or from an initial condition. The training objective is a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is how we get the derivatives inside rᵩ. We don’t approximate them with finite differences. We compute them with automatic differentiation: ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … Then we differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. That’s the whole idea. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold, a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t). Each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large, with color showing the sign. As training runs, those threads go slack across the domain, not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

What are Physics-Informed Neural Networks (PINNs) Physics-Informed Neural Networks (PINNs) are neural nets trained to satisfy a differential equation. The trick is simple. You bake the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work. Meshes. Stencils. Stability tuning. And once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain. Most people first meet PINNs in the least helpful way. A pretty solution surface, almost no clarity on what was enforced to make it appear. In this series we keep the enforcement visible. We pick a PDE, represent the unknown solution as a flexible function, measure how badly that function violates the equation across the domain, and train it to reduce that mismatch at the points we sample. A normal neural net learns from labels. You give it inputs and target outputs. A PINN learns from an equation. You give it inputs (x,t), and it gets penalized whenever its output fails the PDE. Smaller mismatch means smaller loss. Bigger mismatch means bigger loss. That’s all “punish” and “reward” mean here. The network isn’t replacing physics. It’s just a flexible function that we force to obey the same calculus you’d demand from any candidate solution. The math breakdown: We start with a PDE on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we’d have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or from an initial condition. The training objective is a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is how we get the derivatives inside rᵩ. We don’t approximate them with finite differences. We compute them with automatic differentiation: ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … Then we differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. That’s the whole idea. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold, a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t). Each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large, with color showing the sign. As training runs, those threads go slack across the domain, not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

44,806 Aufrufe • vor 6 Monaten

Lecture 2 of our Physics-Informed Neural Networks mini-series. In Lecture 1 we made the idea visible...a neural network isn’t predicting a PDE solution, it is the candidate function uᵩ(x,t), and the PDE residual rᵩ(x,t) is the leash that keeps it honest. Now the natural question follows: How can a neural network be punished for breaking a PDE when nobody ever handed it the true solution, and the equation itself contains derivatives like uᵩₜₜ and uᵩₓₓ? Here’s the satisfying answer: A PINN doesn’t need the true answer to be corrected. It only needs a way to measure how wrong it is according to the PDE! The network outputs uᵩ(x,t). A software called "autodiff" is used to compute the derivatives (uᵩₓ, uᵩₜ, uᵩₓₓ, …) exactly by applying the chain rule through the network. Those derivatives get dropped into the PDE to produce rᵩ(x,t). If rᵩ is big at some point, the loss spikes there, and gradient descent pushes the parameters so that rᵩ shrinks. The math breakdown We want a function u(x,t) that satisfies a PDE on a domain Ω. In this lecture we keep a concrete nonlinear example in mind, the damped sine-Gordon equation uₜₜ(x,t) + γ uₜ(x,t) − c² uₓₓ(x,t) + sin(u(x,t)) = 0. A PINN replaces the unknown function u with a neural network uᵩ(x,t), where ᵩ means all the network parameters (weights and biases). Now we build the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t) + sin(uᵩ(x,t)). If uᵩ were a true solution, rᵩ would be 0 everywhere. So we sample points (xⱼ,tⱼ) inside the domain. These are collocation points. At each one we evaluate rᵩ, and we define a physics loss L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|². This is the punishment mechanism. (Punish just means: if |rᵩ| is big, L_phys is big; training updates ᵩ to make L_phys smaller. Reward means the loss drops, so those parameter changes are kept.) The key question was where the derivatives come from. Since uᵩ is built out of differentiable operations, we can compute uᵩₜ(x,t), uᵩₜₜ(x,t), uᵩₓ(x,t), uᵩₓₓ(x,t), at any input (x,t) we choose. Imagine a simple differentiable model written as a sum of nonlinear features uᵩ(x,t) = Σₖ vₖ σ( wₖx x + wₖt t + bₖ ) + b₀. Then the derivatives are just chain rule uᵩₓ(x,t) = Σₖ vₖ σ′(·) wₖx uᵩₓₓ(x,t) = Σₖ vₖ σ″(·) (wₖx)² uᵩₜ(x,t) = Σₖ vₖ σ′(·) wₖt uᵩₜₜ(x,t) = Σₖ vₖ σ″(·) (wₖt)². So rᵩ(x,t) is an explicit computable number at every (x,t). For the damped sine-Gordon example, it’s the same story, just with one extra nonlinear term: rᵩ(x,t) = [uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t)] + sin(uᵩ(x,t)). A real PINN is a deeper composition of these same building blocks, but it’s still just a chain rule, and autodiff is the machinery that does that bookkeeping reliably for big graphs. Then we train by gradient descent on the total loss. Even if we use only physics for the moment, the update is conceptually just ᵩ ← ᵩ − η ∇ᵩ L_phys(ᵩ), with learning rate η. In practice we also include initial/boundary conditions or data, because PDEs aren’t uniquely determined without them L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ), where L_bc/ic(ᵩ) enforces things like uᵩ(x,0) ≈ u₀(x) and uᵩₜ(x,0) ≈ v₀(x), or boundary conditions at x = ±L. So Lecture 2’s punchline is simple: the PDE becomes a training signal. We keep differentiating uᵩ, measuring rᵩ, and updating ᵩ until the residual goes quiet across Ω. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #AutoDiff #Backpropagation #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

Lecture 2 of our Physics-Informed Neural Networks mini-series. In Lecture 1 we made the idea visible...a neural network isn’t predicting a PDE solution, it is the candidate function uᵩ(x,t), and the PDE residual rᵩ(x,t) is the leash that keeps it honest. Now the natural question follows: How can a neural network be punished for breaking a PDE when nobody ever handed it the true solution, and the equation itself contains derivatives like uᵩₜₜ and uᵩₓₓ? Here’s the satisfying answer: A PINN doesn’t need the true answer to be corrected. It only needs a way to measure how wrong it is according to the PDE! The network outputs uᵩ(x,t). A software called "autodiff" is used to compute the derivatives (uᵩₓ, uᵩₜ, uᵩₓₓ, …) exactly by applying the chain rule through the network. Those derivatives get dropped into the PDE to produce rᵩ(x,t). If rᵩ is big at some point, the loss spikes there, and gradient descent pushes the parameters so that rᵩ shrinks. The math breakdown We want a function u(x,t) that satisfies a PDE on a domain Ω. In this lecture we keep a concrete nonlinear example in mind, the damped sine-Gordon equation uₜₜ(x,t) + γ uₜ(x,t) − c² uₓₓ(x,t) + sin(u(x,t)) = 0. A PINN replaces the unknown function u with a neural network uᵩ(x,t), where ᵩ means all the network parameters (weights and biases). Now we build the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t) + sin(uᵩ(x,t)). If uᵩ were a true solution, rᵩ would be 0 everywhere. So we sample points (xⱼ,tⱼ) inside the domain. These are collocation points. At each one we evaluate rᵩ, and we define a physics loss L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|². This is the punishment mechanism. (Punish just means: if |rᵩ| is big, L_phys is big; training updates ᵩ to make L_phys smaller. Reward means the loss drops, so those parameter changes are kept.) The key question was where the derivatives come from. Since uᵩ is built out of differentiable operations, we can compute uᵩₜ(x,t), uᵩₜₜ(x,t), uᵩₓ(x,t), uᵩₓₓ(x,t), at any input (x,t) we choose. Imagine a simple differentiable model written as a sum of nonlinear features uᵩ(x,t) = Σₖ vₖ σ( wₖx x + wₖt t + bₖ ) + b₀. Then the derivatives are just chain rule uᵩₓ(x,t) = Σₖ vₖ σ′(·) wₖx uᵩₓₓ(x,t) = Σₖ vₖ σ″(·) (wₖx)² uᵩₜ(x,t) = Σₖ vₖ σ′(·) wₖt uᵩₜₜ(x,t) = Σₖ vₖ σ″(·) (wₖt)². So rᵩ(x,t) is an explicit computable number at every (x,t). For the damped sine-Gordon example, it’s the same story, just with one extra nonlinear term: rᵩ(x,t) = [uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t)] + sin(uᵩ(x,t)). A real PINN is a deeper composition of these same building blocks, but it’s still just a chain rule, and autodiff is the machinery that does that bookkeeping reliably for big graphs. Then we train by gradient descent on the total loss. Even if we use only physics for the moment, the update is conceptually just ᵩ ← ᵩ − η ∇ᵩ L_phys(ᵩ), with learning rate η. In practice we also include initial/boundary conditions or data, because PDEs aren’t uniquely determined without them L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ), where L_bc/ic(ᵩ) enforces things like uᵩ(x,0) ≈ u₀(x) and uᵩₜ(x,0) ≈ v₀(x), or boundary conditions at x = ±L. So Lecture 2’s punchline is simple: the PDE becomes a training signal. We keep differentiating uᵩ, measuring rᵩ, and updating ᵩ until the residual goes quiet across Ω. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #AutoDiff #Backpropagation #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

19,977 Aufrufe • vor 7 Monaten

Lecture 3 of our Quantum Mechanics series. Lecture 2 gave us the one clean privilege quantum theory offers: treat ψ(x,t) as the state and ρ(x,t) = |ψ(x,t)|² as probability, because Schrödinger evolution forces ρ to obey a continuity equation. Lecture 3 is what that continuity equation is really telling you. If ρ behaves like a fluid, then the only question that matters is: What is the velocity field? Write ψ(x,t) = r(x,t) exp(i θ(x,t)). The magnitude r sets how much probability is sitting there. The phase θ sets where it tries to go. When you unpack the current j = Im(ψ* ∇ψ), it collapses to j = (ρ/m) ∇θ, which means the flow lines you draw are literally contours of phase geometry. Then the constraint that makes the picture bite: ψ has to be single-valued, so θ can’t wind by an arbitrary amount. Around any closed loop the total phase change must be 2π n, with n an integer. That’s why vortices aren’t features you add...they’re defects the math permits, in quantized units. In the render you see both layers at once...the 3D surface shows |ψ| breathing while the phase skin slides, and the 2D panel exposes the engine...current lines steering around discrete vortex charges. The math breakdown We write the state as a complex field ψ(x,t) on the plane (x in R²). The Born rule defines the probability density ρ(x,t) = |ψ(x,t)|² Schrödinger evolution (ħ = 1 units) is i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Now derive conservation of probability. Start with ρ = ψ*ψ: ∂ρ/∂t = ψ* (∂ψ/∂t) + ψ (∂ψ*/∂t) Use Schrödinger and its complex conjugate: ∂ψ/∂t = (1/i) [ −(1/2m) ∇²ψ + Vψ ] ∂ψ*/∂t = (−1/i) [ −(1/2m) ∇²ψ* + Vψ* ] Substitute. The V terms cancel, and the remaining terms rearrange into the continuity equation ∂ρ/∂t + ∇·j = 0 with probability current j = (1/2mi) ( ψ* ∇ψ − ψ ∇ψ* ) = (1/m) Im(ψ* ∇ψ) So "probability density" really behaves like a conserved fluid density with flux j. Now expose the phase mechanism. Write ψ in polar form ψ(x,t) = r(x,t) exp(i θ(x,t)) Compute the gradient ∇ψ = exp(iθ) (∇r + i r ∇θ) Then ψ* ∇ψ = r (∇r + i r ∇θ) Taking the imaginary part gives Im(ψ* ∇ψ) = r² ∇θ = ρ ∇θ So the current becomes j = (ρ/m) ∇θ That’s the steering-wheel statement: Phase gradient sets the flow direction and speed (modulated by density and m). Finally, quantized vortices. Because ψ must be single-valued, going around any closed loop must return the same complex value. That forces the phase winding to be an integer multiple of 2π: ∮ ∇θ · dl = 2π n with n in Z n is the vortex charge. Vortex cores sit where ρ ≈ 0 (phase is undefined), and the current streamlines circulate around them. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #Vortices #TopologicalDefects #ComplexAnalysis #MathematicalPhysics #Mathematics #Physics
0:34

Lecture 3 of our Quantum Mechanics series. Lecture 2 gave us the one clean privilege quantum theory offers: treat ψ(x,t) as the state and ρ(x,t) = |ψ(x,t)|² as probability, because Schrödinger evolution forces ρ to obey a continuity equation. Lecture 3 is what that continuity equation is really telling you. If ρ behaves like a fluid, then the only question that matters is: What is the velocity field? Write ψ(x,t) = r(x,t) exp(i θ(x,t)). The magnitude r sets how much probability is sitting there. The phase θ sets where it tries to go. When you unpack the current j = Im(ψ* ∇ψ), it collapses to j = (ρ/m) ∇θ, which means the flow lines you draw are literally contours of phase geometry. Then the constraint that makes the picture bite: ψ has to be single-valued, so θ can’t wind by an arbitrary amount. Around any closed loop the total phase change must be 2π n, with n an integer. That’s why vortices aren’t features you add...they’re defects the math permits, in quantized units. In the render you see both layers at once...the 3D surface shows |ψ| breathing while the phase skin slides, and the 2D panel exposes the engine...current lines steering around discrete vortex charges. The math breakdown We write the state as a complex field ψ(x,t) on the plane (x in R²). The Born rule defines the probability density ρ(x,t) = |ψ(x,t)|² Schrödinger evolution (ħ = 1 units) is i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Now derive conservation of probability. Start with ρ = ψ*ψ: ∂ρ/∂t = ψ* (∂ψ/∂t) + ψ (∂ψ*/∂t) Use Schrödinger and its complex conjugate: ∂ψ/∂t = (1/i) [ −(1/2m) ∇²ψ + Vψ ] ∂ψ*/∂t = (−1/i) [ −(1/2m) ∇²ψ* + Vψ* ] Substitute. The V terms cancel, and the remaining terms rearrange into the continuity equation ∂ρ/∂t + ∇·j = 0 with probability current j = (1/2mi) ( ψ* ∇ψ − ψ ∇ψ* ) = (1/m) Im(ψ* ∇ψ) So "probability density" really behaves like a conserved fluid density with flux j. Now expose the phase mechanism. Write ψ in polar form ψ(x,t) = r(x,t) exp(i θ(x,t)) Compute the gradient ∇ψ = exp(iθ) (∇r + i r ∇θ) Then ψ* ∇ψ = r (∇r + i r ∇θ) Taking the imaginary part gives Im(ψ* ∇ψ) = r² ∇θ = ρ ∇θ So the current becomes j = (ρ/m) ∇θ That’s the steering-wheel statement: Phase gradient sets the flow direction and speed (modulated by density and m). Finally, quantized vortices. Because ψ must be single-valued, going around any closed loop must return the same complex value. That forces the phase winding to be an integer multiple of 2π: ∮ ∇θ · dl = 2π n with n in Z n is the vortex charge. Vortex cores sit where ρ ≈ 0 (phase is undefined), and the current streamlines circulate around them. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #Vortices #TopologicalDefects #ComplexAnalysis #MathematicalPhysics #Mathematics #Physics

Mathelirium

37,998 Aufrufe • vor 7 Monaten