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When you let the Mathematics and the Physics do their thing… you get something beautiful.🤩🙌🏻😍 We evolve a two-component quantum field (ψ1, ψ2) under a coupled Gross-Pitaevskii/Schrödinger flow: split-step Fourier time stepping, self-gravity via a Poisson solve, and coherent Rabi mixing between components. Then we render the same state...

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Mathelirium

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The Hidden Flow Inside Schrödinger’s Equation Once you define ρ = |ψ|² and differentiate it, Schrödinger equation rearranges into a continuity law, ∂ρ/∂t + ∇·j = 0, with j = (1/m)Im(ψ*∇ψ). Which means that probability is not just sitting there, it flows. Write ψ = r e^(iθ), and the current becomes j = (ρ/m)∇θ. So, the phase gradient is the steering field. When the phase winds by 2πn around a loop, the circulation is quantized, and the cores appear where ρ collapses so the phase can go undefined. In the animation, the ribbons trace that current, the color field shows phase, and the curls in the streamlines mark quantized vortex defects.
1:00

The Hidden Flow Inside Schrödinger’s Equation Once you define ρ = |ψ|² and differentiate it, Schrödinger equation rearranges into a continuity law, ∂ρ/∂t + ∇·j = 0, with j = (1/m)Im(ψ*∇ψ). Which means that probability is not just sitting there, it flows. Write ψ = r e^(iθ), and the current becomes j = (ρ/m)∇θ. So, the phase gradient is the steering field. When the phase winds by 2πn around a loop, the circulation is quantized, and the cores appear where ρ collapses so the phase can go undefined. In the animation, the ribbons trace that current, the color field shows phase, and the curls in the streamlines mark quantized vortex defects.

Mathelirium

26,511 Aufrufe • vor 5 Monaten

A Bonus render of Electron Holography for our Quantum Mechanics Lecture Series. We’re staging a TEM-style electron holography shot as a single 3D scene. A coherent electron wave ψ(x,y,t) enters from the left as a localized packet, gets split by a thin biprism phase wedge, then bends around a central nanoring that stands in for an enclosed magnetic flux. The wave evolves by Schrödinger dynamics, so its intensity ρ=|ψ|² is literally the electron detection probability density, while the probability current j=(1/m)Im(ψ*∇ψ) acts like a conserved flow of that density across the plane. The main surface is a living whose height tracks |ψ| and whose skin color cycles with arg(ψ). On top of it, luminous current ribbons trace streamlines of v=j/(ρ+ε), so you can see how phase gradients steer the flow around the ring and back into interference. High above, a tilted detector/hologram screen shows the off-axis fringe image you’d measure (intensity after mixing with a reference wave), while a porcelain bead marks the centroid ⟨x⟩(t) and flux threads through the ring pulse in sync with the Aharonov-Bohm phase knob φ_AB(t). See our lecture series for details. #QuantumMechanics #ElectronHolography #MathematicalPhysics #Physics
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A Bonus render of Electron Holography for our Quantum Mechanics Lecture Series. We’re staging a TEM-style electron holography shot as a single 3D scene. A coherent electron wave ψ(x,y,t) enters from the left as a localized packet, gets split by a thin biprism phase wedge, then bends around a central nanoring that stands in for an enclosed magnetic flux. The wave evolves by Schrödinger dynamics, so its intensity ρ=|ψ|² is literally the electron detection probability density, while the probability current j=(1/m)Im(ψ*∇ψ) acts like a conserved flow of that density across the plane. The main surface is a living whose height tracks |ψ| and whose skin color cycles with arg(ψ). On top of it, luminous current ribbons trace streamlines of v=j/(ρ+ε), so you can see how phase gradients steer the flow around the ring and back into interference. High above, a tilted detector/hologram screen shows the off-axis fringe image you’d measure (intensity after mixing with a reference wave), while a porcelain bead marks the centroid ⟨x⟩(t) and flux threads through the ring pulse in sync with the Aharonov-Bohm phase knob φ_AB(t). See our lecture series for details. #QuantumMechanics #ElectronHolography #MathematicalPhysics #Physics

Mathelirium

14,252 Aufrufe • vor 8 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
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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

38,203 Aufrufe • vor 8 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 7 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 8 Monaten

A Quantum Current Chooses a Path A single electron enters a microscopic interferometer, but its wavefunction travels through two routes at once. One branch crosses a resonant quantum dot while the other moves freely around a ring threaded by magnetic flux. The probability current is j = (ℏ/m) Im(ψ*∇ψ) - (q/m)A|ψ|², while the enclosed flux creates the Aharonov-Bohm phase Δφᴬᴮ = qΦ/ℏ. The quantum dot adds the resonant phase δ(E) = atan2(Γ, E - E₀), producing Fano interference when the two paths recombine. Wave packets split, become temporarily trapped, form circulating current vortices, reverse direction after a flux quench, and finally escape through the drain in a coherent burst. The glowing threads are material curves transported by v = j/ρ, revealing how invisible quantum phase controls the flow. #QuantumMechanics #AharonovBohmEffect #QuantumInterference #QuantumDot #ProbabilityCurrent #MesoscopicPhysics #PhysicsAnimation #Physics
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A Quantum Current Chooses a Path A single electron enters a microscopic interferometer, but its wavefunction travels through two routes at once. One branch crosses a resonant quantum dot while the other moves freely around a ring threaded by magnetic flux. The probability current is j = (ℏ/m) Im(ψ*∇ψ) - (q/m)A|ψ|², while the enclosed flux creates the Aharonov-Bohm phase Δφᴬᴮ = qΦ/ℏ. The quantum dot adds the resonant phase δ(E) = atan2(Γ, E - E₀), producing Fano interference when the two paths recombine. Wave packets split, become temporarily trapped, form circulating current vortices, reverse direction after a flux quench, and finally escape through the drain in a coherent burst. The glowing threads are material curves transported by v = j/ρ, revealing how invisible quantum phase controls the flow. #QuantumMechanics #AharonovBohmEffect #QuantumInterference #QuantumDot #ProbabilityCurrent #MesoscopicPhysics #PhysicsAnimation #Physics

Mathelirium

16,684 Aufrufe • vor 2 Monaten

Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics
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Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics

Mathelirium

20,456 Aufrufe • vor 5 Monaten

We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics
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We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

Mathelirium

16,437 Aufrufe • vor 7 Monaten

You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠
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You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠

Mathelirium

109,155 Aufrufe • vor 1 Jahr

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics
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Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Mathelirium

81,608 Aufrufe • vor 8 Monaten

When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics
0:40

When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics

Mathelirium

22,460 Aufrufe • vor 2 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

54,894 Aufrufe • vor 5 Monaten

Quantum Probability Surface Goes Negative A Schrödinger-Cat state is built from two separated coherent wavepackets, |ψ⟩=N(|α⟩+|−α⟩), and its Wigner function reveals the part of Quantum Mechanics that ordinary probability cannot show. The two bright lobes behave like classical possibilities, while the interference between them carves negative valleys into phase space, where W(x,p)<0. In this animation, the surface height is the signed Wigner function. So, gold rises above zero, cyan sinks below it, and the blades between them expose the nonclassical structure of the wavefunction. #QuantumMechanics #WignerFunction #SchrodingerCat #Physics #QuantumArt #ScienceVisualization
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Quantum Probability Surface Goes Negative A Schrödinger-Cat state is built from two separated coherent wavepackets, |ψ⟩=N(|α⟩+|−α⟩), and its Wigner function reveals the part of Quantum Mechanics that ordinary probability cannot show. The two bright lobes behave like classical possibilities, while the interference between them carves negative valleys into phase space, where W(x,p)<0. In this animation, the surface height is the signed Wigner function. So, gold rises above zero, cyan sinks below it, and the blades between them expose the nonclassical structure of the wavefunction. #QuantumMechanics #WignerFunction #SchrodingerCat #Physics #QuantumArt #ScienceVisualization

Mathelirium

12,603 Aufrufe • vor 3 Monaten

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow
0:40

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow

Mathelirium

47,661 Aufrufe • vor 9 Monaten

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation
0:33

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

Mathelirium

22,197 Aufrufe • vor 3 Monaten

Why String Theory Won't Die The string theory idea refuses to die because it's one of the few frameworks that forces a quantum object and spacetime geometry into the same picture. In the render, a closed string isn’t just a loop doing pretty vibrations. Its internal phase pattern is a quantum degree of freedom, shown as color. At the same time, its motion stirs ripples in the surrounding metric, shown as the fabric below. One object, two roles. A wave-like state living on the string, and a curvature-like response spreading through the background. You can literally watch quantum stuff and gravity stuff talk to each other in the same scene. #StringTheory #Physics #QuantumMechanics #GeneralRelativity #MathematicalPhysics #SciComm
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Why String Theory Won't Die The string theory idea refuses to die because it's one of the few frameworks that forces a quantum object and spacetime geometry into the same picture. In the render, a closed string isn’t just a loop doing pretty vibrations. Its internal phase pattern is a quantum degree of freedom, shown as color. At the same time, its motion stirs ripples in the surrounding metric, shown as the fabric below. One object, two roles. A wave-like state living on the string, and a curvature-like response spreading through the background. You can literally watch quantum stuff and gravity stuff talk to each other in the same scene. #StringTheory #Physics #QuantumMechanics #GeneralRelativity #MathematicalPhysics #SciComm

Mathelirium

13,657 Aufrufe • vor 7 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 8 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

17,459 Aufrufe • vor 3 Monaten

Final Lecture of our Statistical Mechanics Series. Lecture 2 showed how we move from the Full Phase-Space Density ρ(q₁, …, qₙ, p₁, …, pₙ, t) to smaller statistical objects by integrating out variables we do not want to keep. That gives reduced descriptions like the One-Particle Density f₁(q₁,p₁,t) and the Two-Particle Density f₂(q₁,p₁,q₂,p₂,t) This was the simplification. Now comes the catch. If the Full Density obeys Liouville’s Equation, the reduced densities do not evolve independently. The equation for one level depends on the next one and this is referred to as the BBGKY hierarchy. The One-Particle Density depends on the Two-Particle Density. The Two-Particle Density depends on the Three-Particle Density. And the chain keeps going. That happens because particles interact. Once one particle feels the rest, one-particle information is no longer enough. Correlations enter, and the lower level is fed from above. If the full Hamiltonian is H = Σᵢ pᵢ²/(2m) + Σᵢ U(qᵢ) + (1/2) Σᵢ Σⱼ≠ᵢ Φ(qᵢ − qⱼ) then reducing the full density does not make the interaction terms disappear. It leaves behind coupling to higher-order reduced densities. So, schematically, ∂f₁/∂t + transport of one particle = interaction term involving f₂ and more generally the heirarchy is such that ∂fₛ/∂t + s-particle transport = interaction term involving fₛ₊₁ Therefore, Lecture 3 is really about the price of reduction. We simplify the description, but the information we remove comes back as coupling to higher-order correlations. So, how do you actually compute anything if every level depends on the next one? This is the so-called Closure Problem. To make the hierarchy usable, you need an extra assumption that cuts the chain. You replace the exact higher-order object by an approximation in terms of lower-order ones. The most basic example is a factorized closure at the pair level, where the exact correlated Two-Particle Density is replaced schematically by a product of One-Particle Densities: f₂(q₁,p₁,q₂,p₂,t) ≈ f₁(q₁,p₁,t) f₁(q₂,p₂,t) That approximation is not exact. It throws away part of the correlation structure. But it gives you something the raw hierarchy does not... a closed equation for the lower-level description. That is why closure matters so much. Without it, the hierarchy is exact but open. With it, the theory becomes approximate but usable. Thus, the combined point of this final Statistical Mechanics post is simple. First, reduced descriptions are not closed because interactions generate correlations across levels. Second, if you want a workable Kinetic Theory, you must close the hierarchy by approximating those higher-order correlations. It is the bridge from formal many-body mechanics to equations people can actually solve. In the render, that is exactly the story you are seeing. The first part shows the hierarchy itself: one reduced level feeding the next, with lower descriptions inheriting structure from higher ones. The second part shows the closure step where the exact correlated pair level is replaced by a factorized ansatz, and that approximation gives back a closed one-particle description. That is, the animation moves from dependence to approximation, and from approximation to solvability. #StatisticalMechanics #BBGKY #ClosureProblem #KineticTheory #PhaseSpace #ReducedDistribution #HamiltonianMechanics #MathematicalPhysics #Mathematics #Physics
1:35

Final Lecture of our Statistical Mechanics Series. Lecture 2 showed how we move from the Full Phase-Space Density ρ(q₁, …, qₙ, p₁, …, pₙ, t) to smaller statistical objects by integrating out variables we do not want to keep. That gives reduced descriptions like the One-Particle Density f₁(q₁,p₁,t) and the Two-Particle Density f₂(q₁,p₁,q₂,p₂,t) This was the simplification. Now comes the catch. If the Full Density obeys Liouville’s Equation, the reduced densities do not evolve independently. The equation for one level depends on the next one and this is referred to as the BBGKY hierarchy. The One-Particle Density depends on the Two-Particle Density. The Two-Particle Density depends on the Three-Particle Density. And the chain keeps going. That happens because particles interact. Once one particle feels the rest, one-particle information is no longer enough. Correlations enter, and the lower level is fed from above. If the full Hamiltonian is H = Σᵢ pᵢ²/(2m) + Σᵢ U(qᵢ) + (1/2) Σᵢ Σⱼ≠ᵢ Φ(qᵢ − qⱼ) then reducing the full density does not make the interaction terms disappear. It leaves behind coupling to higher-order reduced densities. So, schematically, ∂f₁/∂t + transport of one particle = interaction term involving f₂ and more generally the heirarchy is such that ∂fₛ/∂t + s-particle transport = interaction term involving fₛ₊₁ Therefore, Lecture 3 is really about the price of reduction. We simplify the description, but the information we remove comes back as coupling to higher-order correlations. So, how do you actually compute anything if every level depends on the next one? This is the so-called Closure Problem. To make the hierarchy usable, you need an extra assumption that cuts the chain. You replace the exact higher-order object by an approximation in terms of lower-order ones. The most basic example is a factorized closure at the pair level, where the exact correlated Two-Particle Density is replaced schematically by a product of One-Particle Densities: f₂(q₁,p₁,q₂,p₂,t) ≈ f₁(q₁,p₁,t) f₁(q₂,p₂,t) That approximation is not exact. It throws away part of the correlation structure. But it gives you something the raw hierarchy does not... a closed equation for the lower-level description. That is why closure matters so much. Without it, the hierarchy is exact but open. With it, the theory becomes approximate but usable. Thus, the combined point of this final Statistical Mechanics post is simple. First, reduced descriptions are not closed because interactions generate correlations across levels. Second, if you want a workable Kinetic Theory, you must close the hierarchy by approximating those higher-order correlations. It is the bridge from formal many-body mechanics to equations people can actually solve. In the render, that is exactly the story you are seeing. The first part shows the hierarchy itself: one reduced level feeding the next, with lower descriptions inheriting structure from higher ones. The second part shows the closure step where the exact correlated pair level is replaced by a factorized ansatz, and that approximation gives back a closed one-particle description. That is, the animation moves from dependence to approximation, and from approximation to solvability. #StatisticalMechanics #BBGKY #ClosureProblem #KineticTheory #PhaseSpace #ReducedDistribution #HamiltonianMechanics #MathematicalPhysics #Mathematics #Physics

Mathelirium

10,970 Aufrufe • vor 4 Monaten