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A physical system, its phase-space state, and the energy function that governs the flow, all linked together. The point of the animation is that these are not three different objects. They are three views of the same dynamics. The pendulum on the left is the physical motion. The curve... show more

Mathelirium

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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

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

35,596 Aufrufe • vor 27 Tagen

Warmup to Statistical Mechanics What Exactly is a Hamiltonian A System? In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is (q, p) That pair is one point in phase space. Why do we do this? Because in these variables, the equations of motion take a remarkably clean form. Everything is generated by one single function, the Hamiltonian H(q, p) and in the simplest cases this Hamiltonian is just the total energy written in terms of position and momentum. So if you know H, you know the dynamics. You might wonder, but how can one function generate motion? The rule is dqᵢ/dt = ∂H/∂pᵢ dpᵢ/dt = −∂H/∂qᵢ These are Hamilton’s equations. Now read them slowly 😄 The rate of change of position comes from differentiating H with respect to momentum. The rate of change of momentum comes from differentiating H with respect to position, with a minus sign. This constitutes the whole engine. A simple example makes this less abstract: Take one particle of mass m moving in a potential V(q). Then the Hamiltonian is H(q, p) = p²/(2m) + V(q) The first term is kinetic energy. The second term is potential energy. Now apply Hamilton’s equations. First, dq/dt = ∂H/∂p = p/m So momentum tells you how position changes. Second, dp/dt = −∂H/∂q = −dV/dq Thus, momentum changes because of force. If you now combine these two equations, you recover ordinary Newtonian mechanics. Since p = m dq/dt, we get m d²q/dt² = −dV/dq So, Hamiltonian mechanics is not a different theory. It is the same mechanics, written in a form that exposes its geometric structure much more clearly. The animation The full 3D surface is the Hamiltonian itself, the energy landscape H(q, p). The floor underneath is phase space, marked by energy contours and the local flow field. The bright moving point is one actual state (q(t), p(t)) evolving under Hamilton’s equations. Its trail shows that the motion is not arbitrary. It is guided everywhere by the geometry of the same single function H. The render is doing more than illustrating a particle moving, it is showing how one function organizes the whole phase-space motion. The math breakdown: Start with one degree of freedom. The state is described by position q and momentum p. So the system lives in a two-dimensional phase space with coordinates (q, p) Now choose a Hamiltonian H(q, p) Think of H as the energy function. In many standard systems, H(q, p) = kinetic energy + potential energy For a particle of mass m in a potential V(q), this becomes H(q, p) = p²/(2m) + V(q) Hamilton’s equations say dq/dt = ∂H/∂p dp/dt = −∂H/∂q Now substitute this specific H. First compute the p derivative: ∂H/∂p = ∂/∂p (p²/(2m) + V(q)) = p/m So dq/dt = p/m Now compute the q derivative: ∂H/∂q = ∂/∂q (p²/(2m) + V(q)) = dV/dq So dp/dt = −dV/dq These two first-order equations completely determine the motion. Now, connect this back to Newton’s law. From dq/dt = p/m we get p = m dq/dt Differentiate both sides with respect to time: dp/dt = m d²q/dt² But Hamilton’s second equation gives dp/dt = −dV/dq So , together they imply m d²q/dt² = −dV/dq This is exactly Newton’s second law for motion in the potential V(q). Thus, Hamilton’s equations do not replace mechanic, they reorganize it. #HamiltonianMechanics #PhaseSpace #ClassicalMechanics #MathematicalPhysics #DifferentialEquations #Mathematics #Physics

Warmup to Statistical Mechanics What Exactly is a Hamiltonian A System? In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is (q, p) That pair is one point in phase space. Why do we do this? Because in these variables, the equations of motion take a remarkably clean form. Everything is generated by one single function, the Hamiltonian H(q, p) and in the simplest cases this Hamiltonian is just the total energy written in terms of position and momentum. So if you know H, you know the dynamics. You might wonder, but how can one function generate motion? The rule is dqᵢ/dt = ∂H/∂pᵢ dpᵢ/dt = −∂H/∂qᵢ These are Hamilton’s equations. Now read them slowly 😄 The rate of change of position comes from differentiating H with respect to momentum. The rate of change of momentum comes from differentiating H with respect to position, with a minus sign. This constitutes the whole engine. A simple example makes this less abstract: Take one particle of mass m moving in a potential V(q). Then the Hamiltonian is H(q, p) = p²/(2m) + V(q) The first term is kinetic energy. The second term is potential energy. Now apply Hamilton’s equations. First, dq/dt = ∂H/∂p = p/m So momentum tells you how position changes. Second, dp/dt = −∂H/∂q = −dV/dq Thus, momentum changes because of force. If you now combine these two equations, you recover ordinary Newtonian mechanics. Since p = m dq/dt, we get m d²q/dt² = −dV/dq So, Hamiltonian mechanics is not a different theory. It is the same mechanics, written in a form that exposes its geometric structure much more clearly. The animation The full 3D surface is the Hamiltonian itself, the energy landscape H(q, p). The floor underneath is phase space, marked by energy contours and the local flow field. The bright moving point is one actual state (q(t), p(t)) evolving under Hamilton’s equations. Its trail shows that the motion is not arbitrary. It is guided everywhere by the geometry of the same single function H. The render is doing more than illustrating a particle moving, it is showing how one function organizes the whole phase-space motion. The math breakdown: Start with one degree of freedom. The state is described by position q and momentum p. So the system lives in a two-dimensional phase space with coordinates (q, p) Now choose a Hamiltonian H(q, p) Think of H as the energy function. In many standard systems, H(q, p) = kinetic energy + potential energy For a particle of mass m in a potential V(q), this becomes H(q, p) = p²/(2m) + V(q) Hamilton’s equations say dq/dt = ∂H/∂p dp/dt = −∂H/∂q Now substitute this specific H. First compute the p derivative: ∂H/∂p = ∂/∂p (p²/(2m) + V(q)) = p/m So dq/dt = p/m Now compute the q derivative: ∂H/∂q = ∂/∂q (p²/(2m) + V(q)) = dV/dq So dp/dt = −dV/dq These two first-order equations completely determine the motion. Now, connect this back to Newton’s law. From dq/dt = p/m we get p = m dq/dt Differentiate both sides with respect to time: dp/dt = m d²q/dt² But Hamilton’s second equation gives dp/dt = −dV/dq So , together they imply m d²q/dt² = −dV/dq This is exactly Newton’s second law for motion in the potential V(q). Thus, Hamilton’s equations do not replace mechanic, they reorganize it. #HamiltonianMechanics #PhaseSpace #ClassicalMechanics #MathematicalPhysics #DifferentialEquations #Mathematics #Physics

Mathelirium

50,370 Aufrufe • vor 2 Monaten

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

Mathelirium

10,806 Aufrufe • vor 2 Monaten

According to Bentov, consciousness evolves to the 'absolute' which is the source of all consciousness. Matter, composed of quanta of energy, is the vibrating, changing component of pure consciousness. The absolute is fixed, manifest and invisible. Bentov proposed that the brain acts as a transducer, converting the energy of consciousness into the electrical and chemical signals that are used to communicate information within the body. He also believed that the brain has the ability to tune into different frequencies of consciousness, allowing individuals to experience different states of awareness. Bentov's theory is based on the idea that consciousness is a fundamental aspect of the universe and is not limited to the brain or the body. He argued that consciousness is non-local, meaning it is not bound by space and time and can exist independently of the physical body. Bentov's theory has been influential in the field of consciousness studies, but it remains a subject of debate and is not widely accepted by mainstream science.

According to Bentov, consciousness evolves to the 'absolute' which is the source of all consciousness. Matter, composed of quanta of energy, is the vibrating, changing component of pure consciousness. The absolute is fixed, manifest and invisible. Bentov proposed that the brain acts as a transducer, converting the energy of consciousness into the electrical and chemical signals that are used to communicate information within the body. He also believed that the brain has the ability to tune into different frequencies of consciousness, allowing individuals to experience different states of awareness. Bentov's theory is based on the idea that consciousness is a fundamental aspect of the universe and is not limited to the brain or the body. He argued that consciousness is non-local, meaning it is not bound by space and time and can exist independently of the physical body. Bentov's theory has been influential in the field of consciousness studies, but it remains a subject of debate and is not widely accepted by mainstream science.

Vicky Verma

390,472 Aufrufe • vor 1 Jahr

#WATCH | Delhi: Adani Group Chairman Gautam Adani says, "What has become even more obvious from the recent Middle East conflict and targeted attacks on infrastructure is that energy security and digital security are no longer independent dimensions. They are now the twin foundations of national power. And it is increasingly evident that the country that controls its energy will drive its industrial future. The country that controls its compute will drive its intelligence future, and the country that controls both will shape the century ahead. This is the new geometry of power. If 1947 was about our political freedom, true independence in the 21st century will mean owning the energy that lights our homes and the intelligence that guides our minds. This is not a theory. The two largest economies in the world, the United States and China, have understood this deeply. Their systems may be different. Their politics may be different. Their institutions may be different. But their strategic objectives are the same. No nation can call itself truly self-reliant if it depends on others for the energy that powers its economy or the intelligence that shapes its destiny." (Video source: Adani Group)

#WATCH | Delhi: Adani Group Chairman Gautam Adani says, "What has become even more obvious from the recent Middle East conflict and targeted attacks on infrastructure is that energy security and digital security are no longer independent dimensions. They are now the twin foundations of national power. And it is increasingly evident that the country that controls its energy will drive its industrial future. The country that controls its compute will drive its intelligence future, and the country that controls both will shape the century ahead. This is the new geometry of power. If 1947 was about our political freedom, true independence in the 21st century will mean owning the energy that lights our homes and the intelligence that guides our minds. This is not a theory. The two largest economies in the world, the United States and China, have understood this deeply. Their systems may be different. Their politics may be different. Their institutions may be different. But their strategic objectives are the same. No nation can call itself truly self-reliant if it depends on others for the energy that powers its economy or the intelligence that shapes its destiny." (Video source: Adani Group)

ANI

161,308 Aufrufe • vor 1 Monat

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

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 12 Tagen

The House must function. If they give us the opportunity to raise our concerns on the floor of the House, the House will function. It is really incumbent on the government to create an atmosphere for the House to function. They have the majority. The Speaker is from their ranks. The onus is on them to create an atmosphere for the House to function. This is a serious issue that we definitely want to raise on the floor of Parliament. The people are watching this, and they are also noticing that the electoral process is not fair anymore. : Karti P Chidambaram ji

The House must function. If they give us the opportunity to raise our concerns on the floor of the House, the House will function. It is really incumbent on the government to create an atmosphere for the House to function. They have the majority. The Speaker is from their ranks. The onus is on them to create an atmosphere for the House to function. This is a serious issue that we definitely want to raise on the floor of Parliament. The people are watching this, and they are also noticing that the electoral process is not fair anymore. : Karti P Chidambaram ji

Congress

14,153 Aufrufe • vor 10 Monaten

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!

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

21,543 Aufrufe • vor 3 Monaten

A Triple Spring-Mass Pendulum Just for The Fun of Physics We model the mass positions as r₁ = (x₁, y₁), r₂ = (x₂, y₂), r₃ = (x₃, y₃) with y positive downward in the simulation. The three spring segments are r₁ = r₁ − (0,0) r₁₂ = r₂ − r₁ r₂₃ = r₃ − r₂ Each spring applies a Hooke force along its own direction, F(r; k, L) = −k (|r| − L) (r / |r|) and each mass also feels gravity, Fgᵢ = (0, mᵢ g) So the equations of motion are m₁ r̈₁ = Fg₁ + F(r₁; k₁, L₁) − F(r₁₂; k₂, L₂) m₂ r̈₂ = Fg₂ + F(r₁₂; k₂, L₂) − F(r₂₃; k₃, L₃) m₃ r̈₃ = Fg₃ + F(r₂₃; k₃, L₃) We integrate these ODEs forward with RK4 using a small dt. Different scenes come from changing the mass ratios and from giving the third mass a small initial kick. The yellow trail is the path of the third mass. The inset phase portraits show the same motion in state space: (x₃, vₓ₃) and (−y₃, vᵧ₃) When those curves stay tight and loop cleanly, the motion is more regular. When they spread, fold, and smear, energy is being redistributed across the whole chain between gravity, spring stretch, and kinetic motion. #NonlinearDynamics #CoupledOscillators #SpringPendulum #ChaosInMotion #ComputationalPhysics #MathematicalVisualization

A Triple Spring-Mass Pendulum Just for The Fun of Physics We model the mass positions as r₁ = (x₁, y₁), r₂ = (x₂, y₂), r₃ = (x₃, y₃) with y positive downward in the simulation. The three spring segments are r₁ = r₁ − (0,0) r₁₂ = r₂ − r₁ r₂₃ = r₃ − r₂ Each spring applies a Hooke force along its own direction, F(r; k, L) = −k (|r| − L) (r / |r|) and each mass also feels gravity, Fgᵢ = (0, mᵢ g) So the equations of motion are m₁ r̈₁ = Fg₁ + F(r₁; k₁, L₁) − F(r₁₂; k₂, L₂) m₂ r̈₂ = Fg₂ + F(r₁₂; k₂, L₂) − F(r₂₃; k₃, L₃) m₃ r̈₃ = Fg₃ + F(r₂₃; k₃, L₃) We integrate these ODEs forward with RK4 using a small dt. Different scenes come from changing the mass ratios and from giving the third mass a small initial kick. The yellow trail is the path of the third mass. The inset phase portraits show the same motion in state space: (x₃, vₓ₃) and (−y₃, vᵧ₃) When those curves stay tight and loop cleanly, the motion is more regular. When they spread, fold, and smear, energy is being redistributed across the whole chain between gravity, spring stretch, and kinetic motion. #NonlinearDynamics #CoupledOscillators #SpringPendulum #ChaosInMotion #ComputationalPhysics #MathematicalVisualization

Mathelirium

35,597 Aufrufe • vor 3 Monaten

Our general understanding of the characteristics of the physical world are largely restricted by the limited range of our senses. The world appears to be comprised of tangible objects positioned within empty space, separate and distinct. Neither of these characterizations are true. Our unaided senses generate a false interpretation of the true state of reality. Space is note empty, it is substantive with a quantifiable and measurable energy density. Space, in terms of quantum vacuum fluctuations, can be considered as a veritable sea of oscillating energy, like a fluid, quantized at the Planck scale (a billion trillion trillion times smaller than a centimeter) as Planck spherical units, and these tiny oscillators make up the fluid medium of space and comprise the "material" stuff as well. Imagine being able to directly perceive this level of reality. Our photodetector proteins in our eyes are sensitive to electromagnetic radiation in the frequency range of 400 to 800 terahertz (trillions of oscillations per second), and we call this "visible light". If, however, we could see light at the Planck scale, were photons oscillate at the Planck frequency— a mass-energy value that makes the electromagnetic component at order of unity with spacetime curvature— then we would theoretically see directly the substantive fluid medium of space and our sight would relay a world that is integrally interconnected and all one substance; objects would not appear as separate and distinct or even fundamentally different than the substance comprising the bulk space. "Material" objects would appear just as patterned vortices of the fluid that is the very substance of space. So, tangible objects are made of the same substance as space, and only seem physical to our limited senses because of electromagnetic repulsive forces. The electromagnetic repulsive forces are generated by how these PSUs circulate and flow within the structured patterns of space that we call particles and atoms. In this way, the coherent phases, circulation, flow, and pressure forces of this Planck plasma fluid are the source of mass, force, and charge. We are now coming to an understanding of these dynamics at a fundamental level, exemplified in the publication The Origin of Mass and the Nature of Gravity 🔗

Our general understanding of the characteristics of the physical world are largely restricted by the limited range of our senses. The world appears to be comprised of tangible objects positioned within empty space, separate and distinct. Neither of these characterizations are true. Our unaided senses generate a false interpretation of the true state of reality. Space is note empty, it is substantive with a quantifiable and measurable energy density. Space, in terms of quantum vacuum fluctuations, can be considered as a veritable sea of oscillating energy, like a fluid, quantized at the Planck scale (a billion trillion trillion times smaller than a centimeter) as Planck spherical units, and these tiny oscillators make up the fluid medium of space and comprise the "material" stuff as well. Imagine being able to directly perceive this level of reality. Our photodetector proteins in our eyes are sensitive to electromagnetic radiation in the frequency range of 400 to 800 terahertz (trillions of oscillations per second), and we call this "visible light". If, however, we could see light at the Planck scale, were photons oscillate at the Planck frequency— a mass-energy value that makes the electromagnetic component at order of unity with spacetime curvature— then we would theoretically see directly the substantive fluid medium of space and our sight would relay a world that is integrally interconnected and all one substance; objects would not appear as separate and distinct or even fundamentally different than the substance comprising the bulk space. "Material" objects would appear just as patterned vortices of the fluid that is the very substance of space. So, tangible objects are made of the same substance as space, and only seem physical to our limited senses because of electromagnetic repulsive forces. The electromagnetic repulsive forces are generated by how these PSUs circulate and flow within the structured patterns of space that we call particles and atoms. In this way, the coherent phases, circulation, flow, and pressure forces of this Planck plasma fluid are the source of mass, force, and charge. We are now coming to an understanding of these dynamics at a fundamental level, exemplified in the publication The Origin of Mass and the Nature of Gravity 🔗

Nassim Haramein

15,937 Aufrufe • vor 2 Jahren

Whitehouse: There is one point that stands out, and that is that the fakery on the floor is belied by the language of the bill. If these assertions were true, they would not need to raise the debt limit

Whitehouse: There is one point that stands out, and that is that the fakery on the floor is belied by the language of the bill. If these assertions were true, they would not need to raise the debt limit

Acyn

72,223 Aufrufe • vor 1 Jahr

Sir John Curtice, "The public have not reacted against the expansion of the state in the way both the Conservatives and Labour imagine they are doing" "All the evidence shows that the public are much more concerned about the state of the NHS, than they are about lowering taxation" "And frankly the Conservatives would be better spending money on increasing funding for the NHS that they spent on tax cuts, ending the junior doctor strike and getting waiting times down" "The public's priority is with a better NHS, not with the level of taxation"

Sir John Curtice, "The public have not reacted against the expansion of the state in the way both the Conservatives and Labour imagine they are doing" "All the evidence shows that the public are much more concerned about the state of the NHS, than they are about lowering taxation" "And frankly the Conservatives would be better spending money on increasing funding for the NHS that they spent on tax cuts, ending the junior doctor strike and getting waiting times down" "The public's priority is with a better NHS, not with the level of taxation"

Farrukh

675,773 Aufrufe • vor 2 Jahren

Using the straw on the floor of their solitary cell, three Vietnamese nuns wove a Nativity scene to give to the guards of the communist prison at Christmas. They did the same with the Crucifixion on Good Friday, and the guards asked, “Is that the same person in the crib and on the cross? Is that the same mother at the crib and the foot of the cross?” The Good Newsroom

Using the straw on the floor of their solitary cell, three Vietnamese nuns wove a Nativity scene to give to the guards of the communist prison at Christmas. They did the same with the Crucifixion on Good Friday, and the guards asked, “Is that the same person in the crib and on the cross? Is that the same mother at the crib and the foot of the cross?” The Good Newsroom

Cardinal Dolan

36,446 Aufrufe • vor 9 Monaten

Chaos and the double pendulum: the configuration space of the double pendulum is a torus (a circle for each angle coordinate) so we can plot the evolution of such a pendulum in two ways: either as a "real pendulum" in physical space, or as a point in configuration space :)

Chaos and the double pendulum: the configuration space of the double pendulum is a torus (a circle for each angle coordinate) so we can plot the evolution of such a pendulum in two ways: either as a "real pendulum" in physical space, or as a point in configuration space :)

Steve Trettel

149,856 Aufrufe • vor 2 Jahren

String Theory is one of the few frameworks that tries to describe Quantum Physics and Gravity in the same mathematical language. In this animation, the glowing loop is the string, the shifting color along it represent its quantum vibration modes, and the surrounding fabric is a proxy for Spacetime responding. When you quantize a string, one of its vibration modes behaves like a graviton. So, geometry shows up as part of the same quantum system. #StringTheory #QuantumGravity #GeneralRelativity #QuantumPhysics #TheoreticalPhysics #PhysicsVisualization

String Theory is one of the few frameworks that tries to describe Quantum Physics and Gravity in the same mathematical language. In this animation, the glowing loop is the string, the shifting color along it represent its quantum vibration modes, and the surrounding fabric is a proxy for Spacetime responding. When you quantize a string, one of its vibration modes behaves like a graviton. So, geometry shows up as part of the same quantum system. #StringTheory #QuantumGravity #GeneralRelativity #QuantumPhysics #TheoreticalPhysics #PhysicsVisualization

Mathelirium

42,077 Aufrufe • vor 4 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

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

20,411 Aufrufe • vor 2 Monaten

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

Mathelirium

54,411 Aufrufe • vor 7 Monaten

The fluid dynamics of our planet are truly haunting. Watching the global thermohaline circulation at work is a reminder that Earth is less of a rock and more of a massive, interconnected heat engine. The way these eddies dissipate kinetic energy across the Atlantic is pure chaos theory in motion. Nature’s Navier-Stokes solutions are way more beautiful than anything on a chalkboard.

The fluid dynamics of our planet are truly haunting. Watching the global thermohaline circulation at work is a reminder that Earth is less of a rock and more of a massive, interconnected heat engine. The way these eddies dissipate kinetic energy across the Atlantic is pure chaos theory in motion. Nature’s Navier-Stokes solutions are way more beautiful than anything on a chalkboard.

Cosmos Archive

32,421 Aufrufe • vor 2 Monaten

As the atomic bomb unleashed its energy upon Hiroshima, it encountered objects on its path. These objects – human, animal, or inanimate – absorbed this energy and the result looked like a shadow, while really it is the surrounding area that has been dyed

As the atomic bomb unleashed its energy upon Hiroshima, it encountered objects on its path. These objects – human, animal, or inanimate – absorbed this energy and the result looked like a shadow, while really it is the surrounding area that has been dyed

Massimo

181,607 Aufrufe • vor 10 Monaten

''Nigeria is currently in the 1920s of the UK and the US power system and that is how far back they are '' -Energy expert, Idowu Oyebanjo #PoliticsToday

''Nigeria is currently in the 1920s of the UK and the US power system and that is how far back they are '' -Energy expert, Idowu Oyebanjo #PoliticsToday

Channels Television

92,477 Aufrufe • vor 2 Jahren