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We now visualize the complex phase of our earlier self-gravitating quantum fluid twisting into smooth helicoids. Think of phase as the wave's "angle" at each point...mapping it to color lets you see how wavefronts curl and shear. Brightness tracks density. The faint moving bands are phase ribbons...contours of equal... show more

Mathelirium

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Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Mathelirium

30,340 Aufrufe • vor 3 Monaten

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.

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 3 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. 🤩🤗🫠

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 10 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

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 = ψψ: ∂ρ/∂t = ψ (∂ψ/∂t) + ψ (∂ψ/∂t) Use Schrödinger and its complex conjugate: ∂ψ/∂t = (1/i) [ −(1/2m) ∇²ψ + Vψ ] ∂ψ/∂t = (−1/i) [ −(1/2m) ∇²ψ* + Vψ* ] Substitute. The V terms cancel, and the remaining terms rearrange into the continuity equation ∂ρ/∂t + ∇·j = 0 with probability current j = (1/2mi) ( ψ* ∇ψ − ψ ∇ψ* ) = (1/m) Im(ψ* ∇ψ) So "probability density" really behaves like a conserved fluid density with flux j. Now expose the phase mechanism. Write ψ in polar form ψ(x,t) = r(x,t) exp(i θ(x,t)) Compute the gradient ∇ψ = exp(iθ) (∇r + i r ∇θ) Then ψ* ∇ψ = r (∇r + i r ∇θ) Taking the imaginary part gives Im(ψ* ∇ψ) = r² ∇θ = ρ ∇θ So the current becomes j = (ρ/m) ∇θ That’s the steering-wheel statement: Phase gradient sets the flow direction and speed (modulated by density and m). Finally, quantized vortices. Because ψ must be single-valued, going around any closed loop must return the same complex value. That forces the phase winding to be an integer multiple of 2π: ∮ ∇θ · dl = 2π n with n in Z n is the vortex charge. Vortex cores sit where ρ ≈ 0 (phase is undefined), and the current streamlines circulate around them. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #Vortices #TopologicalDefects #ComplexAnalysis #MathematicalPhysics #Mathematics #Physics

Mathelirium

37,998 Aufrufe • vor 6 Monaten

Erwin Schrödinger’s 1926 equation turned quantum mechanics into wave dynamics. Instead of tracking a particle by a sharp trajectory, you evolve a complex wavefunction whose shape and phase carry the physics.

Erwin Schrödinger’s 1926 equation turned quantum mechanics into wave dynamics. Instead of tracking a particle by a sharp trajectory, you evolve a complex wavefunction whose shape and phase carry the physics.

Mathelirium

167,993 Aufrufe • vor 4 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

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 6 Monaten

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

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

35,744 Aufrufe • vor 1 Monat

Factory CEO Matan Grinberg says all companies go through three phases of AI adoption: "Phase Zero is reluctance to adopt because developers have had the same workflows for the last 20 years and they might not want to change it." "Phase One is throw the kitchen sink and use as much AI as possible. Whatever you do, just change your workflows." "Phase Two is that people are now adopting the tech, they're proving that they can change their behavior, and use these tools. But now we need to make sure we're actually doing it efficiently and effectively. Uber and Meta are examples of this." "Yes, you'll overspend in Phase One, but the point is to get to Phase Two where then you're actually really efficient on a per-token basis, moving the needle for the business and delivering software faster."

Factory CEO Matan Grinberg says all companies go through three phases of AI adoption: "Phase Zero is reluctance to adopt because developers have had the same workflows for the last 20 years and they might not want to change it." "Phase One is throw the kitchen sink and use as much AI as possible. Whatever you do, just change your workflows." "Phase Two is that people are now adopting the tech, they're proving that they can change their behavior, and use these tools. But now we need to make sure we're actually doing it efficiently and effectively. Uber and Meta are examples of this." "Yes, you'll overspend in Phase One, but the point is to get to Phase Two where then you're actually really efficient on a per-token basis, moving the needle for the business and delivering software faster."

TBPN

25,747 Aufrufe • vor 3 Monaten

You're watching a Quadratic-Differential Teichmüller flow unfold. We drive the scene with a time-varying quadratic differential built from moving zeros and poles. Its log-magnitude sets the color, and its phase defines the horizontal foliation (the direction is half the phase). Then a Teichmüller stretch biases things along vs. against that foliation...so the surface deforms in a geodesic-like way. 🤩🤗🫠

You're watching a Quadratic-Differential Teichmüller flow unfold. We drive the scene with a time-varying quadratic differential built from moving zeros and poles. Its log-magnitude sets the color, and its phase defines the horizontal foliation (the direction is half the phase). Then a Teichmüller stretch biases things along vs. against that foliation...so the surface deforms in a geodesic-like way. 🤩🤗🫠

Mathelirium

11,938 Aufrufe • vor 8 Monaten

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍

Mathelirium

29,194 Aufrufe • vor 9 Monaten

🚨 John Solomon says that, based on what he’s seen, we are in Phase 2 of 5—just the beginning—as we continue to look into blue areas in states pertaining to election fraud. “This is phase two—where elections are administered the way the law says they’re supposed to be, and as a result the FBI may come in and prosecute people.”

🚨 John Solomon says that, based on what he’s seen, we are in Phase 2 of 5—just the beginning—as we continue to look into blue areas in states pertaining to election fraud. “This is phase two—where elections are administered the way the law says they’re supposed to be, and as a result the FBI may come in and prosecute people.”

MJTruthUltra

125,333 Aufrufe • vor 4 Monaten

President Trump and Israeli Prime Minister Benjamin Netanyahu are meeting at the president's Mar-a-Lago estate in Palm Beach, Florida, Monday, as the Trump administration is intent on moving into the second phase of the Gaza peace plan. Asked how soon the second phase of the peace plan could begin, Mr. Trump, standing alongside Netanyahu, said "as quickly as we can" but "there has to be a disarmament" of Hamas.

President Trump and Israeli Prime Minister Benjamin Netanyahu are meeting at the president's Mar-a-Lago estate in Palm Beach, Florida, Monday, as the Trump administration is intent on moving into the second phase of the Gaza peace plan. Asked how soon the second phase of the peace plan could begin, Mr. Trump, standing alongside Netanyahu, said "as quickly as we can" but "there has to be a disarmament" of Hamas.

CBS News

17,021 Aufrufe • vor 6 Monaten

Aharonov-Bohm Flux Braider A Quantum wave splits around a hidden magnetic flux tube, then recombines with a phase difference that cannot be read from the local magnetic field alone. The core is blocked, the field is trapped near the centre, yet the vector potential A = α(-y, x)/(x² + y² + a²) still changes the phase picked up by paths winding around it. The animation follows i∂ₜψ = 1/2(-i∇ - A)²ψ + Vψ inside an annular interferometer. The density ρ = |ψ|² shows where the wave lives, while the smooth threads trace the probability current j = Im(ψ*∇ψ) - Aρ. Those braided paths are the visual signature of the Aharonov-Bohm effect where the wave feels the hidden flux through phase, not through a classical force. #QuantumMechanics #AharonovBohm #SchrodingerEquation #MathematicalPhysics #QuantumPhase #Wavefunction #PhysicsAnimation #ScientificVisualization #MathematicalArt

Aharonov-Bohm Flux Braider A Quantum wave splits around a hidden magnetic flux tube, then recombines with a phase difference that cannot be read from the local magnetic field alone. The core is blocked, the field is trapped near the centre, yet the vector potential A = α(-y, x)/(x² + y² + a²) still changes the phase picked up by paths winding around it. The animation follows i∂ₜψ = 1/2(-i∇ - A)²ψ + Vψ inside an annular interferometer. The density ρ = |ψ|² shows where the wave lives, while the smooth threads trace the probability current j = Im(ψ*∇ψ) - Aρ. Those braided paths are the visual signature of the Aharonov-Bohm effect where the wave feels the hidden flux through phase, not through a classical force. #QuantumMechanics #AharonovBohm #SchrodingerEquation #MathematicalPhysics #QuantumPhase #Wavefunction #PhysicsAnimation #ScientificVisualization #MathematicalArt

Mathelirium

25,867 Aufrufe • vor 7 Tagen

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

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

47,586 Aufrufe • vor 7 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

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

13,635 Aufrufe • vor 5 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

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 1 Monat

Women are now firmly in the “find out” phase of FAFO. I wonder how long it will take and how bad things will have to get before we see the attitude adjustment?

Women are now firmly in the “find out” phase of FAFO. I wonder how long it will take and how bad things will have to get before we see the attitude adjustment?

Rachel Wilson

45,547 Aufrufe • vor 10 Monaten

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

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

81,501 Aufrufe • vor 6 Monaten

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

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

Mathelirium

54,730 Aufrufe • vor 3 Monaten

Much better visually. We now visualize the 3D version of our stunning Blaschke-Quotient Flow from our previous post.🤗🤩 What follows is a mesmerizing break of dawn in the some realm of complex space with zeros and poles still spinning under SU(1,1) Möbius symmetries on the Poincaré disk…but phase is now shaded by the Fubini-Study metric. 🤯🤯🤯

Much better visually. We now visualize the 3D version of our stunning Blaschke-Quotient Flow from our previous post.🤗🤩 What follows is a mesmerizing break of dawn in the some realm of complex space with zeros and poles still spinning under SU(1,1) Möbius symmetries on the Poincaré disk…but phase is now shaded by the Fubini-Study metric. 🤯🤯🤯

Mathelirium

15,826 Aufrufe • vor 9 Monaten