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Visualization of the hydrogen wave function. Showing how the quantum numbers (n), (l), and (m) determine an electron’s probability distribution, orbital geometry, and nodal structure. The colours represent opposite phases of (ψ), while point concentration represents (|ψ|^2).

Philosophy Of 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

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 просмотров • 6 месяцев назад

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

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 = ψψ. 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 просмотров • 5 месяцев назад

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 просмотров • 1 месяц назад

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

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 просмотров • 6 месяцев назад

Watch hydrogen atom wavefunctions smoothly transition between quantum states (n,l,m) This is the actual probability density of finding the electron; not just static orbitals, but the real quantum dance as it morphs between energy levels.

Watch hydrogen atom wavefunctions smoothly transition between quantum states (n,l,m) This is the actual probability density of finding the electron; not just static orbitals, but the real quantum dance as it morphs between energy levels.

Mathematica

68,910 просмотров • 3 месяцев назад

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 просмотров • 6 месяцев назад

Because it’s basically watching quantum mechanics breathe. This simulation solves the time-dependent Schrödinger equation in a true 1/r Coulomb potential, showing an electron’s probability wave evolve, interfere, and flow in real time, not frozen orbitals.

Because it’s basically watching quantum mechanics breathe. This simulation solves the time-dependent Schrödinger equation in a true 1/r Coulomb potential, showing an electron’s probability wave evolve, interfere, and flow in real time, not frozen orbitals.

Bluntly Put Philosopher (BPP)

34,582 просмотров • 6 месяцев назад

In 1926, Erwin Schrödinger introduced the equation that became the dynamical law of non-relativistic quantum mechanics: iℏ ∂ψ/∂t = [−ℏ²/(2m)∇² + V]ψ It does not describe a particle following one definite path. It describes how the wavefunction evolves and therefore how the probabilities of possible experimental outcomes change with time.

In 1926, Erwin Schrödinger introduced the equation that became the dynamical law of non-relativistic quantum mechanics: iℏ ∂ψ/∂t = [−ℏ²/(2m)∇² + V]ψ It does not describe a particle following one definite path. It describes how the wavefunction evolves and therefore how the probabilities of possible experimental outcomes change with time.

Philosophy Of Physics

123,535 просмотров • 5 дней назад

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 просмотров • 4 месяцев назад

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

26,681 просмотров • 18 дней назад

An inside look at IBM Quantum System Two, showing how qubits, extreme cooling, and quantum physics are shaping the future of computing

An inside look at IBM Quantum System Two, showing how qubits, extreme cooling, and quantum physics are shaping the future of computing

Massimo

37,915 просмотров • 5 месяцев назад

Is an electron a particle or a wave?⁠ ⁠ In this conversation, physicist Sean Carroll explores some of the deepest mysteries in quantum mechanics: the famous double-slit experiment, wave function collapse, the Many Worlds interpretation, entropy and the arrow of time. ⁠ Watch the full interview here:

Is an electron a particle or a wave?⁠ ⁠ In this conversation, physicist Sean Carroll explores some of the deepest mysteries in quantum mechanics: the famous double-slit experiment, wave function collapse, the Many Worlds interpretation, entropy and the arrow of time. ⁠ Watch the full interview here:

New Scientist

17,536 просмотров • 28 дней назад

Bryan Cook highlights starting with some pretty insane range to get outside of the numbers from the opposite hash. Some great processing and anticipation while also showing up as a physical force.

Bryan Cook highlights starting with some pretty insane range to get outside of the numbers from the opposite hash. Some great processing and anticipation while also showing up as a physical force.

mike

20,784 просмотров • 4 месяцев назад

Primes Numbers Are Not Random Noise Prime numbers look scattered when you meet them one by one, but their disorder has structure. In this animation, every integer is placed on a growing spiral, primes ignite as sharp golden events, and the hidden oscillations come from the first Riemann zeta zeros. The scene is based on the von Mangoldt prime signal Λ(n) and the explicit formula for ψ(x), where primes appear as impulses and the zeta zeros behave like frequencies correcting the smooth drift of x. #PrimeNumbers #RiemannHypothesis #NumberTheory #Mathematics #MathAnimation

Primes Numbers Are Not Random Noise Prime numbers look scattered when you meet them one by one, but their disorder has structure. In this animation, every integer is placed on a growing spiral, primes ignite as sharp golden events, and the hidden oscillations come from the first Riemann zeta zeros. The scene is based on the von Mangoldt prime signal Λ(n) and the explicit formula for ψ(x), where primes appear as impulses and the zeta zeros behave like frequencies correcting the smooth drift of x. #PrimeNumbers #RiemannHypothesis #NumberTheory #Mathematics #MathAnimation

Mathelirium

16,018 просмотров • 1 месяц назад

DIVINE MATHEMATICS Number One (1) is the Number of God. Everything is from Number. Number One (1) is the Number of Hydrogen. Your DNA is written in the Divine Mathematics of the Fibonacci Geometric Patterns/Sequences. Your DNA is written in Mathematical Number Patterns. That is why Science is filled with Calculations. Nature is written in the Fibonacci Geometric Patterns/Sequences. You do not need religion to know God. You need to understand the nature of existence from the angle of Science and Mathematics. Galileo Galilei, who deserves to be called the Father of Physics said thus how God created the Universe: "God created the Universe in the language of Mathematics" Everything in the Universe is Governed by Numbers. For this Purpose, Zero or Nothing or the Void is counted, not just as a Number, but as the Source of All Numbers. Zero or Nothing or the Void is mentioned in the Bible as early as Genesis because the Zero or the Void or Nothing is the Basis of All Numbers. In Genesis 1: 1-3 it is written thus: 1. In the Heaven and Earth. 2. And the Earth was without Form and Void; and Darkness was upon the Face of the Deep.... 3. And God said Let there be Light and there was Light. What follows are the 7 Days of Creation. Religion deceives and brainwashed the Masses that these are 7 literal Days. They're really about the Fibonacci Geometric Patterns/Sequences. Numbers govern the Chemical Elements that constitute the Universe, with Hydrogen being Number One to emerge from Zero and is the Same as the Zero with the Zero being the Darkness that was upon the Face of the Great Deep mentioned in Genesis 1:2. Number One became the Light which emerged from the Darkness. Without Darkness, there can be no Light. Everything is essentially a Mirror of the Same Thing. This is what is called Polarity. In other Words, Opposite Things depend on the each other in order to exist. Without Darkness, you cannot know what Light is. This is also represented by Gender, with the Female and the Male being diverse Forms of the Same Thing. The Hermetic Principles explain the relationship between everything in some details. In that regards, I suggest that you should look up the Hermetic Principles in the Kyballion. It is because the Numbers begins from Zero or Nothing or the Void that it is said that God made the Universe from Nothing. As already mentioned Number One is the Number of Hydrogen. The Hydrogen Atom permeates Everything in the Universe. It is because there is Hydrogen that there is a Material Universe. Hydrogen is Number One (1) on the Chemical Periodic Table. There is Nothing in the Material Universe that is not from Hydrogen. Hydrogen is the Basis of all Physical Existence. That is why Number One (1) is also the Number of God. The Sun and the Stars and Planets are All comprised of Hydrogen. Oxygen is constituted by the Thermonuclear Synthesis Hydrogen in the Nuclei of Stars. Hydrogen then COMBUSTS with Hydrogen to produce Dihydrogen Monoxide aka WATER. Your very being is literally powered by Hydrogen. As already indicated, Oxygen first came into Existence through Thermonuclear Fusion of Hydrogen Atoms in Stars. Stars are composed of the Hydrogen. Hydrogen is the Medium of Consciousness and Medium of the Material Universe of which we are an Intrinsic part. All of Nature is powered by Number One (1) aka Hydrogen. You're literally made of Hydrogen that has been transformed in Stars. The Sun, as a Star is composed of Hydrogen Atoms. The Circle which represents Zeros Nothing or the Void or the Darkness and also the Light represents the Hydrogen Atom. That is why the Sun and Moon are Circles. The Circle is the Basic Geometric Pattern. The multiplication of identical Circles create the Patterns and Sequences of the Flower of Life which is in All Indigenous Cultures. ✨🙌🏾💫

DIVINE MATHEMATICS Number One (1) is the Number of God. Everything is from Number. Number One (1) is the Number of Hydrogen. Your DNA is written in the Divine Mathematics of the Fibonacci Geometric Patterns/Sequences. Your DNA is written in Mathematical Number Patterns. That is why Science is filled with Calculations. Nature is written in the Fibonacci Geometric Patterns/Sequences. You do not need religion to know God. You need to understand the nature of existence from the angle of Science and Mathematics. Galileo Galilei, who deserves to be called the Father of Physics said thus how God created the Universe: "God created the Universe in the language of Mathematics" Everything in the Universe is Governed by Numbers. For this Purpose, Zero or Nothing or the Void is counted, not just as a Number, but as the Source of All Numbers. Zero or Nothing or the Void is mentioned in the Bible as early as Genesis because the Zero or the Void or Nothing is the Basis of All Numbers. In Genesis 1: 1-3 it is written thus: 1. In the Heaven and Earth. 2. And the Earth was without Form and Void; and Darkness was upon the Face of the Deep.... 3. And God said Let there be Light and there was Light. What follows are the 7 Days of Creation. Religion deceives and brainwashed the Masses that these are 7 literal Days. They're really about the Fibonacci Geometric Patterns/Sequences. Numbers govern the Chemical Elements that constitute the Universe, with Hydrogen being Number One to emerge from Zero and is the Same as the Zero with the Zero being the Darkness that was upon the Face of the Great Deep mentioned in Genesis 1:2. Number One became the Light which emerged from the Darkness. Without Darkness, there can be no Light. Everything is essentially a Mirror of the Same Thing. This is what is called Polarity. In other Words, Opposite Things depend on the each other in order to exist. Without Darkness, you cannot know what Light is. This is also represented by Gender, with the Female and the Male being diverse Forms of the Same Thing. The Hermetic Principles explain the relationship between everything in some details. In that regards, I suggest that you should look up the Hermetic Principles in the Kyballion. It is because the Numbers begins from Zero or Nothing or the Void that it is said that God made the Universe from Nothing. As already mentioned Number One is the Number of Hydrogen. The Hydrogen Atom permeates Everything in the Universe. It is because there is Hydrogen that there is a Material Universe. Hydrogen is Number One (1) on the Chemical Periodic Table. There is Nothing in the Material Universe that is not from Hydrogen. Hydrogen is the Basis of all Physical Existence. That is why Number One (1) is also the Number of God. The Sun and the Stars and Planets are All comprised of Hydrogen. Oxygen is constituted by the Thermonuclear Synthesis Hydrogen in the Nuclei of Stars. Hydrogen then COMBUSTS with Hydrogen to produce Dihydrogen Monoxide aka WATER. Your very being is literally powered by Hydrogen. As already indicated, Oxygen first came into Existence through Thermonuclear Fusion of Hydrogen Atoms in Stars. Stars are composed of the Hydrogen. Hydrogen is the Medium of Consciousness and Medium of the Material Universe of which we are an Intrinsic part. All of Nature is powered by Number One (1) aka Hydrogen. You're literally made of Hydrogen that has been transformed in Stars. The Sun, as a Star is composed of Hydrogen Atoms. The Circle which represents Zeros Nothing or the Void or the Darkness and also the Light represents the Hydrogen Atom. That is why the Sun and Moon are Circles. The Circle is the Basic Geometric Pattern. The multiplication of identical Circles create the Patterns and Sequences of the Flower of Life which is in All Indigenous Cultures. ✨🙌🏾💫

🧬Maxpein🧬

25,608 просмотров • 1 год назад

🎆 S U N D E R L A N D 🌉 Following the opening of Keel Crossing and the end of England Women’s Rugby World Cup match at the Stadium of Light. People gathered in their numbers to watch an incredible firework display across Sunderland Riverside.

🎆 S U N D E R L A N D 🌉 Following the opening of Keel Crossing and the end of England Women’s Rugby World Cup match at the Stadium of Light. People gathered in their numbers to watch an incredible firework display across Sunderland Riverside.

RJX.MEDIA 📸

12,545 просмотров • 11 месяцев назад

“The amount of points you score doesn’t determine whether you are playing good or bad. Your point total is never an indicator of how you play. It’s how hard you work and how good of a teammate you are,” Caitlin Clark A true difference maker on and off the court.

“The amount of points you score doesn’t determine whether you are playing good or bad. Your point total is never an indicator of how you play. It’s how hard you work and how good of a teammate you are,” Caitlin Clark A true difference maker on and off the court.

The Winning Difference

570,410 просмотров • 2 лет назад

Prion-like amyloid microclots may now be circulating in the majority of human beings as a direct result of mass mRNA injection programs and SARS-CoV-2 gain-of-function experiments. The CDC must do its job and determine just how many people are harboring these amyloids.

Prion-like amyloid microclots may now be circulating in the majority of human beings as a direct result of mass mRNA injection programs and SARS-CoV-2 gain-of-function experiments. The CDC must do its job and determine just how many people are harboring these amyloids.

Nicolas Hulscher, MPH

25,343 просмотров • 8 месяцев назад

This is an important point that I think every AI company CEO needs to listen to The internet promised consumer surplus through access to free platforms and distribution AI is the exact opposite. The messaging has been about putting people out of jobs

This is an important point that I think every AI company CEO needs to listen to The internet promised consumer surplus through access to free platforms and distribution AI is the exact opposite. The messaging has been about putting people out of jobs

Boring_Business

81,496 просмотров • 2 месяцев назад

Footage has been released showing last night's Russian Iskander-M ballistic missile strike on eastern Dnipropetrovsk Oblast. 1 missile (not 2) targeted a concentration of Ukrainian manpower and vehicles near the village of Troitske, just east of Petropavlivka.

Footage has been released showing last night's Russian Iskander-M ballistic missile strike on eastern Dnipropetrovsk Oblast. 1 missile (not 2) targeted a concentration of Ukrainian manpower and vehicles near the village of Troitske, just east of Petropavlivka.

AMK Mapping 🇳🇿

49,228 просмотров • 6 месяцев назад