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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.

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Philosophy Of Physics

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

Mathelirium

54,730 просмотров • 5 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
1:09

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

Mathelirium

20,781 просмотров • 6 месяцев 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
0:40

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

Mathelirium

40,835 просмотров • 7 месяцев назад

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.
0:30

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

168,159 просмотров • 5 месяцев назад

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
1:00

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

Mathelirium

20,456 просмотров • 4 месяцев назад

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

Mathelirium

37,998 просмотров • 7 месяцев назад

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

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

Mathelirium

26,511 просмотров • 5 месяцев назад

A Quantum Current Chooses a Path A single electron enters a microscopic interferometer, but its wavefunction travels through two routes at once. One branch crosses a resonant quantum dot while the other moves freely around a ring threaded by magnetic flux. The probability current is j = (ℏ/m) Im(ψ*∇ψ) - (q/m)A|ψ|², while the enclosed flux creates the Aharonov-Bohm phase Δφᴬᴮ = qΦ/ℏ. The quantum dot adds the resonant phase δ(E) = atan2(Γ, E - E₀), producing Fano interference when the two paths recombine. Wave packets split, become temporarily trapped, form circulating current vortices, reverse direction after a flux quench, and finally escape through the drain in a coherent burst. The glowing threads are material curves transported by v = j/ρ, revealing how invisible quantum phase controls the flow. #QuantumMechanics #AharonovBohmEffect #QuantumInterference #QuantumDot #ProbabilityCurrent #MesoscopicPhysics #PhysicsAnimation #Physics
0:56

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

Mathelirium

16,648 просмотров • 1 месяц 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
1:30

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 • 7 месяцев назад

String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime
0:34

String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime

Mathelirium

31,560 просмотров • 4 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
0:40

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 • 2 месяцев назад

This is a visualization of a Rydberg wavepacket. In simple terms it is a way of showing an electron in a very highly excited quantum state. The Math comes from the Schrödinger equation for hydrogen where the electron is described by a wavefunction rather than as a tiny ball moving on a fixed path. The yellow sphere at the centre is the proton, not the electron. The colored 3D surface around it shows places where the electron is more likely to be, i.e. a probability cloud.
0:56

This is a visualization of a Rydberg wavepacket. In simple terms it is a way of showing an electron in a very highly excited quantum state. The Math comes from the Schrödinger equation for hydrogen where the electron is described by a wavefunction rather than as a tiny ball moving on a fixed path. The yellow sphere at the centre is the proton, not the electron. The colored 3D surface around it shows places where the electron is more likely to be, i.e. a probability cloud.

Mathelirium

15,845 просмотров • 1 день назад

Is it time to quantize time? Everything is quantum in quantum mechanics... almost. Time remains a mysterious flowing quantity slotted in to the equations. Surprisingly, there's a model for making time quantum, with a radical consequence: the overall universe is timeless. How is it possible to experience a flow of time in a timeless universe?! Find out in my latest episode of the Quantum Foundations Podcast with Simone Rijavec, Simone Rijavec! We discuss the Page-Wootters model for the emergence of time in quantum theory, and get an insight into Simone's research on the topic. Here's how physicists are solving the problem of time in quantum theory, explained in 60 minutes: 00:00 Coming Up 00:59 Intro 01:37 Simone's journey into quantum 05:04 The problem of time in quantum theory 07:07 Relational approach to time 11:12 Quantum measurement and time 17:53 Stationary universe and the Wheeler-DeWitt equation 30:18 Multiverse & block universe 31:11 How is time flowing? 34:20 Problems with wave-function collapse 36:11 What are clocks? 37:48 Resolution of time-steps 41:47 Debates on the timeless approach 45:08 Interactions with clocks 51:05 Recap 57:39 Making time fully quantum 1:00:43 Behind the Scenes
1:01:22

Is it time to quantize time? Everything is quantum in quantum mechanics... almost. Time remains a mysterious flowing quantity slotted in to the equations. Surprisingly, there's a model for making time quantum, with a radical consequence: the overall universe is timeless. How is it possible to experience a flow of time in a timeless universe?! Find out in my latest episode of the Quantum Foundations Podcast with Simone Rijavec, Simone Rijavec! We discuss the Page-Wootters model for the emergence of time in quantum theory, and get an insight into Simone's research on the topic. Here's how physicists are solving the problem of time in quantum theory, explained in 60 minutes: 00:00 Coming Up 00:59 Intro 01:37 Simone's journey into quantum 05:04 The problem of time in quantum theory 07:07 Relational approach to time 11:12 Quantum measurement and time 17:53 Stationary universe and the Wheeler-DeWitt equation 30:18 Multiverse & block universe 31:11 How is time flowing? 34:20 Problems with wave-function collapse 36:11 What are clocks? 37:48 Resolution of time-steps 41:47 Debates on the timeless approach 45:08 Interactions with clocks 51:05 Recap 57:39 Making time fully quantum 1:00:43 Behind the Scenes

Dr Maria Violaris

63,981 просмотров • 9 месяцев назад

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!
0:40

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 просмотров • 5 месяцев назад

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

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

Mathelirium

10,970 просмотров • 4 месяцев назад

The double-slit experiment is a key demonstration in quantum physics that reveals the strange behavior of particles like electrons and photons. In the experiment, particles are fired at a barrier with two slits, and a screen records where they land. If both slits are open and no measurement is made, the particles form an interference pattern, like waves overlapping, even when sent one at a time. This suggests that each particle passes through both slits at once in a state called superposition. However, if detectors are placed at the slits to observe which path the particle takes, the interference pattern disappears. The particles then behave like classical objects, going through one slit or the other. This change in behavior simply from observing the system is known as the observer effect, and it highlights a fundamental principle of quantum mechanics: measurement affects the outcome. The experiment challenges our understanding of reality, showing that at the quantum level, particles don't have definite states until they are observed 🤓
0:24

The double-slit experiment is a key demonstration in quantum physics that reveals the strange behavior of particles like electrons and photons. In the experiment, particles are fired at a barrier with two slits, and a screen records where they land. If both slits are open and no measurement is made, the particles form an interference pattern, like waves overlapping, even when sent one at a time. This suggests that each particle passes through both slits at once in a state called superposition. However, if detectors are placed at the slits to observe which path the particle takes, the interference pattern disappears. The particles then behave like classical objects, going through one slit or the other. This change in behavior simply from observing the system is known as the observer effect, and it highlights a fundamental principle of quantum mechanics: measurement affects the outcome. The experiment challenges our understanding of reality, showing that at the quantum level, particles don't have definite states until they are observed 🤓

𝚃𝙷𝙴 𝚆𝙷𝙸𝚃𝙴 𝚁𝙰𝙱𝙱𝙸𝚃

51,041 просмотров • 7 месяцев назад

Labour introduced the Hillsborough Law with no idea of how it would affect the intelligence agencies. So Starmer did what he always does. He made the campaigners one promise… … and he made the intelligence agencies another. The result is a complete mess.
5:43

Labour introduced the Hillsborough Law with no idea of how it would affect the intelligence agencies. So Starmer did what he always does. He made the campaigners one promise… … and he made the intelligence agencies another. The result is a complete mess.

Nick Timothy MP

50,905 просмотров • 7 месяцев назад

How does the study of the very smallest objects in our universe shed light on Caleb and MC’s relationship? An analysis of Caleb's Decoherence myth with a focus on quantum mechanics 🔗 #LoveandDeepspace #CalebUltimateWeapon
0:20

How does the study of the very smallest objects in our universe shed light on Caleb and MC’s relationship? An analysis of Caleb's Decoherence myth with a focus on quantum mechanics 🔗 #LoveandDeepspace #CalebUltimateWeapon

natto 🪱🍎

36,250 просмотров • 1 год назад