Quantum Mechanics Series Lecture 3 Lecture 1 earned the right to call ρ(x,t) = |ψ(x,t)|² a probability density by showing that Schrödinger evolution forces a continuity equation, ∂ρ/∂t ∇·j = 0. Lecture 2 then answered the next question: What actually steers that flow? Writing ψ = r exp(iθ) turns the current into density times phase gradient. So, phase geometry literally drives the streamlines. Quantized vortices appear where that phase winds in discrete units. Lecture 3 asks the next natural followup question If probability flows, does the bulk of that flow have a trajectory? Ehrenfest says yes. The centroid of the wavepacket can move almost classically when the packet is tight enough and the potential is smooth enough. In this render, the surface height is |ψ(x,t)| and the surface skin is colored by arg(ψ(x,t)). The porcelain bead rides on the surface at the quantum centroid ⟨x⟩(t). The amber bead tracks a classical centroid evolved under the same V(x,t). The steel arrow points along ⟨p⟩. The glowing threads trace the probability current itself. It is still the same ψ from the earlier lectures. We are just pushing the same objects one step further and asking whether their collective motion begins to look classical. The math breakdown: We treat the state as a complex field ψ(x,t) on the plane, with x in R². The Born rule defines the probability density ρ(x,t) = |ψ(x,t)|² and Schrödinger evolution, in units with ħ = 1, is i ∂ψ/∂t = [ −(1/2m) ∇² V(x,t) ] ψ From Lecture 1 we already have the continuity equation ∂ρ/∂t ∇·j = 0 with probability current j = (1/2mi) ( ψ* ∇ψ − ψ ∇ψ* ) = (1/m) Im(ψ* ∇ψ) From Lecture 2, writing ψ = r exp(iθ) turns this into j = (ρ/m) ∇θ So now we ask another question. What happens to the centroid of that flowing density? Define the position centroid and mean momentum by ⟨x⟩(t) = ∫ x ρ(x,t) dx ⟨p⟩(t) = ∫ ψ*(x,t) [ −i ∇ ] ψ(x,t) dx Ehrenfest’s theorem states d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V(x,t)⟩ The first identity drops straight out of the continuity equation. So it really is built from the same probability-flow picture as the earlier lectures. Start from the centroid: d⟨x⟩/dt = ∫ x (∂ρ/∂t) dx = −∫ x (∇·j) dx Integrate by parts. The boundary term vanishes because the wavefunction decays at the edges. This leaves −∫ x (∇·j) dx = ∫ j dx Hence d⟨x⟩/dt = ∫ j dx Now substitute the expression for j: ∫ j dx = (1/m) ∫ (1/2i)(ψ* ∇ψ − ψ ∇ψ*) dx = (1/m) ∫ ψ*(−i ∇)ψ dx = ⟨p⟩/m So we get d⟨x⟩/dt = ⟨p⟩/m The second identity comes from differentiating ⟨p⟩, inserting Schrödinger’s equation for ∂ψ/∂t and ∂ψ*/∂t, and integrating by parts. The kinetic pieces cancel. The potential contributes the force term: d⟨p⟩/dt = −⟨∇V⟩ Then comes the classical bridge. If the packet is narrow and the potential does not change much across it, the mean force is close to the force at the mean position: ⟨∇V⟩ ≈ ∇V(⟨x⟩) so the centroid approximately obeys m d²⟨x⟩/dt² ≈ −∇V(⟨x⟩) That is the point of the animation. It takes the same ingredients from Lectures 1 and 2, namely ρ, j, and phase, and asks whether their collective motion begins to follow Newton’s law. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #Vortices #EhrenfestTheorem #ExpectationValues #MathematicalPhysics #Mathematics #Physics

Things are heating up 🔥 Lecture 4 of our Quantum Mechanics series. Lecture 2 earned the right to call ρ(x,t) = |ψ(x,t)|² a probability density by showing Schrödinger evolution forces a continuity equation, ∂ρ/∂t ∇·j = 0. Lecture 3 then revealed what actually steers that flow...writing ψ = r exp(iθ) makes the current j become "density × phase-gradient," so phase geometry literally draws the streamlines and quantized vortices show up as phase winding defects. Lecture 4 is the next link in that same chain...once you accept that probability flows, you can ask whether the bulk of that flowing density has a trajectory...and Ehrenfest says yes, its centroid moves almost classically when the packet is tight and the potential is smooth. In the 3D panel, the surface height is |ψ(x,t)| and the surface skin is colored by arg(ψ(x,t)). The porcelain bead rides on the surface at the quantum centroid ⟨x⟩(t), the amber bead is a classical centroid integrated under the same V(x,t), and the teal arrow points along ⟨p⟩. In the 2D panel, the background uses the same encoding (phase as color, amplitude as brightness), but now you can see the current field...the glowing ribbons are streamlines of j, and the small charged defects are vortices where the phase winds by ±2π around a low-density core. Both panels are the same ψ, just two ways of seeing the same objects from Lectures 2 and 3. The math breakdown We treat 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 From Lecture 2 we have the continuity equation ∂ρ/∂t ∇·j = 0 with probability current j = (1/2mi) ( ψ* ∇ψ − ψ ∇ψ* ) = (1/m) Im(ψ* ∇ψ) From Lecture 3, writing ψ = r exp(iθ) turns that into j = (ρ/m) ∇θ Now define the centroid and mean momentum: ⟨x⟩(t) = ∫ x ρ(x,t) dx ⟨p⟩(t) = ∫ ψ*(x,t) [ −i ∇ ] ψ(x,t) dx Ehrenfest’s theorem states d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V(x,t)⟩ Derive the first identity directly from the continuity equation (so it visibly depends on Lecture 2). Start from ⟨x⟩: d⟨x⟩/dt = ∫ x (∂ρ/∂t) dx = −∫ x (∇·j) dx Integrate by parts (boundary term vanishes because ρ → 0 at the edges): −∫ x (∇·j) dx = ∫ j dx So d⟨x⟩/dt = ∫ j dx Now substitute j and rewrite using Im(z) = (z − z*)/(2i): ∫ j dx = (1/m) ∫ (1/2i)(ψ* ∇ψ − ψ ∇ψ*) dx = (1/m) ∫ ψ*(−i ∇)ψ dx = ⟨p⟩/m Therefore d⟨x⟩/dt = ⟨p⟩/m The second identity comes from differentiating ⟨p⟩, using Schrödinger’s equation to replace ∂ψ/∂t and ∂ψ*/∂t, and integrating by parts so the kinetic contributions cancel while the potential contributes a force term: d⟨p⟩/dt = −⟨∇V⟩ When V is nearly linear across the wavepacket (narrow packet, smooth potential), the mean force is approximately the force at the mean position: ⟨∇V⟩ ≈ ∇V(⟨x⟩) and you get the Newton-like approximation m d²⟨x⟩/dt² ≈ −∇V(⟨x⟩) That’s what the animation is testing live using the same objects from Lectures 2 and 3...ρ, j, and phase. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #Vortices #EhrenfestTheorem #ExpectationValues #MathematicalPhysics #Mathematics #Physics

Quantum Switching in Resistor Networks #TechRxiv #ExpectationValues #QuantumSwitches #Resistornetworks #Superposition techrxiv.org/articles/prepri…