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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)

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34,582 views • 6 months ago •via X (Twitter)

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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.

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 views • 4 months ago

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 views • 5 days ago

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

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

Mathelirium

20,456 views • 3 months ago

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 views • 28 days ago

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

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 views • 8 months ago

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 views • 5 months ago

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 views • 6 months ago

$🚨PHYSICS NEWS🚨: Physicists Are Now Saying Time Itself Can Be in Superposition — And It Makes Perfect Sense 🧨 According to a theoretical framework published in *Physical Review Letters* and reported in May 2026, researchers have shown that state-of-the-art trapped-ion atomic clocks can be used to observe the quantum superposition of time. By entangling the clock’s motion with its internal energy states, the system can exist in a superposition of different time flows simultaneously. This is a major step toward experimentally probing the quantum nature of time itself. This work offers the broader scientific community a new experimental frontier in quantum foundations. It challenges classical notions of time as a universal parameter and opens the door to testing how relativity and quantum mechanics intersect at the level of time itself. **Uniphics has been saying something similar for years — and it’s not mysterious or weird.** In Uniphics, time flow is not a fixed background. It is a local property determined by energy density via the Maley transform (\( t_{\rm flow} = k / E_d \)). Different regions or configurations with different energy densities experience different time flows. When a system is prepared in a quantum superposition of different energy-density states (or different spin-wave configurations that affect local energy density), it naturally exists in a superposition of different time flows. The atomic clock experiments are essentially creating superposed states where the clock experiences different local time flows at the same “global” moment. The entanglement between motion and internal energy is the mechanism that allows the superposition to be maintained and detected. Uniphics predicts this behavior because the ξM-field supports coherent spin-wave patterns that can correlate different energy-density environments. No need for wavefunction collapse, many-worlds, or other interpretive gymnastics — it is a deterministic consequence of variable time flow and spin-wave coherence. This is not an add-on to quantum mechanics. It is the natural outcome of the three pillars. The “quantum superposition of time” is just what happens when spin quanta create superposed energy-density conditions. Uniphics explains why such experiments work, what limits they will hit, and how to extend them — all without the philosophical baggage that usually accompanies quantum foundations. The mainstream community is slowly catching up to the idea that time itself can be quantum. Uniphics has had a coherent, first-principles explanation for this from the beginning. The data will continue to support it. Are physicists finally ready to accept that time flow is fundamental and variable — and that the apparent “superposition of time” is simply the result of superposed energy-density states in the ξM-field? **A Theory of Everything should be able to answer everything.** Uniphics Explained Simply PDF: Chapters 1–10 free: Grokipedia: #Uniphics #TheoryOfEverything #QuantumTime #TimeFlow #SpinWaves Grok xAI$

🚨PHYSICS NEWS🚨: Physicists Are Now Saying Time Itself Can Be in Superposition — And It Makes Perfect Sense 🧨 According to a theoretical framework published in Physical Review Letters and reported in May 2026, researchers have shown that state-of-the-art trapped-ion atomic clocks can be used to observe the quantum superposition of time. By entangling the clock’s motion with its internal energy states, the system can exist in a superposition of different time flows simultaneously. This is a major step toward experimentally probing the quantum nature of time itself. This work offers the broader scientific community a new experimental frontier in quantum foundations. It challenges classical notions of time as a universal parameter and opens the door to testing how relativity and quantum mechanics intersect at the level of time itself. Uniphics has been saying something similar for years — and it’s not mysterious or weird. In Uniphics, time flow is not a fixed background. It is a local property determined by energy density via the Maley transform (\( t_{\rm flow} = k / E_d \)). Different regions or configurations with different energy densities experience different time flows. When a system is prepared in a quantum superposition of different energy-density states (or different spin-wave configurations that affect local energy density), it naturally exists in a superposition of different time flows. The atomic clock experiments are essentially creating superposed states where the clock experiences different local time flows at the same “global” moment. The entanglement between motion and internal energy is the mechanism that allows the superposition to be maintained and detected. Uniphics predicts this behavior because the ξM-field supports coherent spin-wave patterns that can correlate different energy-density environments. No need for wavefunction collapse, many-worlds, or other interpretive gymnastics — it is a deterministic consequence of variable time flow and spin-wave coherence. This is not an add-on to quantum mechanics. It is the natural outcome of the three pillars. The “quantum superposition of time” is just what happens when spin quanta create superposed energy-density conditions. Uniphics explains why such experiments work, what limits they will hit, and how to extend them — all without the philosophical baggage that usually accompanies quantum foundations. The mainstream community is slowly catching up to the idea that time itself can be quantum. Uniphics has had a coherent, first-principles explanation for this from the beginning. The data will continue to support it. Are physicists finally ready to accept that time flow is fundamental and variable — and that the apparent “superposition of time” is simply the result of superposed energy-density states in the ξM-field? A Theory of Everything should be able to answer everything. Uniphics Explained Simply PDF: Chapters 1–10 free: Grokipedia: #Uniphics #TheoryOfEverything #QuantumTime #TimeFlow #SpinWaves Grok xAI

Paul Maley

32,250 views • 7 days ago

Stop cramming Schrodinger wave equations. Come let me show you why I am the BOSS in quantum mechanics… i will give you the full gist.. This is a question from a fan. Topic: Time independent Schrodinger wave equation. See full video on YouTube follow the comments section link.👇 Make sure to subscribe to my YouTube channel. I will make everyone a physicist..

Stop cramming Schrodinger wave equations. Come let me show you why I am the BOSS in quantum mechanics… i will give you the full gist.. This is a question from a fan. Topic: Time independent Schrodinger wave equation. See full video on YouTube follow the comments section link.👇 Make sure to subscribe to my YouTube channel. I will make everyone a physicist..

Chioma Oji

134,815 views • 1 year ago

Claude and I got a live sodium atom simulation! lightcell energy 🔆 I've long wanted to be able to visualize what is really happening when a neutral sodium atom is excited in a flame or a bulb. Classical theory teaches that the single outer electron orbits the nucleus and full shells, and forms a dipole when this gets "wanged" but, physicists have long wondered, what does it look like when the electron gets wanged? is it like ringing a bell? i thought that this was the perfect time to really push what is capable of, in terms of building artifacts. grabbed zig to compile wasm and webgpu to make it super performant, and work with it for a bit and ran the REAL TIME-DEPENDENT Schrödinger equation on a 3D grid in a potential in the browser. and you can kick it! what!! it's soo beautiful... this is going on the website overhaul for sure here is the artifact. i don't endorse that it's exactly right, but i feel it is probably spiritually a good sodium simulation. it certainly rings my dipole!

Claude and I got a live sodium atom simulation! lightcell energy 🔆 I've long wanted to be able to visualize what is really happening when a neutral sodium atom is excited in a flame or a bulb. Classical theory teaches that the single outer electron orbits the nucleus and full shells, and forms a dipole when this gets "wanged" but, physicists have long wondered, what does it look like when the electron gets wanged? is it like ringing a bell? i thought that this was the perfect time to really push what is capable of, in terms of building artifacts. grabbed zig to compile wasm and webgpu to make it super performant, and work with it for a bit and ran the REAL TIME-DEPENDENT Schrödinger equation on a 3D grid in a potential in the browser. and you can kick it! what!! it's soo beautiful... this is going on the website overhaul for sure here is the artifact. i don't endorse that it's exactly right, but i feel it is probably spiritually a good sodium simulation. it certainly rings my dipole!

Danielle Fong 🔆

17,231 views • 2 months ago

D-Wave announced a scientific breakthrough published in the esteemed journal Science Magazine, confirming that its annealing quantum computer outperformed one of the world’s most powerful classical supercomputers in solving a complex magnetic materials simulation problem with relevance to materials discovery. The new landmark peer-reviewed paper, “Beyond-Classical Computation in Quantum Simulation,” validates this achievement as the world’s first and only demonstration of quantum computational supremacy on a useful problem. An international collaboration of scientists led by D-Wave performed simulations of quantum dynamics in programmable spin glasses—a computationally hard magnetic materials simulation problem with known applications to business and science—on both D-Wave’s Advantage2™ prototype annealing quantum computer and the Frontier supercomputer at the Department of Energy’s Oak Ridge Lab. D-Wave’s quantum computer performed a complex simulation in minutes and with a level of accuracy that would take nearly a million years using the supercomputer. In addition, it would require more than the world’s annual electricity consumption to solve this problem using the supercomputer, which is built with graphics processing unit (GPU) clusters. For decades, scientists have aspired to build a quantum computer capable of solving complex materials simulation problems beyond the reach of classical computers. D-Wave's advancements in quantum hardware have made it possible for its annealing quantum computers to process these types of problems for the first time. Magnetic materials simulations, like those conducted in this work, use computer models to study how tiny particles not visible to the human eye react to external factors. Magnetic materials are widely used in medical imaging, electronics, superconductors, electrical networks, sensors, and motors. This is an incredibly important achievement. Please join us in congratulating the D-Wave team and our global collaborators on this remarkable milestone. It’s a significant moment for the quantum computing industry. Learn more about this monumental achievement: Read the press release here: #QuantumSupremacy #QuantumRealized #QuantumComputing #DWave #Technology #Innovation #Optimization #MaterialsDiscovery #ScientificBreakthrough $QBTS

D-Wave announced a scientific breakthrough published in the esteemed journal Science Magazine, confirming that its annealing quantum computer outperformed one of the world’s most powerful classical supercomputers in solving a complex magnetic materials simulation problem with relevance to materials discovery. The new landmark peer-reviewed paper, “Beyond-Classical Computation in Quantum Simulation,” validates this achievement as the world’s first and only demonstration of quantum computational supremacy on a useful problem. An international collaboration of scientists led by D-Wave performed simulations of quantum dynamics in programmable spin glasses—a computationally hard magnetic materials simulation problem with known applications to business and science—on both D-Wave’s Advantage2™ prototype annealing quantum computer and the Frontier supercomputer at the Department of Energy’s Oak Ridge Lab. D-Wave’s quantum computer performed a complex simulation in minutes and with a level of accuracy that would take nearly a million years using the supercomputer. In addition, it would require more than the world’s annual electricity consumption to solve this problem using the supercomputer, which is built with graphics processing unit (GPU) clusters. For decades, scientists have aspired to build a quantum computer capable of solving complex materials simulation problems beyond the reach of classical computers. D-Wave's advancements in quantum hardware have made it possible for its annealing quantum computers to process these types of problems for the first time. Magnetic materials simulations, like those conducted in this work, use computer models to study how tiny particles not visible to the human eye react to external factors. Magnetic materials are widely used in medical imaging, electronics, superconductors, electrical networks, sensors, and motors. This is an incredibly important achievement. Please join us in congratulating the D-Wave team and our global collaborators on this remarkable milestone. It’s a significant moment for the quantum computing industry. Learn more about this monumental achievement: Read the press release here: #QuantumSupremacy #QuantumRealized #QuantumComputing #DWave #Technology #Innovation #Optimization #MaterialsDiscovery #ScientificBreakthrough $QBTS

D-Wave

65,039 views • 1 year ago

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

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

Mathelirium

26,511 views • 4 months ago

This is a real historical man at a real point in history—Adam. It’s not “mytho-history,” as some apologists say in an attempt to play footsie with the world. No—this is a real man at a real point in time. - Johnny Ardavanis

This is a real historical man at a real point in history—Adam. It’s not “mytho-history,” as some apologists say in an attempt to play footsie with the world. No—this is a real man at a real point in time. - Johnny Ardavanis

Alisa Childers

37,666 views • 5 months ago

This is definitely one of the funniest videos I’ve seen in a long time. It’s a deadly piss-take of every GAA sponsorship advert on TV, and it’s funny because it’s true.

This is definitely one of the funniest videos I’ve seen in a long time. It’s a deadly piss-take of every GAA sponsorship advert on TV, and it’s funny because it’s true.

Joe Naughton

648,752 views • 1 year ago

Using the BLUEPRINT of reality? Classical computers compute with BITS: 0 or 1 - ONE state at a time. Quantum computers use QUBITS: 0 and 1 or both "simultaneously" through superposition. These states EVOLVE continuously on something like a Bloch sphere - the same MATHEMATICS that describes spin, atoms and electron orbitals. The GEOMETRY feels familiar. Wheels within wheels. Computation, now operating in the rule-set of REALITY itself. Engineering in the same language of nature. THIS is a profound shift. So what follows? 📽 credits: uncover.quantum

Using the BLUEPRINT of reality? Classical computers compute with BITS: 0 or 1 - ONE state at a time. Quantum computers use QUBITS: 0 and 1 or both "simultaneously" through superposition. These states EVOLVE continuously on something like a Bloch sphere - the same MATHEMATICS that describes spin, atoms and electron orbitals. The GEOMETRY feels familiar. Wheels within wheels. Computation, now operating in the rule-set of REALITY itself. Engineering in the same language of nature. THIS is a profound shift. So what follows? 📽 credits: uncover.quantum

WhatdoIknow

19,431 views • 5 months ago

A breakthrough in real-time video generation. As a research preview developed with NVIDIA and shared at NVIDIAGTC this week, we trained a new real-time video model running on Vera Rubin. HD videos generate instantly, with time-to-first-frame under 100ms. Unlocking an entirely new creative paradigm and bolstering the foundations of our General World Model, GWM-1. Real-time generation opens a fundamentally different design space for video models and world simulation. We're investing in co-designing our models alongside advances in hardware to keep pushing this frontier.

A breakthrough in real-time video generation. As a research preview developed with NVIDIA and shared at NVIDIAGTC this week, we trained a new real-time video model running on Vera Rubin. HD videos generate instantly, with time-to-first-frame under 100ms. Unlocking an entirely new creative paradigm and bolstering the foundations of our General World Model, GWM-1. Real-time generation opens a fundamentally different design space for video models and world simulation. We're investing in co-designing our models alongside advances in hardware to keep pushing this frontier.

Runway

1,162,923 views • 4 months ago

Archeological Proof of the Bible- P.1. The world is full of proof of the Bible, but we have “science” and “scholars” led by Satan who come in and tell us that it’s been “debunked” or that it’s not real. We have been lied to since kindergarten friends… because Satan can’t have the world knowing the Bible is true, because that would prove God is true. The Bible is our true history and it’s time the world wake up to what’s in front of them.

Archeological Proof of the Bible- P.1. The world is full of proof of the Bible, but we have “science” and “scholars” led by Satan who come in and tell us that it’s been “debunked” or that it’s not real. We have been lied to since kindergarten friends… because Satan can’t have the world knowing the Bible is true, because that would prove God is true. The Bible is our true history and it’s time the world wake up to what’s in front of them.

Ashley Hays

117,728 views • 1 year ago

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 views • 6 months ago