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This is Lithium, visualized as a quantum wavepacket. The bright structures are 3D isosurfaces of the outer electron’s probability density, while colour represents the complex phase. Unlike Hydrogen, Lithium’s valence electron moves in a screened potential created by the nucleus and the closed 1s² core. The animation solves that... show more

Mathelirium

82,284 subscribers

176,633 просмотров • 1 месяц назад •via X (Twitter)

Искусство Образование Наука и технологии

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Комментарии: 31

Фото профиля Tobie Venne

Tobie Venne1 месяц назад

Lithium... top tier nucleator.. Why do you think there was over 200kg of it on the MH370 flight....

Фото профиля Larry Panozzo

Larry Panozzo1 месяц назад

Reality is wild, and most of it is invisible to us

Фото профиля Pode

Pode1 месяц назад

Neat. Now do the nucleus 😜

Фото профиля Janne Särkelä

Janne Särkelä1 месяц назад

@DanielleFong The real disturbance in the fo… field

Фото профиля M

M1 месяц назад

how much of this visualization is possible from tying down the pde to random timevalues

Фото профиля drdonhen

drdonhen1 месяц назад

cf (from first principles):

Фото профиля Mythic: CEO of Gnoticing

Mythic: CEO of Gnoticing1 месяц назад

"I know Gravity is Electromagnetic, but I can't prove it yet..."

Фото профиля Stone Kaiju ᯅ ⚡ ³³º¹

Stone Kaiju ᯅ ⚡ ³³º¹1 месяц назад

Trace amounts may be a cure for Alzheimer’s Lithium orate has been used successfully in tests with rats as this form is able to pass through the blood brain barrier. Study Reference: Published in Nature (August 6, 2025) by Dr. Bruce Yankner’s lab at Harvard Medical School (Title: Lithium deficiency drives Alzheimer’s disease pathology and cognitive decline). Key Findings: Mass spectrometry of human post-mortem brains revealed selective early depletion of trace lithium in the prefrontal cortex during MCI. Pathological Aβ plaques act as a physical sink, sequestering free ionic lithium. Mechanism: Lithium depletion disinhibits GSK3β, driving tau phosphorylation, microglial dysfunction, and synaptic loss. Pharmacokinetics: Among 16 tested salts, low-dose lithium orotate bypassed plaque sequestration (unlike lithium carbonate) and restored cognitive function in mouse models at roughly 1/1000th of psychiatric dosages. Clinical Status: Human pilot and biomarker trials (e.g., Johns Hopkins) are currently underway. Patients/family should be advised against OTC self-medication due to unverified human safety dosing, unregulated supplement purity, and potential renal/thyroid monitoring needs.

Фото профиля Chad DiGirolamo

Chad DiGirolamo1 месяц назад

No wonder our “science” is jacked

Фото профиля Farmer Skot

Farmer Skot1 месяц назад

Incredible work.

Фото профиля stunspot | ⟨🤩⨯📍⟩ |

stunspot | ⟨🤩⨯📍⟩ |1 месяц назад

Now: Why the FUCK is there so MUCH of the damned stuff all over the place?!?!

Фото профиля LaserEyeRobot

LaserEyeRobot1 месяц назад

I knew it!

Фото профиля N.A. Cottrell

N.A. Cottrell1 месяц назад

matter

Фото профиля fFaceCharli and Mr. E present YOUR DOOM, or 4given

fFaceCharli and Mr. E present YOUR DOOM, or 4given1 месяц назад

#farout , man.

Фото профиля Astro 🏴

Astro 🏴1 месяц назад

Brother, no need for digital animations of atoms and elementary particles. The ancient hidden mountain tribe “Afareet Al Jibaal” in modern day Iraq/Iran have a custom blend of components that enables directed higher dimensional journeys. But it’s not for the faint of heart 🤣

Фото профиля Pai Mei

Pai Mei1 месяц назад

Dude that’s a jawbreaker.

Фото профиля Dale Cloudman

Dale Cloudman1 месяц назад

Why does anyone refer to electrons as particles anymore... it is just not a particle.

Фото профиля Mantis Shrimp

Mantis Shrimp1 месяц назад

Beyond physical intuition. So weird.

Фото профиля Frankenbyte

Frankenbyte1 месяц назад

Really nice!🌟

Фото профиля wuchengying

wuchengying1 месяц назад

alive

Фото профиля Jeff

Jeff1 месяц назад

115 please 🙂

Фото профиля Avisek kumar

Avisek kumar1 месяц назад

Hi , I wanted to know if we can extend this to cluster of atoms to see the quantum fluctuations of the larger body

Фото профиля MathPolice

MathPolice1 месяц назад

Stunning and elucidating as usual. Tangentially related:

Фото профиля Mauro

Mauro1 месяц назад

Adore the math behind it

Фото профиля CD

CD1 месяц назад

Beating like a living organ. Does this mean that matter is alive ?

Фото профиля BROOKLYDAMUS

BROOKLYDAMUS1 месяц назад

Closed eye visuals

Фото профиля FaraOne

FaraOne1 месяц назад

Amazing!

Фото профиля Dale Cloudman

Dale Cloudman1 месяц назад

And they call it 'inert' matter

Фото профиля Jose Angel Solorzano Luna

Jose Angel Solorzano Luna1 месяц назад

@grok I love to eat edible lithium lollipops as a quantum wavepacket

Фото профиля Alchemy

Alchemy1 месяц назад

Is this real or simulation image @grok ?

Фото профиля winter ❄️

winter ❄️1 месяц назад

boundary conditions

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Mathelirium

30,529 просмотров • 7 месяцев назад

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁ + a₂) = |a₁|² + |a₂|² + a₁ a₂ + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂ = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Mathelirium

81,784 просмотров • 10 месяцев назад

We evolve a complex quantum field ψ(x,y,t) with a Schrödinger-Poisson flow: i ψ_t = -(1/2)Δψ + (V_conf(x,y) + G Φ(x,y,t)) ψ, with ρ=|ψ|² and ΔΦ = ρ - ⟨ρ⟩. That bracket (V_conf + GΦ) is just the potential energy landscape. V_conf is the trap you choose by hand (a smooth bowl that keeps the wave from drifting away). Φ is the field generated by the wave’s own density...compute ρ, solve the Poisson equation, feed Φ back into the evolution. G is the knob that sets how strongly the wave feels its own Φ. Visually Hue is phase φ=arg(ψ), brightness is density ρ^γ, the banded helicoids are phase ribbons cos(mφ+θ(t)), the bright ridges are boosted by |∇ρ|, and the tiny white pearls mark vortices where the phase winding satisfies ∮∇φ·dl = 2πk. #QuantumMechanics #SchrodingerPoisson #ComputationalPhysics #PhysicsVisualization

We evolve a complex quantum field ψ(x,y,t) with a Schrödinger-Poisson flow: i ψ_t = -(1/2)Δψ + (V_conf(x,y) + G Φ(x,y,t)) ψ, with ρ=|ψ|² and ΔΦ = ρ - ⟨ρ⟩. That bracket (V_conf + GΦ) is just the potential energy landscape. V_conf is the trap you choose by hand (a smooth bowl that keeps the wave from drifting away). Φ is the field generated by the wave’s own density...compute ρ, solve the Poisson equation, feed Φ back into the evolution. G is the knob that sets how strongly the wave feels its own Φ. Visually Hue is phase φ=arg(ψ), brightness is density ρ^γ, the banded helicoids are phase ribbons cos(mφ+θ(t)), the bright ridges are boosted by |∇ρ|, and the tiny white pearls mark vortices where the phase winding satisfies ∮∇φ·dl = 2πk. #QuantumMechanics #SchrodingerPoisson #ComputationalPhysics #PhysicsVisualization

Mathelirium

18,700 просмотров • 9 месяцев назад

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

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

Mathelirium

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