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What Does a Quantum Particle's Wavefunction Look Like? A quantum wavefunction is not simply an ordinary wave oscillating up and down. It is a complex-valued mathematical function. Its real and imaginary parts can be visualized as two perpendicular oscillations, forming a rotating helix when plotted together. This is a... representation of quantum phase, not the physical shape of a particle. For a free, non-relativistic particle, the dispersion relation is ω(k) = ℏk²/2m. Its wavepacket travels at the group velocity, twice the phase velocity. Because different momentum components travel at different speeds, a localized Gaussian wavepacket spreads over time: σ(t) = σ₀√[1 + (t/τd)²]. This mathematics has experimental foundations. Electron diffraction, C₆₀ fullerene interference and ultracold atomic gases demonstrate the wave nature and evolution of quantum matter. What we observe experimentally are interference patterns and probability distributions, not the wavefunction itself.show more

Philosophy Of Physics

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27,182 views • 3 days ago •via X (Twitter)

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tmallard3 days ago

How two gyroscopic_pair partners fly into each other by spiraling past each other provides the StdModel "gluon"force ... U238 ~~~ From Stearnglass-Einstein's work 70yrs ago not my theory, it quantifies it to near_deterministic results to 9-decimals no probability 🦆

Prranshu Yadav3 days ago

So what do they actually look like then?

David5D3 days ago

non relativistic of course

entropic astrophysical united architecture3 days ago

hey Philosophy of Physics! has anyone told you that you are fake and gay today?

Dr Piyush chaudhary2 days ago

Probably because Quantum particles protect the source material.

Tenotes2 days ago

The output signal of a simple mechanical resolver is precisely this waveform. It enters the ledger like the momentum of a spinning top, yet nothing in the particle is spinning.

O Observador2 days ago

"Probabilidade" e "função de onda complexa" é apenas um artifício estatístico para mascarar o que é puramente mecânico: a hélice de fase é a torção real do meio contínuo, e o espalhamento do pacote de ondas é a dissipação física de tensão de uma cavidade sob pressão.

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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

20,964 views • 9 months ago

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 effective Schrödinger problem and follows a coherent mixture of s- and p-type Rydberg states as they beat, breathe, shift and reform around the core.

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 effective Schrödinger problem and follows a coherent mixture of s- and p-type Rydberg states as they beat, breathe, shift and reform around the core.

Mathelirium

176,789 views • 2 months ago

Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.

Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.

Eron

43,247 views • 2 months ago