Physics · Viv posts

Statics

articleAIPhysicswritten with an agent#0395f15bda

Adam Zettel@adamSep 24, 20261 min read

The tension in each of two ropes holding up a box

T_{1} = \frac{W cos θ _{2}}{sin ( θ _{1} + θ _{2} )}, T_{2} = \frac{W cos θ _{1}}{sin ( θ _{1} + θ _{2} )}, T = \frac{W}{2 sin θ} when both angles are θ

A box of weight W hangs from two ropes that make angles θ₁ and θ₂ with the ceiling. Balancing the sideways pulls and the upward pulls gives the tension in each rope in terms of W and the angles. The surprise is how large it gets: at 30° each rope holds half the weight, 49 N, but its tension is 98 N, because only half of the tension points up; a nearly flat rope needs a tension many times the weight.

A 98 N box hanging from two ropes at 30 degrees

The knot at 30 degrees: each pull split into a sideways and an upward part

The same 98 N box hung two ways; the colored arrows are the ropes' pulls, to scale

Both ropes at 30°

Left rope at 60°, right rope at 30°

both ropes at the same angletension equal to the weight

2 paragraphs · 10 functions · 1 min

✓ 5 tests passchecked when the site was built#statics #forces #tension #free-body-diagram

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Solids

demoAIPhysics #e1882d2329

Adam Zettel@adamSep 25, 2026

Graphene's Dirac cones

E_{\pm} (k) = \pm t j = 1 \sum 3 e^{i k \cdot δ_{j}}, v_{F} = \frac{3 t a}{2ℏ}

An electron hopping between neighboring carbon atoms of a honeycomb has two energy bands, and they touch at the corners of the Brillouin zone. Near a corner the energy rises in a straight line in every direction, a cone, with slope $v_{F} = 3 t a /2ℏ$ . A centered finite difference at the corner gives $1.5$ to eight digits in every direction.

Graphene's two bands, nearest-neighbor tight binding

upper bandlower band

The upper band over the k plane

61 × 61 grid

✓ 6 tests passchecked when the site was built#graphene #tight-binding #band-structure #dirac-cones #fermi-velocity

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Quantum mechanics

demoAIPhysicswritten with an agent#6665325a9c

Adam Zettel@adamSep 25, 2026

Tunneling through one and two barriers by transfer matrices

(A_{out} B_{out}) = M (1 r), t = \frac{det M}{M _{22}}, T + R = 1

An electron meets a barrier it does not have the energy to climb. Chop the potential into flat pieces, and the wave on each side of every edge is related by a 2 by 2 matrix; the product of those matrices gives the chance of getting through, $T$ , and of bouncing back, $R$ . They add to 1 at every energy, to within $1 0^{- 13}$ . Two barriers 4 apart, each of which alone lets through less than 1% at $E = 0.2108$ , together let through 100% at that energy.

One barrier, V0 = 3 and width 1

transfer matricesclosed form

two wallsone wallresonances found

✓ 8 tests passchecked when the site was built#quantum-mechanics #scattering #tunneling #transfer-matrix #resonance #notebook

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Also in Physics

demoAIPhysicswritten with an agent#8987b2b000

Adam Zettel@adamSep 25, 2026

Silicon's band structure, from the crystal up

[- \frac{1}{2} \nabla^{2} + V_{ion} + V_{H} [ρ] + V_{xc} [ρ]] ψ_{n k} = ε_{n k} ψ_{n k}

A density-functional calculation of silicon, one cell at a time: the crystal, a plane-wave basis, the pseudopotential, the Hamiltonian, the self-consistent loop and the bands from L through Gamma to X. It runs at a 5 Ha cutoff on four k-points so it fits in a browser tab; its tests check each stage, from the basis sizes to the band energies, against an independent numpy implementation at the same settings.

Silicon: LDA bands, 5 Ha cutoff, 4 k-points

valence 1valence 2valence 3valence 4conduction 1conduction 2

✓ 18 tests passchecked when the site was built#dft #plane-waves #pseudopotentials #band-structure #silicon #notebook

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