Quantum Physics Vol. 3 Pixel Sticker Sheet

11 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Quantum Physics Vol. 3. Every character below has its own sticker page with the full physics panel.

Condensa

Below a critical temperature, a gas of bosons starts collecting in its lowest quantum state. The fraction in that state grows as the temperature drops, following 1 − (T/Tc)^(3/2), and the critical temperature depends on the gas's density and particle mass.

N₀/N = 1 − (T/T_c)^3/2 (T ≤ T_c)
N₀: number of bosons in the ground state; N: total number of bosons; T: temperature (K); T_c: critical temperature for condensation (K); n: number density (m⁻³); m: boson mass (kg)

Fogplate

Henri Becquerel suspected that uranium salts, made to glow by sunlight, might give off X-rays. Paris stayed cloudy in late February 1896, so he put plates wrapped in black paper, with uranium salt on top, in a drawer. On 1 March he developed them anyway and found strong images: the uranium emitted penetrating rays with no sunlight at all (Comptes rendus 122, 501, 1896). It was radioactivity.

uranium → rays, no light needed
uranium salt: potassium uranyl sulfate in his first tests; plate: photographic plate wrapped in black paper

Bohra

In hydrogen's ground state (1s), the radial probability density r²|ψ|² peaks at exactly a₀ (with the proton treated as infinitely heavy). This is the atomic unit of length.

a₀ = 4πε₀ħ²/(m_ee²) = 5.291 772 105 44(82) × 10⁻¹¹ m ≈ 52.9 pm
a₀: Bohr radius (m); ε₀: vacuum permittivity (measured since 2019); ħ: reduced Planck constant (exact); m_e: electron mass (measured); e: elementary charge (exact)

Massratio

The proton is about 1836 times heavier than the electron. This dimensionless ratio sets why atoms have a tiny heavy nucleus with a light electron cloud, why nuclei barely move in chemistry (the Born–Oppenheimer approximation), and the ratio of vibrational to electronic energies in molecules. It is measured with Penning-trap mass comparisons and precision spectroscopy of molecular hydrogen ions such as HD⁺.

μ = m_p/m_e = 1836.152 673 426(32)
μ: proton-to-electron mass ratio (dimensionless); m_p: proton rest mass, measured; m_e: electron rest mass, measured; (32): standard uncertainty in the last two digits, about 1.7 × 10⁻¹¹ relative

Vevette

The Higgs field's potential has its lowest point away from zero, so the field takes a nonzero value v of about 246 GeV everywhere. Particles that interact with it gain masses proportional to that value and their coupling strength.

V(φ) = −μ²|φ|² + λ|φ|⁴
V: Higgs potential (energy density; natural units) (GeV⁴); φ: Higgs field; on the art the horizontal axis is the field's size (natural units) (GeV); μ²: parameter of the potential, positive (GeV²); λ: self-coupling of the field, positive; ⟨φ⟩: vacuum expectation value: where the field settles, in the valley (GeV); v: electroweak scale, about 246.22 GeV (GeV)

Weinberga

In the electroweak theory (Glashow, Weinberg, Salam), the neutral gauge fields B and W³ mix. Rotating them by the weak mixing angle θ_W gives the massless photon and the massive Z boson. The angle also links the couplings: e = g sin θ_W.

sin²θ_W ≈ 0.231 (MS-bar at M_Z)
θ_W: weak mixing (Weinberg) angle, about 28.7°; MS-bar: modified minimal-subtraction renormalisation scheme; M_Z: Z boson mass scale, about 91.19 GeV

Kapitzadirac

Two laser beams travelling in opposite directions form a standing wave, with bright and dark bands half a wavelength apart. Pyotr Kapitza and Paul Dirac predicted in 1933 (“The reflection of electrons from standing light waves”, Proc. Camb. Phil. Soc. 29, 297) that electrons crossing such a wave would be diffracted, as if by a grating. The electrons are pushed away from the bright bands by the ponderomotive force, and the regular pattern of bands acts as a periodic grating.

d = λ / 2
d: period of the light grating (m); λ: wavelength of the laser light (m)

Casimira

Put two uncharged, perfectly conducting parallel plates a small distance a apart in vacuum and they attract. Between the plates only electromagnetic modes that fit the boundary conditions are allowed; outside, every mode is. Summing the zero-point energies ½ħω of the modes, and subtracting the free-space value, leaves a finite energy that falls as the plates approach. Hendrik Casimir derived it in 1948.

P = π²ħc / (240 a⁴)
P: attractive pressure, force per unit plate area (Pa); ħ: reduced Planck constant, h/2π = 1.054 571 817 × 10⁻³⁴ J s; c: speed of light, 299 792 458 m s⁻¹; a: gap between the plates (m); π²/240: numerical factor from summing the allowed modes

Delaychoice

In a Mach–Zehnder interferometer, a single photon meets a first beam splitter and then two paths. If a second beam splitter recombines the paths, the photon shows interference: which detector clicks depends on the phase difference. If the second splitter is removed, each detector clicks half the time, as if the photon took one path. John Wheeler proposed in 1978 making this choice after the photon has passed the first splitter. Jacques et al. (Science 315, 966, 2007) did it with single photons and a random choice that was space-like separated from the photon's entry.

with splitter: P₁ = cos²(φ/2), P₂ = sin²(φ/2); without: P₁ = P₂ = ½
P₁, P₂: probabilities that detector 1 or 2 clicks; φ: phase difference between the two arms (rad)

Bifurcor

A magnetic field that changes with height pushes on an atom's magnetic moment. Because an electron's spin can only point up or down, silver atoms in the Stern–Gerlach magnet split into exactly two beams instead of smearing out.

F_z = μ_z ∂B_z/∂z
F_z: force on the atom along the field gradient (N); μ_z: z-component of the atom's magnetic moment (for silver, set by its one outer electron) (J/T); B_z: z-component of the magnetic field (T); z: position along the gradient, up the page (m); g_s: electron spin g-factor, about 2.0023; μ_B: Bohr magneton (J/T)

Revolvix

Rotating a spin-½ particle by one full turn multiplies its wavefunction by −1; only two full turns bring it back to its original state. Neutron interferometers measured this sign change directly in 1975.

R(2π)ψ = −ψ
R(θ): operator that rotates a spin-1/2 state by angle θ about the axis n; ψ: two-component spinor state of a spin-1/2 particle; n̂: unit vector along the rotation axis; σ: the three Pauli matrices (σx, σy, σz); 2π, 4π: one full turn (360°) and two full turns (720°) (rad)