Weird Quantum Pixel Sticker Sheet

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

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)

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)

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)

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)

Feeblea

In 1909 a young G. I. Taylor photographed the shadow of a needle with light so dim that, in modern terms, rarely more than one photon was in the box at a time (Proc. Camb. Phil. Soc. 15, 114). His longest exposure ran about 2000 hours, roughly three months. The diffraction fringes were just as sharp as with bright light.

t_exposure ≈ 2000 h
t_exposure: exposure time of the faintest photograph

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)

Invertia

Temperature is defined by how entropy changes when you add energy. In a system with a top energy level, like nuclear spins, putting more particles in the upper state than the lower one makes that slope negative: a negative absolute temperature, which is actually 'hotter than infinity'.

1/T = ∂S/∂E < 0
T: absolute temperature, in the standard (Boltzmann) definition (K); S: entropy (Boltzmann form, from the number of states at energy E) (J/K); E: total energy of the spin system (J); N_↑, N_↓: number of particles in the upper and the lower level; Δ E: energy gap between the two levels (J); k_B: Boltzmann constant (J/K)

Expulsieur

A superconductor expels magnetic field from its interior, which a merely perfect conductor would not do. The field only penetrates a thin layer, dying away exponentially over the London penetration depth.

B_inside = 0
B_(inside): magnetic field in the bulk of the superconductor (applied field below the critical field) (T); B(x): field at depth x below the surface (T); B₀: field at the surface (T); x: depth below the surface (m); λ_L: London penetration depth (m); m: electron mass (kg)

Berryphase

Carry a quantum system slowly round a closed loop of its settings and it returns to its starting state, apart from a phase. Michael Berry showed in 1984 that part of this phase depends only on the geometry of the loop, not on how long it takes (Proc. R. Soc. A 392, 45). For a spin-½ in the state aligned with a magnetic field (m = +½), with the field direction swept slowly round a closed loop, the geometric phase is minus half the solid angle that loop encloses: γ = −mΩ = −Ω/2. The anti-aligned state (m = −½) picks up +Ω/2.

γ = −Ω/2
γ: geometric (Berry) phase of the aligned state (rad); Ω: solid angle enclosed by the loop traced by the field direction, seen from the origin (sr); m: spin projection along the field in units of ħ (dimensionless; +½ aligned, −½ anti-aligned)

Dicka

N excited atoms that radiate independently give N times the light of one. But if they are packed close together, within about a wavelength, or otherwise coupled to the same light field, they can lock into phase and radiate as one giant dipole. Then the peak intensity grows as N², and the light comes out in a short, bright burst. Robert Dicke predicted this in “Coherence in Spontaneous Radiation Processes” (Phys. Rev. 93, 99, 1954).

I_peak ∝ N²
I_peak: peak emitted intensity (W); N: number of emitters radiating in phase

Wolfgang Pauli

Two identical fermions must have a wavefunction that changes sign when they are swapped. If both tried to occupy the same state, the wavefunction would be zero everywhere, so the arrangement is forbidden: one fermion per state.

ψ(1,2) = [φ_a(1)φ_b(2) − φ_a(2)φ_b(1)]/√2
ψ: two-particle antisymmetrised wavefunction; φₐ, φ_b: single-particle orbitals (including spin); a, b: full sets of quantum numbers (n,ℓ,m_ℓ,m_s)