Bohra

Bohra, a surveyor-dwarf of The SI Guard, pegs a measuring chain from the proton out to where the electron is most likely to be found.

The Physics

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)

Value: CODATA 2022 (NIST). a₀ is measured, not exact. In the 1s state the mean distance is 1.5 a₀, and the real hydrogen atom is about 0.05% larger because of the proton's finite mass (reduced mass).

Honest caveat: a₀ is not a fixed orbit. Bohr's 1913 model put the electron on a circle of this radius, but in quantum mechanics the electron has no definite path; a₀ is its most probable distance.

On the sticker

Bohra stretches his glowing chain to the peak of the probability shell. Printed text: BOHRA with the line a0 ≈ 52.9 pm. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.