{"product_id":"bohra-bohr-radius","title":"Bohra – Bohr Radius \/ The Size of the Hydrogen Atom Physics Sticker","description":"\u003cp\u003eBohra, 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.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn hydrogen's ground state (1s), the radial probability density r²|ψ|² peaks at exactly a\u003csub\u003e0\u003c\/sub\u003e (with the proton treated as infinitely heavy). This is the atomic unit of length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ea\u003csub\u003e0\u003c\/sub\u003e = 4πε\u003csub\u003e0\u003c\/sub\u003eħ²\/(m\u003csub\u003ee\u003c\/sub\u003ee²) = 5.291 772 105 44(82) × 10⁻¹¹ m ≈ 52.9 pm\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ea\u003csub\u003e0\u003c\/sub\u003e: Bohr radius (m)\u003c\/li\u003e\n\u003cli\u003eε\u003csub\u003e0\u003c\/sub\u003e: vacuum permittivity (measured since 2019)\u003c\/li\u003e\n\u003cli\u003eħ: reduced Planck constant (exact)\u003c\/li\u003e\n\u003cli\u003em\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured)\u003c\/li\u003e\n\u003cli\u003ee: elementary charge (exact)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eValue: CODATA 2022 (NIST). a\u003csub\u003e0\u003c\/sub\u003e is measured, not exact. In the 1s state the mean distance is 1.5 a\u003csub\u003e0\u003c\/sub\u003e, and the real hydrogen atom is about 0.05% larger because of the proton's finite mass (reduced mass).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e a\u003csub\u003e0\u003c\/sub\u003e 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; a\u003csub\u003e0\u003c\/sub\u003e is its most probable distance.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eBohra stretches his glowing chain to the peak of the probability shell. Printed text: \u003cstrong\u003eBOHRA\u003c\/strong\u003e with the line \u003cstrong\u003ea0 ≈ 52.9 pm\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50963655131239,"sku":"31169519135515258584","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50963655164007,"sku":"28140481847856542835","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50963655196775,"sku":"33180000715780400100","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50963655229543,"sku":"33424040580744908858","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/5506454125801195974_1200.jpg?v=1793651146","url":"https:\/\/pixelatedphysics.com\/fr\/products\/bohra-bohr-radius","provider":"Pixelated Physics","version":"1.0","type":"link"}