{"product_id":"adiabatix-adiabatic-invariant","title":"Adiabatix – The Adiabatic Invariant \/ J = ∮p dq Physics Sticker","description":"\u003cp\u003eAdiabatix, a white-bearded sage, slowly draws up the string of a glowing pendulum.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eAt the 1911 Solvay conference Lorentz asked what happens to a pendulum if you shorten its string very slowly. Einstein answered: the energy changes, but the ratio of energy to frequency stays the same. Paul Ehrenfest built this into a principle (1916): for slow changes, the action J stays constant. That is why quantum numbers stay put when conditions change gently.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJ = ∮p dq\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eJ: action over one cycle (J·s)\u003c\/li\u003e\n\u003cli\u003ep: momentum\u003c\/li\u003e\n\u003cli\u003eq: position coordinate\u003c\/li\u003e\n\u003cli\u003e∮: integral over one full oscillation\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eFor a harmonic oscillator J = E\/f, which is why the old quantum rule set E = nhf.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e J is only approximately conserved; it holds better the slower the change compared with one period.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eAdiabatix pinches a thin string from which a small glowing bob hangs. Printed text: \u003cstrong\u003eADIABATIX\u003c\/strong\u003e with the line \u003cstrong\u003eJ = ∮p dq\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":50964270219367,"sku":"18372733272431524979","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964270252135,"sku":"25926292825269668903","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964270284903,"sku":"17146217689410735086","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964270317671,"sku":"27081779628282022638","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/10886878321522334611_1200.jpg?v=1793664030","url":"https:\/\/pixelatedphysics.com\/de\/products\/adiabatix-adiabatic-invariant","provider":"Pixelated Physics","version":"1.0","type":"link"}