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Den gamle kvanteteori (Fysik/Matematik)
[redigér]
Bohr-Sommerfeld kvantiseringslov:
for en periodisk bevægelse for hver koordinat
og den tilsvarende impuls
skal integralet
![{\displaystyle \oint \limits _{H(p,q)=E}p_{i}dq_{i}=(2\pi \hbar )n_{i}\;,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31664ad6d0ecd912f13f3f5c8d4d98d1a67a353f)
hvor
er en heltal (som kaldes kvantetal),
er Plancks konstant, og hvor integralet tages over én periode.
Matematik: integral regning.
Elektromagnetisk felt i et hulrum: sum af oscillatorer.
Oscillator:
Plancks strålingslov.
Eksperiment: varmestrålings spektrum (Pasco: blackbody radiation experiment).
Rydberg spektrum:
![{\displaystyle H={p^{2} \over 2}+{l^{2} \over 2r^{2}}-{1 \over r}\;,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b088f5c84a57fe9c503390c4963c00e2d31a89a1)
![{\displaystyle 2\oint {\sqrt {2E-{l^{2} \over r^{2}}+{2 \over r}}}\ dr=kh\;,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/925f00027e41a0f7fb4f094e642380991f56956b)
![{\displaystyle E=-{1 \over 2(k+l)^{2}}\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8900fabda05f970c795de563f7b6469f0f26f8f6)
Eksperiment: brint-atom linjespektrum.
Arnold Sommerfeld:[1]
![{\displaystyle {\frac {mv^{2}}{2}}\rightarrow {\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-mc^{2}\;,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/743817f8ae215ec83ea079ab0c12d8310e62df94)
![{\displaystyle \Delta E=E_{n}{\frac {Z^{2}\alpha ^{2}}{n}}\left({\frac {1}{j+{\frac {1}{2}}}}-{\frac {3}{4n}}\right)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0132691f8fd1bac9fc1e19c856bfe0ec0b65b84b)
Eksperiment: fin struktur (data).
Ultrakolde neutroner i tyngdefelt
[redigér]
Ultrakolde neutroner i tyngdefelt:[2][3]
![{\displaystyle V(x)=mgx\;,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9dc606c248e39048f370ef828df163934c646bb)
![{\displaystyle 2\int _{0}^{\frac {E}{mg}}{\sqrt {2m(E-mgx)}}\ dx=n(2\pi \hbar )\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd218afc686d275c6fbdbc2dc056634e683ddda)
Eksperiment: data fra artiklen.
- ↑ Arnold Sommerfeld, Atombau und Spektrallinien, Braunschweig 1921 [1]
- ↑ V.V. Nesvizhevsky et.al., Measurement of quantum states of neutrons in the Earth’s gravitational field, arXiv:hep-ph/0306198, [2]
- ↑ E. Cartlidge, Neutrons reveal quantum effects of gravity, PhysicsWorld News 2002 Jan 17, [3]