Su'aalo Tusaale ah oo Ka Hadlaya Aragtida Quantum ee Planck
Aragtida Quantum-ka ee Planck waxay ahayd meel muhiim ah oo laga beddelo fiisigiska casriga ah, iyadoo beddeshay fahamkeenna ku saabsan shucaaca jirka madow iyo farsamada quantum-ka. Waxaa soo bandhigay Max Planck sanadkii 1900, aragtidani waxay ka caawisaa sharraxaadda dhacdooyinka aysan sharxi karin fiisigiska caadiga ah. Maqaalkani wuxuu sahamin doonaa aragtida quantum-ka ee Planck iyada oo loo marayo dood ku saabsan tusaalooyinka dhibaatooyinka, laga bilaabo fikradaha aasaasiga ah ilaa codsiyada.
Asalka Aragtida Quantum ee Planck
Kahor inta aan laga hadlin dhibaatada tusaalaha ah, waa muhiim in la fahmo asalka Aragtida Quantum ee Planck. Dabayaaqadii qarnigii 19aad, fiisikiska caadiga ah wuxuu la kulmay caqabad weyn oo ku saabsan sharraxaadda noocyada shucaaca jirka madow. Shucaaca jirka madow waa shucaaca elektromagnetic-ka ee ay soo saaraan walxaha heerkul gaar ah.
Fiisikiska caadiga ah, isagoo adeegsanaya sharciga Rayleigh-Jeans, wuxuu saadaaliyay in tamarta shucaaca ay si aan xad lahayn u kordhi doonto marka ay timaado hirarka sare, oo loo yaqaan "masiibada ultraviolet." Halkan ayay Max Planck ku soo bandhigtay xal kacaan ah: wuxuu soo jeediyay in tamarta la sii daayo ama lagu nuugo baakado kala go'an oo loo yaqaan "quanta."
Qaacidada Aasaasiga ah ee Aragtida Quantum ee Planck
Qaacidada aasaasiga ah ee tamarta kuantumka sida waafaqsan aragtida Planck waa:
\[ E = h \nu \]
diima:
– \( E \) waa tamarta baakadda kuntum (oo sidoo kale loo yaqaan quanta),
– \( h \) waa joogtada Planck (\(6.626 \times 10^{-34} \, \text{Js}\)),
– \( \nu \) waa soo noqnoqoshada shucaaca.
Su'aalo iyo Doodo Tusaale ah
Su'aal 1aad: Xisaabinta Tamarta Quantum-ka
Su'aal:
Fotoonku wuxuu leeyahay soo noqnoqosho ah oo ah \( 5 \jeer 10^{14} \, \text{Hz} \). Xisaabi tamarta fotoonku sida waafaqsan aragtida Planck.
Dood:
Waa la ogyahay:
– Soo noqnoqoshada \( \nu = 5 \jeer 10^{14} \, \qoraalka{Hz} \)
– Joogtada Planck \( h = 6.626 \times 10^{-34} \, \text{Js} \)
Isticmaalka qaacidada tamarta kuntum ee Planck:
\[ E = h \nu \]
\[ E = (6.626 \jeer 10^{-34} \, \qoraalka{Js}) \jeer (5 \jeer 10^{14} \, \qoraalka{Hz}) \]
\[ E = 3.313 \jeer 10^{-19} \, \qoraal{J} \]
Markaa, tamarta sawirkan waa \( 3.313 \times 10^{-19} \, \text{J} \).
Su'aal 2aad: Xiriirka ka dhexeeya Mowjadaha iyo Tamarta
Su'aal:
Go'aami tamarta photon-ka oo leh hirar dhererkoodu yahay \( 600 \, \text{nm} \).
Dood:
Waa la ogyahay:
– Mowjadaha hirarka \( \lambda = 600 \, \text{nm} = 600 \jeer 10^{-9} \, \text{m} \)
– Xawaaraha iftiinka \( c = 3 jeer 10^{8} \, \text{m/s} \)
– Joogtada Planck \( h = 6.626 \times 10^{-34} \, \text{Js} \)
Marka hore, waxaan u baahanahay inaan helno soo noqnoqoshada \( \nu \) annagoo adeegsanayna xiriirka ka dhexeeya mowjadda iyo soo noqnoqoshada:
\[ \nu = \frac{c}{\lambda} \]
\[ \nu = \frac{3 \jeer 10^{8} \, \text{m/s}}{600 \jeer 10^{-9} \, \text{m}} \]
\[ \nu = 5 \jeer 10^{14} \, \qoraal{Hz} \]
Hadda, waxaan isticmaali karnaa qaacidada tamarta kuntum ee Planck:
\[ E = h \nu \]
\[ E = (6.626 \jeer 10^{-34} \, \qoraalka{Js}) \jeer (5 \jeer 10^{14} \, \qoraalka{Hz}) \]
\[ E = 3.313 \jeer 10^{-19} \, \qoraal{J} \]
Markaa, tamarta fotonka leh hirarka hirarka \( 600 \, \text{nm} \) waa \( 3.313 \times 10^{-19} \, \text{J} \).
Su'aal 3aad: Tamarta la xiriirta Shucaaca Jirka Madow
Su'aal:
Jir madow wuxuu ku jiraa heerkul dhan 3000 K. Waa maxay inta jeer ee shucaaca uu soo saaro shaygu?
Dood:
Waa la ogyahay:
– Heerkulka \( T = 3000 \, \text{K} \)
– Joogtada Boltzmann \( k = 1.38 \times 10^{-23} \, \text{J/K} \)
Sida waafaqsan sharciga Wien, hirarka ugu sarreeya ee shucaaca jirka madow \( \lambda_{\text{max}} \) waxaa bixiya:
\[ \lambda_{\text{max}} T = 2.898 \jeer 10^{-3} \, \text{m K} \]
Sidaas darteed:
\[ \lambda_{\text{max}} = \frac{2.898 \times 10^{-3} \, \text{m K}}{3000 \, \text{K}} \]
\[ \lambda_{\text{max}} = 9.66 \jeer 10^{-7} \, \text{m} \]
Si loo helo soo noqnoqoshada ugu sarreysa \( \nu_{\text{max}} \), waxaan isticmaalnaa:
\[ \nu_{\text{max}} = \frac{c}{\lambda_{\text{max}}} \]
\[ \nu_{\text{max}} = \frac{3 \times 10^{8} \, \text{m/s}}{9.66 \times 10^{-7} \, \text{m}} \]
\[ \nu_{\text{max}} \qiyaastii 3.10 \times 10^{14} \, \text{Hz} \]
Sidaas darteed, inta jeer ee ugu sarreysa ee shucaaca uu soo saaro jirka madow heerkulkiisu yahay 3000 K waa qiyaastii \( 3.10 \times 10^{14} \, \text{Hz} \).
Su'aal 4: Qaybinta Tamarta Shucaaca
Su'aal:
Xisaabi tamarta shucaaca guud ee ay soo saarto jirka madow halkii unugba heerkulkiisu yahay 5000 K.
Dood:
Waa la ogyahay:
– Heerkulka \( T = 5000 \, \text{K} \)
– Stefan-Boltzmann joogto ah \( \sigma = 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \)
Qaacidada qaybinta tamarta shucaaca ee guud ee uu soo saaro jirka madow waa:
\[ E = \sigma T^4 \]
\[ E = (5.67 \jeer 10^{-8} \, \qoraalka{W/m}^2\qoraalka{K}^4) \jeer (5000 \, \qoraalka{K})^4 \]
\[ E = 5.67 \jeer 10^{-8} \jeer 625 \jeer 10^{12} \]
\[ E \qiyaastii 3.54375 \times 10^{7} \, \text{W/m}^2 \]
Markaa, tamarta shucaaca guud ee jirka madow uu soo saaro heerkul dhan 5000 K waa \( 3.54375 \times 10^{7} \, \text{W/m}^2 \).
Gabagabo
Aragtida Quantum-ka ee Planck waxay bixisaa aasaas muhiim ah oo loogu talagalay fiisigiska casriga ah, iyadoo la fahmayo sida tamarta loo soo daayo loona nuugo qaabka quanta. Annagoo adeegsanayna qaacidada aasaasiga ah \( E = h \nu \), waxaan xisaabin karnaa macluumaad muhiim ah oo kala duwan, oo ay ku jiraan tamarta photon-ka, soo noqnoqoshada iyo hirarka hirarka ee la xiriira shucaaca elektromagnetic-ka, iyo qaybinta tamarta ee shucaaca ka yimaada jirka madow. Daraasaddani ma aha oo kaliya inay jebisay xuduudaha fiisigiska caadiga ah laakiin waxay sidoo kale u gogol xaartay horumarinta farsamada quantum-ka iyo hal-abuurka tignoolajiyada kala duwan.
