Mehlala ea Lipotso Tse Buang ka Phenomena ea Quantum
Liketsahalo tsa quantum, kapa liketsahalo tse laoloang ke mechanics ea quantum, li akaretsa likhopolo le melao-motheo e mengata e hlokang kutloisiso e tebileng le ho rarahana ha lipalo. Mechanics ea quantum ke lekala la fisiks le hlalosang boitšoaro ba likaroloana tse ka tlase ho athomo, tse kang lielektrone le li-photon, tse ke keng tsa hlalosoa ke fisiks ea khale. Sehloohong sena, re tla hlahloba mathata a 'maloa a mehlala le litharollo tsa' ona tse amanang le liketsahalo tsa quantum ho thusa ho utloisisa melao-motheo ea motheo ea mechanics ea quantum.
Mohlala oa Potso ea 1: Molao-motheo oa Heisenberg oa ho se Tiisehe
Potso:
Hoa tsebahala hore boemo ba elektrone ka har'a athomo bo lekanngoa ka ho nepahala ha \( \Delta x = 0.1 \text{ nm} \). Fumana ho se tsitse ho fokolang ha ho lekanyetsoa momentum oa elektrone (\( \Delta p \)) ho sebelisoa molao-motheo oa Heisenberg oa ho se tsitse.
Karabo:
Molao-motheo oa Heisenberg oa ho se tsitse o re:
\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]
moo \( \hbar \) e leng kamehla ya Planck e fokotsehileng, ka boleng \( \hbar \hoo e ka bang 1.054 \times 10^{-34} \text{ Js} \).
Nkela sebaka \( \Delta x = 0.1 \text{ nm} = 0.1 \times 10^{-9} \text{ m} \):
\[ \Delta p \geq \frac{\hbar}{2 \Delta x} \]
\[ \Delta p \geq \frac{1.054 \times 10^{-34}}{2 \times 0.1 \times 10^{-9}} \]
\[ \Delta p \geq \frac{1.054 \times 10^{-34}}{2 \times 10^{-10}} \]
\[ \Delta p \geq \frac{1.054 \times 10^{-34}}{2 \times 10^{-10}} = 5.27 \times 10^{-25} \text{ kg m/s} \]
Kahoo ho se tsitse ho fokolang ha ho lekanyetsoa momentum ea elektrone ke \( 5.27 \times 10^{-25} \text{ kg m/s} \).
Mohlala Potso ea 2: Matla a ka bang teng ka har'a Lebokose (Karoloana ka har'a Lebokose)
Potso:
Karoloana e nang le boima ba m e qabeletsoe ka lebokoseng la bolelele ba L le lekanang le le leng. Matla a motheo (matla a boemo ba mobu) a karoloana ke afe?
Karabo:
Matla a motheo (matla a boemo ba mobu) a karoloana ka lebokoseng le nang le tekanyo e le 'ngoe a fanoa ke equation:
\[ E_n = \frac{n^2 h^2}{8mL^2} \]
Bakeng sa boemo ba fatše (\( n=1 \)):
\[ E_1 = \frac{h^2}{8mL^2} \]
moo \( h \) e leng ntho e sa fetoheng ya Planck \( (h \hoo e ka bang 6.626 \makgetlo a 10^{-34} \mongolo{ Js}) \).
A re re \( m = 9.109 \times 10^{-31} \text{ kg} \) (boima ba elektrone) le \( L = 1 \times 10^{-9} \text{ m} \):
\[ E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (1 \times 10^{-9})^2} \]
\[ E_1 = \frac{4.39 \times 10^{-67}}{7.287 \times 10^{-50}} \]
\[ E_1 = 6.02 \makgetlo a 10^{-18} \mongolo{ J} \]
Kahoo matla a motheo a karoloana ke \( 6.02 \times 10^{-18} \text{ J} \).
Mohlala oa 3: Ts'ebetso ea Motsamaisi oa Hamiltonian Mesebetsing ea Maqhubu
Potso:
Mosebetsi wa leqhubu la karoloana ka lebokoseng le nang le tekanyo e le nngwe ke \( \psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) \) bakeng sa \( n=1,2,3,\ldots \). Fumana matla a karoloana ka ho sebedisa opereishene ya Hamiltonian \( \hat{H} \).
Karabo:
Mokhanni oa Hamiltonian ka tekanyo e le 'ngoe ke:
\[ \hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \]
Re tlameha ho sebelisa opareitara ea Hamiltonian mosebetsing oa maqhubu \( \psi(x) \):
\[ \hat{H} \psi(x) = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) \]
Tšimoloho ea pele ea \( \psi(x) \):
\[ \frac{d}{dx} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) = \sqrt{\frac{2}{L}} \left( \frac{n\pi}{L} \cos\left( \frac{n\pi x}{L} \right) \right) \]
Ntho ea bobeli e nkiloeng ho eona:
\[ \frac{d^2}{dx^2} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) = \sqrt{\frac{2}{L}} \left( -\left( \frac{n\pi}{L} \right)^2 \sin\left( \frac{n\pi x}{L} \right) \right) \]
\[ \frac{d^2}{dx^2} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) = -\frac{n^2 \pi^2}{L^2} \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \]
Jwale, kgutlisetsa sephetho sebakeng sa opereishene ya Hamiltonian:
\[ \hat{H} \psi(x) = -\frac{\hbar^2}{2m} \left( -\frac{n^2 \pi^2}{L^2} \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) \]
\[ \hat{H} \psi(x) = \frac{\hbar^2 n^2 \pi^2}{2m L^2} \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \]
Ho tloha mona, rea bona hore:
\[ \hat{H} \psi(x) = \frac{\hbar^2 n^2 \pi^2}{2m L^2} \psi(x) \]
Ka hona, matla a likaroloana ke:
\[ E_n = \frac{\hbar^2 n^2 \pi^2}{2m L^2} \]
A re re re batla ho fumana matla bakeng sa \( n=1 \):
\[ E_1 = \frac{\hbar^2 \pi^2}{2m L^2} \]
Qetello
Ho rarolla mathata a amanang le liketsahalo tsa quantum ho hloka kutloisiso e tiileng ea melao-motheo ea motheo ea mechanics ea quantum, joalo ka molao-motheo oa ho se tsitse ha Heisenberg le matla a likaroloana ka lebokoseng le ka bang teng. Ka mehlala e 'maloa ea mathata le lipuisano tsa bona, re tšepa ho thusa ho matlafatsa likhopolo tsa motheo tsa mechanics ea quantum le ts'ebeliso ea eona maemong a fapaneng a fisiks. Le hoja mechanics ea quantum e ka bonahala e rarahane, mathata a ho itloaetsa le kutloisiso ea mohopolo li tla thusa haholo ho tseba boitsebiso bona ba motheo.
