Nā nīnau hoʻohālike e kūkākūkā ana i nā hanana Quantum
ʻO nā hanana Quantum, a i ʻole nā hanana i hoʻomalu ʻia e ka mechanics quantum, e hoʻopuni ana i kahi ākea o nā manaʻo a me nā loina e pono ai ka hoʻomaopopo hohonu a me ka paʻakikī makemakika. ʻO ka mechanics Quantum kahi lālā o ka physics e wehewehe ana i ke ʻano o nā ʻāpana subatomic, e like me nā electrons a me nā photons, ʻaʻole hiki ke wehewehe ʻia e ka physics kuʻuna. Ma kēia ʻatikala, e ʻimi mākou i kekahi mau pilikia hoʻohālike a me kā lākou mau hoʻonā e pili ana i nā hanana quantum e kōkua i ka hoʻomaopopo ʻana i nā loina kumu o ka mechanics quantum.
Laʻana Nīnau 1: Ke Kumumanaʻo o ka Hoʻomaopopo ʻole o Heisenberg
Nīnau:
Ua ʻike ʻia ua ana ʻia ke kūlana o kahi electron i loko o kahi ʻātoma me ka pololei o \( \Delta x = 0.1 \text{ nm} \). E hoʻoholo i ka maopopo ʻole loa i ke ana ʻana i ka momentum electron (\( \Delta p \)) me ka hoʻohana ʻana i ke kumumanaʻo maopopo ʻole o Heisenberg.
Pane:
ʻŌlelo ke kumumanaʻo kānalua o Heisenberg:
\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]
kahi ʻo \( \hbar \) ke kūpaʻa Planck i hoʻemi ʻia, me ka waiwai \( \hbar \approx 1.054 \times 10^{-34} \text{ Js} \).
Pani \( \Delta x = 0.1 \text{ nm} = 0.1 \times 10^{-9} \text{ m} \):
\[ \Delta p \geq \frac{\hbar}{2 \Delta x} \]
\[ \Delta p \geq \frac{1.054 \times 10^{-34}}{2 \times 0.1 \times 10^{-9}} \]
\[ \Delta p \geq \frac{1.054 \times 10^{-34}}{2 \times 10^{-10}} \]
\[ \Delta p \geq \frac{1.054 \times 10^{-34}}{2 \times 10^{-10}} = 5.27 \times 10^{-25} \text{ kg m/s} \]
No laila, ʻo ka maopopo ʻole loa i ke ana ʻana i ka momentum electron ʻo \( 5.27 \times 10^{-25} \text{ kg m/s} \).
Laʻana Nīnau 2: Ikehu Hiki i loko o kahi Pahu (ʻĀpana i loko o kahi Pahu)
Nīnau:
Ua paʻa kahi ʻāpana me ke kaumaha m i loko o kahi pahu hoʻokahi-dimensional o ka lōʻihi L. He aha ka ikehu kumu (ikehu kūlana honua) o ka ʻāpana?
Pane:
Ua hāʻawi ʻia ka ikehu kumu (ka ikehu moku'āina honua) o kahi ʻāpana i loko o kahi pahu hoʻokahi-dimensional e ka hoohalike:
\[ E_n = \frac{n^2 h^2}{8mL^2} \]
No ke kūlana honua (\( n=1 \)):
\[ E_1 = \frac{h^2}{8mL^2} \]
kahi ʻo \( h \) ke kūpaʻa o Planck \( (h \approx 6.626 \times 10^{-34} \text{ Js}) \).
Manaʻo ʻia \( m = 9.109 \times 10^{-31} \text{ kg} \) (ka nuipa o ka electron) a me \( L = 1 \times 10^{-9} \text{ m} \):
\[ E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (1 \times 10^{-9})^2} \]
\[ E_1 = \frac{4.39 \times 10^{-67}}{7.287 \times 10^{-50}} \]
\[ E_1 = 6.02 \times 10^{-18} \text{ J} \]
No laila, ʻo ka ikehu kumu o ka ʻāpana he \( 6.02 \times 10^{-18} \text{ J} \).
Laʻana 3: Nā Hana Hana Hamiltonian ma nā Hana Nalu
Nīnau:
ʻO ka hana nalu o kahi ʻāpana i loko o kahi pahu hoʻokahi-dimensional ʻo \( \psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) \) no \( n=1,2,3,\ldots \). E hoʻoholo i ka ikehu o ka ʻāpana me ka hoʻohana ʻana i ka mea hana Hamiltonian \( \hat{H} \).
Pane:
ʻO ke ʻano hana Hamiltonian ma kekahi ana:
\[ \hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \]
Pono mākou e hoʻopili i ka mea hana Hamiltonian i ka hana nalu \( \psi(x) \):
\[ \hat{H} \psi(x) = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) \]
ʻO ka hua mua o \( \psi(x) \):
\[ \frac{d}{dx} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) = \sqrt{\frac{2}{L}} \left( \frac{n\pi}{L} \cos\left( \frac{n\pi x}{L} \right) \right) \]
Ka lua o ka derivative:
\[ \frac{d^2}{dx^2} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) = \sqrt{\frac{2}{L}} \left( -\left( \frac{n\pi}{L} \right)^2 \sin\left( \frac{n\pi x}{L} \right) \right) \]
\[ \frac{d^2}{dx^2} \left( \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) = -\frac{n^2 \pi^2}{L^2} \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \]
I kēia manawa, e hoʻololi i ka hopena i loko o ka mea hana Hamiltonian:
\[ \hat{H} \psi(x) = -\frac{\hbar^2}{2m} \left( -\frac{n^2 \pi^2}{L^2} \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \right) \]
\[ \hat{H} \psi(x) = \frac{\hbar^2 n^2 \pi^2}{2m L^2} \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right) \]
Mai ʻaneʻi mai, ʻike mākou:
\[ \hat{H} \psi(x) = \frac{\hbar^2 n^2 \pi^2}{2m L^2} \psi(x) \]
No laila, ʻo ka ikehu o ka ʻāpana:
\[ E_n = \frac{\hbar^2 n^2 \pi^2}{2m L^2} \]
Manaʻo mākou e makemake mākou e ʻimi i ka ikehu no \( n=1 \):
\[ E_1 = \frac{\hbar^2 \pi^2}{2m L^2} \]
Ka hopena
ʻO ka hoʻoponopono ʻana i nā pilikia e pili ana i nā hanana quantum e pono ai ka ʻike paʻa o nā kumumanaʻo kumu o ka mechanics quantum, e like me ke kumumanaʻo maopopo ʻole o Heisenberg a me ka ikehu o nā ʻāpana i loko o kahi pahu hiki. Ma o kekahi mau pilikia hoʻohālike a me kā lākou mau kūkākūkā ʻana, ke manaʻolana nei mākou e kōkua i ka hoʻoikaika ʻana i nā manaʻo kumu o ka mechanics quantum a me kāna mau noi i nā kūlana physics like ʻole. ʻOiai hiki ke ʻano paʻakikī ka mechanics quantum, ʻo nā pilikia hoʻomaʻamaʻa a me ka hoʻomaopopo ʻana i ka manaʻo e kōkua nui i ka hoʻopaʻa ʻana i kēia mea kumu.
