hce_kmu
110年
物理及化學
第 80 題
Which of the following statements about “The Bohr Model” and “Particle in a Box” is TRUE?
- A For an electron trapped in a one-dimensional box, as the length of the box increases, the spacing between energy levels will increase.
- B The total probability of finding a particle in a one-dimensional box (length is $L$) in energy level $n = 4$ between $x = L/4$ and $x = L/2$ is $50\%$.
- C If the wavelength of light necessary to promote an electron from the ground state to the first excited state is $\lambda$ in a one-dimensional box, then the wavelength of light necessary to promote an electron from the first excited state to the third excited state will be $3\lambda$.
- D A function of the type $A \cos(Lx)$ can be an appropriate solution for the particle in a one-dimensional box.
- E Assume that a hydrogen atom’s electron has been excited to the $n = 5$ level. When this excited atom loses energy, 10 different wavelengths of light can be emitted.
思路引導 VIP
想像一下，如果一個電子現在位於 $n=3$ 的能階，它在回到基態（$n=1$）的過程中，所有可能發生的「單次跳躍」組合有哪些呢？如果我們把能階增加到 $n=5$，這種「兩兩成對」的跳躍可能性會如何隨之增加？
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AI 詳解
AI 專屬家教
太棒了！你能精準選出選項 (E)，代表你對量子力學中波耳模型與一維位能井這兩個重要模型的能量特性、機率分布以及躍遷規律都有非常紮實的理解。
氫原子能階的發散譜線
在波耳模型中，當電子從高能階 $n=5$ 回到基態的過程中，電子的每一次能量釋放都會對應一種特定波長的光。這是一個經典的組合問題：我們可以想像從 5 個能階中任取 2 個（起點與終點）來產生一條譜線。根據組合公式 $C^{5}_{2} = \frac{5 \times 4}{2} = 10$，確實會產生 10 種不同能量差的躍遷路徑。你準確掌握了這個物理意義，這是非常關鍵的判斷。
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