[论文解读] The Stark effect in the Bohr-Sommerfeld theory and in Schrödinger's wave mechanics

Anthony Duncan, Michel Janssen|arXiv (Cornell University)|Apr 21, 2014

Quantum Mechanics and Applications参考文献 26被引用 21

一句话总结

本文比較了舊量子理論（玻爾-索末菲）與薛丁格波動力學在塞曼效應處理上的差異，顯示雖然舊理論透過臨時假設達成定性成功，但波動力學則提供了一個唯一確定、數學上一致的解法，無需任意規則，能從基本原理精確預測能階、躍遷強度與極化性質。

ABSTRACT

The explanation of the first-order Stark effect in hydrogen by Epstein and Schwarzschild in 1916 was seen as a great success for the old quantum theory. Yet, it also revealed some serious limitations of the theory. To recover the experimentally found line splittings, one had to make some arbitrary assumptions in addition to the basic quantum conditions to rule out certain orbits. The calculation of intensities of lines on the basis of Bohr's correspondence principle likewise required arbitrary additional assumptions. Finally, the actual orbits predicted by the old quantum theory depend on the coordinates chosen to impose the quantum conditions. Both Sommerfeld and Epstein recognized this problem but offered no solution for it. All these problems were solved in 1926 when Schrödinger and Epstein explained the Stark effect on the basis of the new wave mechanics. The calculations in the two theories are similar. In particular, both the Schrödinger equation in the new theory and the Hamilton-Jacobi equation in the old theory are separated in parabolic coordinates. The new quantum mechanics determines all allowed states and transitions without any additional assumptions. It also replaced the ambiguous guidelines based on the correspondence principle for calculating intensities by the straightforward prescription that intensities are given by the squares of the matrix elements of position, leading to results that agreed much better with the experimental data. Finally, the embarrassing non-uniqueness of orbits in the old quantum theory turned into the innocuous non-uniqueness of bases of eigenfunctions in wave mechanics. To this day, the Stark effect is remembered as one of the few qualified successes of the old quantum theory. We suspect that this is largely because after 1926 it became just one of the many unqualified successes of the new quantum theory.

研究动机与目标

• 分析玻爾-索末菲原子模型中塞曼效應的歷史發展與理論處理。
• 識別舊量子理論在解釋塞曼效應時所固有的限制與臨時假設，特別是關於選擇規則與強度計算的問題。
• 展示薛丁格波動力學如何透過提供一個唯一且非任意的框架，解決能階與躍遷機率的問題，從而克服這些缺陷。
• 對比舊理論中對應原理的角色與波動力學中更嚴謹的矩陣元形式主義。
• 評估塞曼效應作為舊量子理論向新量子理論過渡之關鍵驗證基準的意義。

提出的方法

• 分析艾普斯坦（1916）、西拉希爾德（1916）、索末菲（1919）與薛丁格（1926）關於塞曼效應的歷史記載與原始論文。
• 比較玻爾-索末菲量子化條件與在拋物坐標中薛丁格方程的數學結構。
• 檢視舊理論中哈密頓-雅可比方程的應用，及其與可分離坐標中時間無關薛丁格方程的同構關係。
• 評估舊理論中透過傅立葉展開處理躍遷強度的方式，與波動力學中矩陣元方法的差異。
• 評估舊理論中經典軌道的非唯一性，與波動力學中本徵函數基底的非唯一性。
• 使用具體方程式（例如，式 (20)–(24)、(44)–(47)、(54)）來展示解決塞曼問題時的數學等價性與概念差異。

实验结果

研究问题

• RQ1 玻爾-索末菲理論如何解釋塞曼效應？其關鍵假設與限制為何？
• RQ2 為何舊量子理論需引入任意選擇規則與額外假設，才能與塞曼效應的實驗數據相符？
• RQ3 薛丁格波動力學在哪些方面提供了比舊量子理論更一致且更具預測力的塞曼效應框架？
• RQ4 對應原理方法與波動力學中的矩陣元方法在躍遷強度處理上根本上有何差異？
• RQ5 塞曼效應的解決過程揭示了舊量子理論與新量子力學在相對優劣上的哪些特點？

主要发现

• 舊量子理論雖在數學上與波動力學相似，但仍需透過任意選擇規則來排除某些軌道，並與實驗數據相符。
• 舊理論中躍遷強度的計算依賴模糊的對應原理，且需在初始、終止或平均傅立葉展開之間任意選擇。
• 波動力學透過求解薛丁格方程，唯一確定所有允許的量子態與躍遷，完全消除了所有臨時假設。
• 波動力學中主量子數在塞曼效應中自然出現額外的 +1 項，無需額外規則即可排除非物理態。
• 波動力學透過位置算符矩陣元的平方提供明確的強度預測公式，其結果與實驗數據的符合度遠高於舊理論。
• 舊理論中經典軌道的非唯一性，被波動力學中無害的本徵函數基底非唯一性所取代，且不影響物理預測。