# 舊量子論

（重定向自量子论

## 基本原理

${\displaystyle \oint p_{i}dq_{i}=n_{i}h\,\!}$

## 範例

### 諧振子

${\displaystyle H={p^{2} \over 2m}+{m\omega ^{2}q^{2} \over 2}\,\!}$

${\displaystyle A=\pi ab=\pi {\sqrt {2mE}}{\sqrt {2E/m\omega ^{2}}}=2\pi E/\omega =nh\,\!}$

${\displaystyle E=n\hbar \omega \,\!}$

${\displaystyle U=3N{\cfrac {\sum _{n}\hbar \omega ne^{-n\hbar \omega /k_{B}T}}{\sum _{n}e^{-n\hbar \omega /k_{B}T}}}=3N{\cfrac {\hbar \omega }{e^{\hbar \omega /k_{B}T}-1}}\,\!}$

${\displaystyle C_{V}={\frac {\partial U}{\partial T}}=3N{\frac {(\hbar \omega )^{2}}{k_{B}T^{2}}}{\frac {e^{\hbar \omega /k_{B}T}}{(e^{\hbar \omega /k_{B}T}-1)^{2}}}\,\!}$

${\displaystyle U\to 3Nk_{B}T\,\!}$
${\displaystyle C_{V}\to 3Nk_{B}\,\!}$

${\displaystyle k_{B}T\,\!}$超低，${\displaystyle k_{B}T\ll \hbar \omega \,\!}$的時候，系統非常冷，諧振子的熱能量${\displaystyle U\,\!}$會以指數函數趨向零，比熱的物理行為也一樣。在1900年前後，很多氣體、液體、固體的比熱實驗都得到了這非經典結果，證明了理論的正確性。

### 一維位勢

${\displaystyle p={\sqrt {2m(E-V(q))}}\,\!}$

${\displaystyle 2\int _{0}^{L}pdq=nh\,\!}$

${\displaystyle p={nh \over 2L}\,\!}$

${\displaystyle E={n^{2}h^{2} \over 8mL^{2}}\,\!}$

${\displaystyle V(q)=-Fq\,\!}$

${\displaystyle F=-{\frac {\partial V(q)}{\partial q}}\,\!}$

${\displaystyle 2\int _{0}^{E/F}\ {\sqrt {2m(E-Fq)}}\ dq=nh\,\!}$

${\displaystyle {\frac {4{\sqrt {2m}}E^{3/2}}{3F}}=nh\,\!}$

${\displaystyle E=\left({\frac {3nhF}{4{\sqrt {2m}}}}\right)^{2/3}\,\!}$

### 旋轉子

${\displaystyle L={\frac {1}{2}}MR^{2}{\dot {\theta }}^{2}\,\!}$

${\displaystyle J=MR^{2}{\dot {\theta }}\,\!}$

${\displaystyle 2\pi J=nh\,\!}$

${\displaystyle L={\frac {1}{2}}MR^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}MR^{2}(\sin \theta {\dot {\phi }})^{2}\,\!}$

${\displaystyle p_{\theta }=MR^{2}{\dot {\theta }}\,\!}$
${\displaystyle p_{\phi }=MR^{2}\sin ^{2}\theta {\dot {\phi }}\,\!}$

${\displaystyle p_{\phi }=l_{\phi }\,\!}$

${\displaystyle 2\pi l_{\phi }=mh\,\!}$

### 氫原子

${\displaystyle H={p^{2} \over 2}+{L^{2} \over 2r^{2}}-{1 \over r}\,\!}$

${\displaystyle p={\sqrt {2E-{L^{2} \over r^{2}}+{2 \over r}}}\,\!}$

${\displaystyle r_{1}=(-1+{\sqrt {1+2L^{2}E}})/2E\,\!}$
${\displaystyle r_{2}=(-1-{\sqrt {1+2L^{2}E}})/2E\,\!}$

${\displaystyle \oint {\sqrt {2E-{L^{2} \over r^{2}}+{2 \over r}}}\ dr=2\int _{r_{1}}^{r_{2}}{\sqrt {2E-{L^{2} \over r^{2}}+{2 \over r}}}\ dr=kh\,\!}$

${\displaystyle 2\pi \left({\frac {1}{\sqrt {-2E}}}-L\right)=kh\,\!}$

${\displaystyle E=-{\frac {1}{2(k+l)^{2}\hbar ^{2}}}\,\!}$

${\displaystyle n=k+l\,\!}$

## 光子與物質波

1905年，愛因斯坦發覺在同樣一個盒子內，假若波長很短，則量子化的電磁場諧振子的等於一群呈氣體態的粒子的熵[9]。粒子的數量等於量子的數量。愛因斯坦因此推斷，這量子是實際存在於空間某個位置的物體，即光的粒子。他將這量子取名為光子

${\displaystyle p={\frac {h}{\lambda }}\,\!}$

1924年，路易·德布羅意還正在攻讀博士學位的時候，他提出了一個新的詮釋。他建議所有的物質，電子或光子，都是物質波，遵守關係式

${\displaystyle \lambda ={h \over p}\,\!}$

${\displaystyle \oint pdx=\oint {\frac {h}{\lambda }}dx=nh\,\!}$

${\displaystyle n\lambda /2=L\,\!}$

${\displaystyle p={\frac {nh}{2L}}\,\!}$

## 克拉莫躍遷關係

${\displaystyle X_{n}(t)=\sum _{k=-\infty }^{\infty }e^{ik\omega t}X_{n;\,k}\,\!}$

## 歷史

1913年，波耳發表了對應原理。應用這原理，他又建構了氫原子波耳模型，成功地解釋出氫原子的發射譜線

## 參考文獻

1. ^ ter Haar, D. The Old Quantum Theory. Pergamon Press. 1967: pp. 206.
2. ^ Gutzwiller, Martin, Effect of correlation on the ferromagnetism of transition metals, Physical Review Letters, 1963, 10 (5): pp. 159–162
3. ^ Gutzwiller, Martin, Correlation of Electrons in a Narrow s Band, Physical Review, 1965, 137: pp. A1726–A1735
4. ^ Wilson, W., The quantum theory of radiation and line spectra, Philosophical Magazine, 1915, 29: pp. 795–802
5. 索末菲, 阿諾, Zur Quantentheorie der Spektrallinien, Annalen der Physik, 1916, 51: pp. 1–94
6. ^ Aruldhas, G. Quantum Mechanics. Prentice-Hall of India Learning Pvt. Ltd. 2004: pp. 11. ISBN 978-8120319622.
7. ^ 索末菲, 阿諾, Atombau und Spektrallinien, Braunschweig, Friedrich Vieweg und Sohn, 1919，德文原文。
8. ^ 索末菲, 阿諾; Brose, Henry Leopold, Atomic structure and spectral lines 3rd., Methuen & Co., 1934，英文翻譯。
9. ^ 愛因斯坦, 阿爾伯特, Die Planckshe Theorie der Strahlung und die Theorie der Spezifischen Wärme, Annalen der Physik, 1907, 22: pp.180–190
10. 克拉莫, 亨德里克, The Quantum Theory of Dispersion, Nature, 1924, 114: pp. 310–311
11. ^ 愛因斯坦, 阿爾伯特; 施特恩, 奧托, Einige Argumente für die Annahme einer molekular Agitation beim absoluten Nullpunkt, Annalen der Physik, 1913, 40: pp. 551–560
12. ^ 克拉莫, 亨德里克, Über den Einfluß eines elektrischen Feldes auf die Feinstruktur der Wasserstofflinien (On the influence of an electric field on the fine structure of hydrogen lines), Zeitschrift für Physik, 1920, 3: pp. 199–233
13. ^ 克拉莫, 亨德里克; 海森堡, 維爾納, Über die Streuung von Strahlen durch Atome, Zeitschrift fur Physik, 1925, 31: pp.681–708
14. ^ 海森堡, 維爾納, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik, 1925, 33: pp. 879–893