# 盒中氣體

## 量子數極大近似

${\displaystyle \mathbf {k} ={\frac {\mathbf {n} \pi }{L}}\,\!}$

${\displaystyle k={\cfrac {{\sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}\ \pi }{L}}={\frac {n\pi }{L}}\,\!}$

${\displaystyle g=f\left({\frac {1}{8}}\right)\left({\frac {4\pi k^{3}}{3}}\right)\left({\frac {L}{\pi }}\right)^{3}={\frac {fV}{6\pi ^{2}}}k^{3}\,\!}$

${\displaystyle dg={\frac {fV}{2\pi ^{2}}}k^{2}dk\,\!}$

${\displaystyle N_{i}={\frac {g_{i}}{\Phi }}\,\!}$

 *馬克士威-玻茲曼統計： ${\displaystyle \Phi =e^{\beta (\epsilon _{i}-\mu )}\,\!}$， *玻色-愛因斯坦統計： ${\displaystyle \Phi =e^{\beta (\epsilon _{i}-\mu )}-1\,\!}$， *費米-狄拉克統計： ${\displaystyle \Phi =e^{\beta (\epsilon _{i}-\mu )}+1\,\!}$。

${\displaystyle dN={\frac {dg}{\Phi }}={\frac {fV}{2\pi ^{2}}}\,{\frac {k^{2}}{\phi }}~dk\,\!}$(1)

## 能量分佈函數

${\displaystyle P_{A}~dA={\frac {dN}{N_{T}}}={\frac {dg}{N_{T}\Phi }}\,\!}$

${\displaystyle \int _{A}P_{A}~dA=1\,\!}$

${\displaystyle P_{k}~dk={\frac {fV}{2\pi ^{2}N_{T}}}\,{\frac {k^{2}}{\phi }}~dk\,\!}$

${\displaystyle P_{E}~dE=P_{k}{\frac {dk}{dE}}~dE\,\!}$(2)

## 正質量粒子

${\displaystyle E={\frac {\hbar ^{2}k^{2}}{2m}}\,\!}$
${\displaystyle dE={\frac {\hbar ^{2}k}{m}}dk\,\!}$

${\displaystyle E\,\!}$${\displaystyle dE\,\!}$的公式代入公式(2)，再稍加運算，可得到

${\displaystyle P_{E}~dE={\cfrac {2fV\beta ^{3/2}}{{\sqrt {\pi }}\,\Lambda ^{3}N_{T}}}~{\frac {E^{1/2}}{\Phi }}~dE\,\!}$(3)

## 零質量粒子

${\displaystyle E=\hbar kc\,\!}$
${\displaystyle dE=\hbar c~dk\,\!}$

${\displaystyle E\,\!}$${\displaystyle dE\,\!}$的公式代入公式(2)，再稍加運算，可得到

${\displaystyle P_{E}~dE={\frac {fV\beta ^{3}}{2\Lambda ^{3}N_{T}}}~{\frac {E^{2}}{\Phi }}~dE\,\!}$(4)

## 範例

### 正質量馬克士威-玻茲曼粒子

${\displaystyle \Phi =e^{\beta (E-\mu )}\,\!}$

${\displaystyle 1={\cfrac {2fV\beta ^{3/2}}{{\sqrt {\pi }}\,\Lambda ^{3}N_{T}}}~\int _{0}^{\infty }\,{\frac {E^{1/2}}{e^{\beta (E-\mu )}}}~dE={\cfrac {2fV\beta ^{3/2}}{{\sqrt {\pi }}\,\Lambda ^{3}N_{T}}}\,e^{\beta \mu }~\int _{0}^{\infty }\,{\frac {E^{1/2}}{e^{\beta E}}}~dE\,\!}$

${\displaystyle \int _{0}^{\infty }\,{\frac {E^{1/2}}{e^{\beta E}}}~dE={\frac {1}{2\beta }}{\sqrt {\frac {\pi }{\beta }}}\,\!}$

${\displaystyle 1={\cfrac {2fV\beta ^{3/2}}{{\sqrt {\pi }}\,\Lambda ^{3}N_{T}}}\,e^{\beta \mu }{\frac {1}{2\beta }}{\sqrt {\frac {\pi }{\beta }}}={\cfrac {fV}{\Lambda ^{3}N_{T}}}\,e^{\beta \mu }\,\!}$

${\displaystyle N_{T}={\cfrac {fV}{\Lambda ^{3}}}e^{\beta \mu }\,\!}$

${\displaystyle P_{E}=2{\sqrt {\frac {\beta ^{3}E}{\pi }}}~e^{-\beta E}\,\!}$

### 正質量費米-狄拉克粒子

${\displaystyle \Phi =e^{\beta (E-\mu )}+1\,\!}$

${\displaystyle 1=\left({\frac {fV}{\Lambda ^{3}N_{T}}}\right)\left[-{\textrm {Li}}_{3/2}(-z)\right]\,\!}$

${\displaystyle N_{T}=\left({\frac {fV}{\Lambda ^{3}}}\right)\left[-{\textrm {Li}}_{3/2}(-z)\right]\,\!}$

### 正質量玻色-愛因斯坦粒子

${\displaystyle \Phi =e^{\beta (E-\mu )}-1\,\!}$

${\displaystyle 1=\left({\frac {fV}{\Lambda ^{3}N_{T}}}\right){\textrm {Li}}_{3/2}(z)\,\!}$

${\displaystyle N_{T}=\left({\frac {fV}{\Lambda ^{3}}}\right){\textrm {Li}}_{3/2}(z)\,\!}$(5)

${\displaystyle N_{T}=\left({\frac {fV}{\Lambda _{c}^{3}}}\right)\zeta (3/2)\,\!}$

${\displaystyle \Lambda \,\!}$等於${\displaystyle \Lambda _{c}\,\!}$的溫度稱為臨界溫度。當溫度低於臨界溫度時，公式(5)沒有解。臨界溫度是玻色-愛因斯坦凝聚開始形成的溫度。可是前面講述的連續近似，忽略了基態。還好，這並不是很嚴重的問題，公式(5)能夠相當正確地求算出的受激態玻色子的數量。因此，

${\displaystyle N_{T}={\frac {g_{0}z}{1-z}}+\left({\frac {fV}{\Lambda ^{3}}}\right){\textrm {Li}}_{3/2}(z)\,\!}$

### 零質量玻色-愛因斯坦粒子

${\displaystyle \Phi =e^{\beta (E-\mu )}-1\,\!}$

${\displaystyle \Phi \,\!}$代入公式(4)，為了方便計算，轉換為頻率的公式：

${\displaystyle P_{\nu }~d\nu ={\frac {fV\beta ^{3}}{2\Lambda ^{3}N_{T}}}{\frac {1}{2}}~{\frac {h^{3}\nu ^{2}}{e^{(h\nu -\mu )/kT}-1}}~d\nu \,\!}$(6)

${\displaystyle 1={\frac {16\,\pi V}{c^{3}h^{3}\beta ^{3}N_{T}}}\,\mathrm {Li} _{3}\left(e^{\mu /kT}\right)\,\!}$

${\displaystyle N_{T}={\frac {16\,\pi V}{c^{3}h^{3}\beta ^{3}}}\,\mathrm {Li} _{3}\left(e^{\mu /kT}\right)\,\!}$

${\displaystyle U_{\nu }~d\nu =\left({\frac {N_{T}\,h\nu }{V}}\right)P_{\nu }~d\nu ={\frac {4\pi fh\nu ^{3}}{c^{3}}}~{\frac {1}{e^{(h\nu -\mu )/kT}-1}}~d\nu \,\!}$

${\displaystyle U_{\nu }={\frac {8\pi h\nu ^{3}}{c^{3}}}~{\frac {1}{e^{h\nu /kT}-1}}\,\!}$

## 參考文獻

• Huang, Kerson. Statistical Mechanics. New York: John Wiley & Sons. 1967.
• Isihara, A. Statistical Physics. New York: Academic Press. 1971.
• Landau, L. D.; E. M. Lifshitz. Statistical Physics 3rd Edition Part 1. Oxford: Butterworth-Heinemann. 1996.
• Yan, Zijun. General thermal wavelength and its applications (PDF). Eur. J. Phys. 2000, 21: 625–631 [2006-11-20]. doi:10.1088/0143-0807/21/6/314.