$
\newcommand{\I}{\mathrm{i}}
\newcommand{\E}{\mathrm{e}}
\newcommand{\bvec}[1]{\boldsymbol{\mathbf{#1}}}
\newcommand{\mat}[1]{\boldsymbol{\mathbf{#1}}}
\newcommand{\ten}[1]{\boldsymbol{\mathbf{#1}}}
\newcommand{\Nabla}{\boldsymbol{\nabla}}
\renewcommand{\Tr}{^{\mathrm{T}}}
\newcommand{\uvec}[1]{\hat{\boldsymbol{\mathbf{#1}}}}
\renewcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}
\newcommand{\ali}[1]{\begin{aligned}#1\end{aligned}}
\newcommand{\leftgroup}[1]{\left\{\begin{aligned}#1\end{aligned}\right.}
\newcommand{\vmat}[1]{\begin{vmatrix}#1\end{vmatrix}}
\newcommand{\Cj}{^*}
\newcommand{\Her}{^\dagger}
\newcommand{\Q}[1]{\hat #1}
\newcommand{\Qv}[1]{\uvec #1}
\newcommand{\sinc}{\operatorname{sinc}}
\newcommand{\Arctan}{\operatorname{Arctan}}
\newcommand{\erfi}{\operatorname{erfi}}
\newcommand{\Arctan}{\operatorname{Arctan}}
\newcommand{\Si}[1]{\,\mathrm{#1}}
\newcommand{\les}{\leqslant}
\newcommand{\ges}{\geqslant}
\newcommand{\qty}[1]{\left\{{#1}\right\}}
\newcommand{\qtyRound}[1]{\left({#1}\right)}
\newcommand{\qtySquare}[1]{\left[{#1}\right]}
\newcommand{\abs}[1]{\left\lvert{#1}\right\rvert}
\newcommand{\eval}[1]{\left.{#1}\right\rvert}
\newcommand{\comm}[2]{\left[{#1},{#2}\right]}
\newcommand{\commStar}[2]{[{#1},{#2}]}
\newcommand{\pb}[2]{\left\{{#1},{#2}\right\}}
\newcommand{\pbStar}[2]{\{{#1},{#2}\}}
\newcommand{\vdot}{\boldsymbol\cdot}
\newcommand{\cross}{\boldsymbol\times}
\newcommand{\grad}{\boldsymbol\nabla}
\newcommand{\div}{\boldsymbol{\nabla}\boldsymbol{\cdot}}
\newcommand{\curl}{\boldsymbol{\nabla}\boldsymbol{\times}}
\newcommand{\laplacian}{\boldsymbol{\nabla}^2}
\newcommand{\sinRound}[2][{}]{\sin^{#1}\left(#2\right)}
\newcommand{\cosRound}[2][{}]{\cos^{#1}\left(#2\right)}
\newcommand{\tanRound}[2][{}]{\tan^{#1}\left(#2\right)}
\newcommand{\cscRound}[2][{}]{\csc^{#1}\left(#2\right)}
\newcommand{\secRound}[2][{}]{\sec^{#1}\left(#2\right)}
\newcommand{\cotRound}[2][{}]{\cot^{#1}\left(#2\right)}
\newcommand{\sinhRound}[2][{}]{\sinh^{#1}\left(#2\right)}
\newcommand{\coshRound}[2][{}]{\cosh^{#1}\left(#2\right)}
\newcommand{\tanhRound}[2][{}]{\tanh^{#1}\left(#2\right)}
\newcommand{\arcsinRound}[2][{}]{\arcsin^{#1}\left(#2\right)}
\newcommand{\arccosRound}[2][{}]{\arccos^{#1}\left(#2\right)}
\newcommand{\arctanRound}[2][{}]{\arctan^{#1}\left(#2\right)}
\newcommand{\expRound}[1]{\exp\left(#1\right)}
\newcommand{\logRound}[2][{}]{\log^{#1}\left(#2\right)}
\newcommand{\lnRound}[2][{}]{\ln^{#1}\left(#2\right)}
\renewcommand{\Re}{\mathrm{Re}}
\renewcommand{\Im}{\mathrm{Im}}
\newcommand{\dd}[1][]{\,\mathrm{d}^{#1}}
\newcommand{\dv}[2][{}]{\frac{\mathrm{d}^{#1}}{\mathrm{d}{#2}^{#1}}}
\newcommand{\dvStar}[2][{}]{\mathrm{d}^{#1}/\mathrm{d}{#2}^{#1}}
\newcommand{\dvTwo}[3][{}]{\frac{\mathrm{d}^{#1}{#2}}{\mathrm{d}{#3}^{#1}}}
\newcommand{\dvStarTwo}[3][{}]{\mathrm{d}^{#1}{#2}/\mathrm{d}{#3}^{#1}}
\newcommand{\pdv}[2][{}]{\frac{\partial^{#1}}{\partial{#2}^{#1}}}
\newcommand{\pdvStar}[2][{}]{\partial^{#1}/\partial{#2}^{#1}}
\newcommand{\pdvTwo}[3][{}]{\frac{\partial^{#1}{#2}}{\partial{#3}^{#1}}}
\newcommand{\pdvStarTwo}[3][{}]{\partial^{#1}{#2}/\partial{#3}^{#1}}
\newcommand{\pdvThree}[3]{\frac{\partial^2{#1}}{\partial{#2}\partial{#3}}}
\newcommand{\pdvStarThree}[3]{\partial^2{#1}/\partial{#2}\partial{#3}}
\newcommand{\bra}[1]{\left\langle{#1}\right\rvert}
\newcommand{\braStar}[1]{\langle{#1}\rvert}
\newcommand{\ket}[1]{\left\lvert{#1}\right\rangle}
\newcommand{\ketStar}[1]{\lvert{#1}\rangle}
\newcommand{\braket}[1]{\left\langle{#1}\middle|{#1}\right\rangle}
\newcommand{\braketStar}[1]{\langle{#1}|{#1}\rangle}
\newcommand{\braketTwo}[2]{\left\langle{#1}\middle|{#2}\right\rangle}
\newcommand{\braketStarTwo}[2]{\langle{#1}|{#2}\rangle}
\newcommand{\ev}[1]{\left\langle{#1}\right\rangle}
\newcommand{\evStar}[1]{\langle{#1}\rangle}
\newcommand{\evTwo}[2]{\left\langle{#2}\middle|{#1}\middle|{#2}\right\rangle}
\newcommand{\evStarTwo}[2]{\langle{#2}|{#1}|{#2}\rangle}
\newcommand{\mel}[3]{\left\langle{#1}\middle|{#2}\middle|{#3}\right\rangle}
\newcommand{\melStar}[3]{\langle{#1}|{#2}|{#3}\rangle}
\newcommand{\order}[1]{\mathcal{O}\left(#1\right)}
\newcommand{\bmat}[1]{\begin{bmatrix}#1\end{bmatrix}}
\newcommand{\Bmat}[1]{\left\{\begin{matrix}#1\end{matrix}\right\}}
\newcommand{\sumint}[1]{\int\kern-1.4em\sum}
\newcommand{\Q}[1]{\hat{#1}}
\newcommand{\opn}{\operatorname}
\newcommand{\norm}[1]{\left\lVert{#1}\right\rVert}
$
理想气体单粒子能级密度
相空间法
\begin{equation}
\Omega_0 = \frac{1}{h^3}\int\limits_{\sum {p^2} \leqslant 2mE} \dd[3]{q} \dd[3]{p}
= \frac{V}{h^3}\frac43 \pi {p^3}
= \frac{V}{h^3}\frac43 \pi (2m\varepsilon)^{3/2}
\end{equation}
\begin{equation}
a(\varepsilon) = \dvTwo{\Omega_0}{\varepsilon} = \frac{2\pi V(2m)^{3/2}}{h^3} \varepsilon^{1/2}
\end{equation}
量子力学法
由盒中粒子得, 单粒子的能级为
\begin{equation}
\varepsilon = \frac{\hbar ^2}{2m} \qtySquare{\qtyRound{\frac{\pi n_x}{L_x}}^2 + \qtyRound{\frac{\pi n_y}{L_y}}^2 + \qtyRound{\frac{\pi n_z}{L_z}}^2} = \frac{\hbar ^2}{2m} (k_x^2 + k_y^2 + k_z^2)
\end{equation}
在 $k$ 空间中, 每个能级所占的体积为
\begin{equation}
V_1 = \frac{\pi^3}{L_x L_y L_z} = \frac{\pi^3}{V}
\end{equation}
$K$ 空间中, 能量小于 $E$ 的量子态数为(注意 $n$ 为正值, 所以只求一个卦限的体积, 加 $1/8$ 系数)
\begin{equation}
\Omega_0 = \left. \frac18\cdot \frac{\hbar^2}{2m}\frac43 \pi k^3 \middle/ \frac{\pi^2}{V} \right. = \frac{V}{h^3}\frac43 \pi(2m\varepsilon)^{3/2}
\end{equation}
\begin{equation}
a(\varepsilon) = \dvTwo{\Omega_0}{\varepsilon} = \frac{2\pi V(2m)^{3/2}}{h^3} \varepsilon^{1/2}
\end{equation}
致读者: 小时物理百科一直以来坚持所有内容免费且不做广告,这导致我们处于日渐严重的亏损状态。长此以往很可能会最终导致我们不得不选择商业化,例如大量广告,内容付费,会员制,甚至被收购。因此,我们鼓起勇气在此请求广大读者
热心捐款,使网站得以健康发展。如果看到这条信息的每位读者能慷慨捐助 10 元,我们几天内就能脱离亏损状态,并保证网站能在接下来的一整年里向所有读者继续免费提供优质内容。感谢您的支持。
—— 小时(项目创始人)