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第713行：
第713行： − + − − 《数学》 第728行：
第726行： − − = \int \langle x,p| \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \frac{dx \,dp}{h}. 第749行：
第745行： − − 《数学》 − − − P _ i = frac { Omega _ b (e-e _ i)}{ Omega _ {(s，b)}(e)}. − − :<math> − p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}. 第770行：
第758行： − 《数学》 − − − − \begin{align} − − 开始{ align } − − :<math> − K ln p _ i & = k ln Omega _ b (e-e _ i)-k ln Omega _ (s，b)}(e)[5 pt ] − \begin{align} − 大约-frac { partial big (k ln Omega _ b (e) big)}{ partial e } e _ i + k ln Omega _ b (e)-k ln Omega _ {(s，b)}(e)) − − k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_{(S,B)}(E) \\[5pt] − − \\[5pt] − − [5 pt ] − &\approx -\frac{\partial S_B}{\partial E} E_i + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)} \\[5pt] − − 大约-frac { partial s _ b }{ partial e } e _ i + k ln frac { Omega _ b (e)}{ Omega _ {(s，b)}(e)}[5 pt ] − \\[5pt] − 约-frac { e _ i }{ t } + k ln frac { Omega _ b (e)}{ Omega _ {(s，b)}(e)} − − &\approx -\frac{\partial S_B}{\partial E} E_i + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)} \\[5pt]
→Connection to probability theory 与概率论的结合
where x, p is a normalised Gaussian wavepacket centered at
where x, p is a normalised Gaussian wavepacket centered at
其中 x，p 是一个被位置 x 和动量 p包围的正态高斯波包被
其中 x，p 是一个被位置 x 和动量 p包围的正态高斯波包
<math>
<math>
Z = \int \operatorname{tr} \left( \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \langle x, p| \right) \frac{dx \,dp}{h}
Z = \int \operatorname{tr} \left( \mathrm{e}^{-\beta\hat{H}} |x, p\rangle \langle x, p| \right) \frac{dx \,dp}{h}
</math>
</math>
A coherent state is an approximate eigenstate of both operators <math> \hat{x} </math> and <math> \hat{p} </math>, hence also of the Hamiltonian Ĥ, with errors of the size of the uncertainties. If Δx and Δp can be regarded as zero, the action of Ĥ reduces to multiplication by the classical Hamiltonian, and Z reduces to the classical configuration integral.
A coherent state is an approximate eigenstate of both operators <math> \hat{x} </math> and <math> \hat{p} </math>, hence also of the Hamiltonian Ĥ, with errors of the size of the uncertainties. If Δx and Δp can be regarded as zero, the action of Ĥ reduces to multiplication by the classical Hamiltonian, and Z reduces to the classical configuration integral.
<math>
<math>
p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}.
p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}.
</math>
</math>
Assuming that the heat bath's internal energy is much larger than the energy of S (E ≫ E<sub>i</sub>), we can Taylor-expand <math>\Omega_B</math> to first order in E<sub>i</sub> and use the thermodynamic relation <math>\partial S_B/\partial E = 1/T</math>, where here <math>S_B</math>, <math>T</math> are the entropy and temperature of the bath respectively:
Assuming that the heat bath's internal energy is much larger than the energy of S (E ≫ E<sub>i</sub>), we can Taylor-expand <math>\Omega_B</math> to first order in E<sub>i</sub> and use the thermodynamic relation <math>\partial S_B/\partial E = 1/T</math>, where here <math>S_B</math>, <math>T</math> are the entropy and temperature of the bath respectively:
<math>
<math>
k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_{(S,B)}(E) \\[5pt]
k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_{(S,B)}(E) \\[5pt]
&\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i + k \ln\Omega_B(E) - k \ln \Omega_{(S,B)}(E)
&\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i + k \ln\Omega_B(E) - k \ln \Omega_{(S,B)}(E)
&\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i + k \ln\Omega_B(E) - k \ln \Omega_{(S,B)}(E)
&\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i + k \ln\Omega_B(E) - k \ln \Omega_{(S,B)}(E)
&\approx -\frac{E_i}{T} + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)}
&\approx -\frac{E_i}{T} + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)}
\end{align}
\end{align}