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물리:조화_고체 [2016/05/18 20:19] – admin | 물리:조화_고체 [2023/09/05 15:46] (current) – external edit 127.0.0.1 | ||
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더 자세한 설명은 [[: | 더 자세한 설명은 [[: | ||
- | ======운동 에너지====== | + | ======에너지====== |
+ | =====운동 에너지===== | ||
- | \begin{eqnarray*} | + | \begin{eqnarray} |
\frac{1}{2} m \sum_j \dot{x}_j^2 &=& \frac{1}{2}m \sum_j \dot{x}_j \dot{x}_j\\ | \frac{1}{2} m \sum_j \dot{x}_j^2 &=& \frac{1}{2}m \sum_j \dot{x}_j \dot{x}_j\\ | ||
&=& \frac{1}{2} m \sum_j \left( N^{-1/2} \sum_k \dot{\hat{x}}_k e^{ikR_j} \right)\left( N^{-1/2} \sum_{k' | &=& \frac{1}{2} m \sum_j \left( N^{-1/2} \sum_k \dot{\hat{x}}_k e^{ikR_j} \right)\left( N^{-1/2} \sum_{k' | ||
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&=& \frac{1}{2} m N^{-1} \sum_{kk' | &=& \frac{1}{2} m N^{-1} \sum_{kk' | ||
&=& \frac{1}{2} m \sum_k \dot{\hat{x}}_k \dot{\hat{x}}_{-k}\\ | &=& \frac{1}{2} m \sum_k \dot{\hat{x}}_k \dot{\hat{x}}_{-k}\\ | ||
- | &=& \frac{1}{2} m \sum_k \left| \dot{\hat{x}}_k \right|^2 | + | &=& \frac{1}{2} m \sum_k \left| \dot{\hat{x}}_k \right|^2. |
- | \end{eqnarray*} | + | \end{eqnarray} |
+ | |||
+ | 식 (2)를 적을 때에 허깨비 변수끼리 같을 이유가 없으므로 $k$, $k' | ||
+ | |||
+ | 식 (3)으로부터 (4)로 넘어올 때에 위에서 언급한 [[: | ||
+ | |||
+ | |||
+ | =====위치 에너지===== | ||
+ | |||
+ | \begin{eqnarray} | ||
+ | \frac{1}{2}K \sum_j \left( x_j - x_{j+1} \right)^2 &=& \frac{1}{2}K \sum_j \left( x_j - x_{j+1} \right) \left( x_j - x_{j+1} \right)\\ | ||
+ | & | ||
+ | & | ||
+ | &=& \frac{1}{2}K N^{-1} \sum_{kk' | ||
+ | &=& \frac{1}{2}K \sum_k \hat{x}_k \hat{x}_{-k} \left( 1-e^{ika} \right) \left( 1-e^{-ika} \right)\\ | ||
+ | &=& \frac{1}{2} \sum_k \hat{K}(k) \left| \hat{x}_k \right|^2. | ||
+ | \end{eqnarray} | ||
+ | 이 때에 $\hat{K}(k) = K\left( 1-e^{ika} \right) \left( 1-e^{-ika} \right) = 4K \sin^2 \frac{ka}{2}$로서, | ||
+ | |||
+ | =====전체 에너지===== | ||
+ | 따라서 전체 에너지는 | ||
+ | $$E = \sum_k \left[ \frac{1}{2} m \left| \dot{\hat{x}}_k \right|^2 + \frac{1}{2} \hat{K}(k) \left| \hat{x}_k \right|^2 \right]$$ | ||
+ | 로서 독립적인 [[: | ||
+ | |||
+ | ======같이 보기====== | ||
+ | [[: | ||
======참고문헌====== | ======참고문헌====== | ||
- | *Robert H. Swendsen, An Introduction to Statistical Mechanics and Thermodynamics (Oxford Univ. Press, Oxford, 2012). | + | *Robert H. Swendsen, |