# 布洛赫波

${\displaystyle \psi ({\boldsymbol {r}})=\mathrm {e} ^{\mathrm {i} {\boldsymbol {k}}\cdot {\boldsymbol {r}}}u({\boldsymbol {r}}).}$

${\displaystyle \psi ({\boldsymbol {r}}+{\boldsymbol {R_{n}}})=\mathrm {e} ^{\mathrm {i} {\boldsymbol {k}}\cdot {\boldsymbol {R_{n}}}}\psi ({\boldsymbol {r}})}$

## 布洛赫定理的证明

${\displaystyle {\hat {T}}_{{\boldsymbol {R}}_{n}}f({\boldsymbol {r}})=f({\boldsymbol {r}}+{\boldsymbol {R}}_{n})}$

${\displaystyle {\hat {H}}({\boldsymbol {r}}+{\boldsymbol {R}}_{n})={\hat {H}}({\boldsymbol {r}})}$

${\displaystyle {\hat {T}}_{{\boldsymbol {R}}_{n}}{\hat {H}}\psi ({\boldsymbol {r}})={\hat {H}}({\boldsymbol {r}}+{\boldsymbol {R}}_{n})\psi ({\boldsymbol {r}}+{\boldsymbol {R}}_{n})={\hat {H}}({\boldsymbol {r}})\psi ({\boldsymbol {r}}+{\boldsymbol {R}}_{n})={\hat {H}}{\hat {T}}_{{\boldsymbol {R}}_{n}}\psi ({\boldsymbol {r}})}$

${\displaystyle \psi ({\boldsymbol {r}})}$ 为这两个算符的共同本征函数，${\displaystyle \lambda _{{\boldsymbol {R}}_{n}}}$ 是对应本征值，那么有：

${\displaystyle {\hat {T}}_{{\boldsymbol {R}}_{n}}\psi ({\boldsymbol {r}})=\psi ({\boldsymbol {r}}+{\boldsymbol {R}}_{n})=\lambda _{{\boldsymbol {R}}_{n}}\psi ({\boldsymbol {r}})}$

${\displaystyle \int |\psi ({\boldsymbol {r}})|^{2}{\text{d}}{\boldsymbol {r}}=\int |\psi ({\boldsymbol {r}}+{\boldsymbol {R}}_{n})|^{2}{\text{d}}{\boldsymbol {r}}=1}$

${\displaystyle \lambda _{{\boldsymbol {R}}_{n}}=e^{i\beta _{{\boldsymbol {R}}_{n}}}}$

${\displaystyle \lambda _{{\boldsymbol {R}}_{n}+{\boldsymbol {R}}_{m}}=\lambda _{{\boldsymbol {R}}_{m}}\lambda _{{\boldsymbol {R}}_{n}}}$

${\displaystyle \beta _{{\boldsymbol {R}}_{n}+{\boldsymbol {R}}_{m}}=\beta _{{\boldsymbol {R}}_{n}}+\beta _{{\boldsymbol {R}}_{m}}}$

${\displaystyle \lambda _{{\boldsymbol {R}}_{n}}=e^{i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{n}}}$

${\displaystyle \psi ({\boldsymbol {r}}+{\boldsymbol {R}}_{n})={\hat {T}}_{{\boldsymbol {R}}_{n}}\psi ({\boldsymbol {r}})=\lambda _{{\boldsymbol {R}}_{n}}\psi ({\boldsymbol {r}})=e^{i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{n}}\psi ({\boldsymbol {r}})}$

## 参考资料

• 黄昆原著，韩汝琦改编，《固体物理学》，高等教育出版社，北京，1988，ISBN 7-04-001025-9
• 阎守胜编著，《固体物理基础》（第三版），北京大学出版社，ISBN 978-7-301-18863-7
• Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).
• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
• Felix Bloch, "Über die Quantenmechanik der Elektronen in Kristallgittern," Z. Physik 52, 555-600 (1928).
• George William Hill, "On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon," Acta. Math. 8, 1-36 (1886).（本文初版于1877年，后曾被私下传阅）。
• Gaston Floquet, "Sur les équations différentielles linéaires à coefficients périodiques," Ann. École Norm. Sup. 12, 47-88 (1883).
• Alexander Mikhailovich Lyapunov, The General Problem of the Stability of Motion (London: Taylor and Francis, 1992).（李雅普洛夫的博士论文，1892年完稿，稳定性理论的奠基之作）