2016量子-37 - *3.7*3.7.1(Floquet-Bloch U x a U x a 1 2 H...

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1 *§3.7 周期性势场中的能带结构 *3.7.1 有限平移不变性 弗洛盖 - 布洛赫 (Floquet-Bloch) 定理 粒子在周期性势场中运动时,它的状态介于束缚态与非束缚态之间,而能谱具有带式的结构。 以一维情况为例,周期性势场就是 ) ( ) ( x U a x U , 其中 a 是满足此式的最小正数,称为势场的周期,所以粒子的哈密顿量 2 1 ˆ ˆ ( ) 2 H p U x m 在有限平移变换 x x a 下保持不变。这也是一种对称性。虽然它并不导致什么守恒量,但仍然有非常丰富的物理结果。 问题仍然是求解能量本征方程 2 2 ( ) 0. m E U x   Floquet 定理:周期性势场中的波函数 ) ( x 满足条件 i ( ) e ( ), Ka x a x 其中 K 是常数。由于 K 2 / a 为周期,所以通常选它在区间 ( / , / ) a a 中变化,这个区间称为 第一布里渊 (Brillouin) 区,简写为 1st BZ 。满足上述条件的函数称为 准周期函数 。定理的严格证明这里从 略。直观来看,粒子在周期场中出现的几率也是周期性的,所以 2 2 ) ( ) ( x a x ,因此 ( ) x a ( ) x 有一个相位因子的差别。 Bloch 定理:周期性势场中的波函数可以写为如下形式: i ( ) e ( ), Kx K x x 其中 ) ( x K
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