- 2,100

- 16

V(x) = \left\{\begin{array}{ll}

0, & \exists n\in\mathbb{Z},\; x\in [2nL, (2n+1)L]\\

\infty, &\exists n\in\mathbb{Z},\; x\in\; ](2n-1)L, 2nL[\\

\end{array}\right.

[/tex]

This is a periodic potential. L is some constant. Is a solution

[tex]

\psi(x) = \chi_{[0,L]}(x)\;\sin\big(\frac{\pi x}{L}\big)

[/tex]

of the Schrödinger's equation

[tex]

\Big(-\frac{\hbar^2}{2m}\partial_x^2 + V(x)\Big)\psi(x) = E\psi(x)

[/tex]

a counter example to the Bloch's theorem?

[tex]\chi_{[0,L]}[/tex] is a characteristic function, 1 when [tex]x\in [0,L][/tex] and 0 otherwise.