**1. The problem statement, all variables and given/known data**

In Kittel’s ‘Introduction to solid state physics’ (8th ed.), on page 167, it says "The wave functions at the Brillouin zone boundary ##k=\pi/a## are ##\sqrt{2} cos (\pi x/a)## and ##\sqrt{2} sin(\pi x/a)##, normalized over unit length of line."

Here I cannot understand what is the meaning of "normalized over unit length of line".

Does that mean that, when I integrate the square of the wave function from 0 to 1, the result should be 1? But

$$ \int_0^1 2cos^2 \frac{\pi x}{a} dx = \int_0^1 (1+cos \frac{2\pi x}{a})dx $$

$$ = 1+\frac{a}{2\pi} sin\frac{2\pi}{a} \neq 1. $$

Please help me find out what is wrong in my reasoning.

Thanks.

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

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