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Regarding Kittel's solid state physics 167 page

  1. Apr 25, 2014 #1
    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
     
  2. jcsd
  3. Apr 25, 2014 #2
    Is ##a## given, or is it something you can specify at will? In the latter case, can you find ##a## that would satisfy the requirement?
     
  4. Apr 25, 2014 #3
    Thank you for the responce.
    I think ##a## is a given quantity. ##a## is the size of the unit cell in one-dimensional lattice.
    The above equation is satisfied only when ##a=1##.
     
  5. Apr 26, 2014 #4
    No, not only. The sine is a periodic function.
     
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