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Homework Help: Bandgap in a weak periodic potential

  1. Nov 6, 2008 #1
    Using the Schrodinger equation for an electron in a periodic potential where U(r +R) [R is the translation vector R=n1a1+n2a2+n3a3 and ni are intergers and ai are teh primitive lattice vectors, G is for reciprocal lattice G=n1b1+n2b2+n3b3 and ni are intergers and bi= (2PI*aj x ak)/(a1 . a2 x a3)]

    a)Show that the periodic potential can be expanded as

    U(r)=SUM (over G) exp(iG.r) . U(G)

    show the potential is real and is reflection symmetric U(-r)=U(r)
    show that implies U(-G)=U(complex conjiguate)(G)=U(G)
    potential is chosen as U(G=0)=0

    2. Relevant equations
    I hope you know the SE for a periodic potential......

    3. The attempt at a solution
    can i jump straight to the fourier transform for the U(r+R)
    f(x)=SUM(over m) exp(imx)f(m)
    bcause after that it's just loosing the
    exp (iR.G)=1

    I don't get the complex conjugation of U(r) = U(r) unless the "i" in the exponential isn't changed then there wouldn't be a complex part......

    U(-r) = U(r) cos it's jsut a translation in the real lattice which is stated in teh question.

    It implies that U(-G)=U(complex conjiguate)(G)=U(G) cos the form of U(r) has U(G) in it
  2. jcsd
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