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Matrix elements of non-normalizable states

  1. Jan 13, 2016 #1
    Although strictly quantum mechanics is defined in ##L_2## (square integrable function space), non normalizable states exists in literature.

    In this case, textbooks adopt an alternative normalization condition. for example, for ##\psi_p(x)=\frac{1}{2\pi\hbar}e^{ipx/\hbar}##
    ##
    \langle\psi_p|\psi_{p'}\rangle=\delta(p-p')
    ##

    However, it is not easy calculating matrix elements this way. For example, how to calculate
    ##
    A(k)=i\langle u(k)|\partial_k|u(k)\rangle
    ##
    ##A(k)## is actually berry connection in solid state band theory and ##u(k)## is periodic part of bloch wave function.

    Can anyone tell me how to define this matrix elements?
     
    Last edited: Jan 13, 2016
  2. jcsd
  3. Jan 13, 2016 #2

    vanhees71

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    I do not understand your notation, defining ##A(k)##. It simply doesn't make any sense to me. Where does this come from?
     
  4. Jan 13, 2016 #3
    According to bloch theorem, wave function in crystals should be like ##\psi_k(x)=e^{ikx}u_k(x)##, where ##u_k(x+a)=u_k(x)## and ##a## is lattice constant.

    So ##\langle u(k)|\partial_k|u(k)\rangle## should be something like ##\int u^*_k(x)\partial_k u_k(x)dx##, although it doesn't make sense because this integral is infinite.

    ##A## is berry connection where the adiabatic parameter is ##k##.(https://en.wikipedia.org/wiki/Berry_connection_and_curvature)This quantity is heavily used in topological insulators
     
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