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I'm following the book of Kaxiras on solid state physics and I'm a bit confused about Brillouin zone and solving Schroedinger equation in BZ. Please let me to write the logical statements I've understood (maybe correct ot not):

1. Crystals are made up of atoms located periodically in 3-D. This space is real space.

2. There is an abstract space called reciprocal space whose lattice vectors are defined in terms of real space vectors. The points in reciprocal space are k-vectors. k-vectors represent momenta of electrons.

3. In periodic potential, electron wavefunctions can be expressed as plane waves as:

http://img145.imageshack.us/img145/5051/94472449is2.th.jpg [Broken]http://g.imageshack.us/thpix.php [Broken]

4. The Schroedinger equation for each electron for each k is given as:

http://img379.imageshack.us/img379/7445/50770285gk9.th.jpg [Broken]http://g.imageshack.us/thpix.php [Broken]

It is said that "Solving this last equation determines uk(r), which with the factor exp(i.k.r) makes up the solution to the original single particle equation."

First question is: After we have got the solution of the equation, we get a wavefunction Psi k (r) where k is a subscript. Does this mean that: "The probability of finding an electron at r which has a momentum of k is |Psi k (r)|^2"?

The second question is: Do we have to solve the above equation for a set of k-vectors in the first Brillouin zone and then will we sum all the wavefunctions to get the actual wavefunction in the real space?

Regards,

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# K-space and Brillouin zone

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