A curious student wrote on 04/15/2011 11:18 AM:
> Hi Professor!
> About part b) of problem 3, where we are to show that the integral
> equals 1 when i=j and 0 when i does not equal j, should I go about
> solving each integral for i=1, 2, 3, 4; or is there a simpler approach
> where I can be general that will suffice? Thanks and have a great weekend!

That is a good question.

Please do not do the four cases separately. Possible, but not optimal.

You can work out the integral with x_i, y_i and z_i left as symbols. The integral result will, then, be a (simple) function of these symbols. Once you have that result, it should be apparent that it reduces to a simple number for all cases of i, i.e. for all possible sets of values for x_i, y_i, z_i.

Another curious student wrote on 04/18/2011 10:58 PM:
> Hello Professor,
>
> For problem 1 on Homework 3, I proved (or at least I hope I did) that
> -2t cos(ka) - 2t(sub2)cos(ka/2) is not a valid band dispersion by
> showing that it does not have k - space periodicity. ... snip ...
> Is this a valid method for solving the problem? What concerns me is that
> I didn't need to use the information that t > 0 and t(sub2) > 0. 

Yes, I understand that not using all information could be a concern. However, in this case, only the periodicity matters. So, I think you are doing it perfectly. The integrals such as t and t_2 can be positive or negative depending on materials and bands.