I have some questions that I was not able to ask during the ‘werkcollege’ could mayby someone help me understand these things?
-
X-ray diffraction exercise 1.2. I don’t understand how to use the hint. A×(B×C)=B(A⋅C)−C(A⋅B)
I understand that from $a_i ⋅ b_i= 2 \pi \delta_{ij} we can derive a_i in terms of b_i in the same way as we derived it for b_i, but I don’t use the hint for this. How can the problem be solved using the hint? -
NFEM op3.3 how is <n-1| K|n> calculated? In previous exercises we could assume <n-1|n>=0, so then <n-1| K|n>=0, why can’t we do this now, why are the different states no longer orthonormal? Furthermore I tried to calculate it by splitting the integral in 3 parts to take care of the absolute values, but I can only do this for n>0, and I get to the answer <n-1| K|n>=-\frac{m \lambda^2}{2 \hbar^2}(1+ \frac{m \lambda}{\hbar^2} a) exp(-\frac{-m a \lambda}{\hbar^2}). If I compare this with the answers I miss a factor \frac{1}{\hbar^2} and my first component is negative instead of positive. I looked over my solution many times but I can’t see where I went wrong, or how I could calculate this otherwise.
I also didn’t manage to calculate <n-1|V|n>. I tried writing it out.
But I didn’t know how to proceed, I think you should use a geometric series, but I didn’t manage to do this, because then you need a lower limit of 0, and I don’t know how to get this because n is unknown. Does someone know how to solve <n-1|V|n>?
- Bandstructures in 2D exercise 3. First of all what is meant by state q?
I tried to use the same procedure as in NFEM exercise 2 and 3 and use
To this purpose I used y=0 and tried to calculate the Fourier coëfficients of V(x,0), I then get \mathcal{F}_0=2 V_{10} and \mathcal{F}_1= \mathcal{F}_0=V_{10}+ 2V_{11}. I then get the matrix:
with eigen energies E_{\pm}=\frac{\hbar^2 \pi^2}{2 m a^2}+2V_{10} \pm |V_{10}+2 V_{11}|
I don’t understand why in the answers there is no +2 V_{10} and why it independent of V_{11}
Is my method correct or how should I apply the NFEM in 2D?
How can you calculate a 2D fourier transform?
And how can you apply this for sections of the plane not parallel to the x or y axis, such as in question 3.2. I did not know how to start with 3.2 and 3.3 Could somebody give me a hint how to approach/solve these quetsions.
