Fourier Transforms: Find Bounds, Dirac Delta, Euler's Identity

[scigal89]

My teacher worked out the following and said it's a unitary transformation (how?) of exp(ikx). He said we're supposed to find the periodic bounds of integration - but I thought for Fourier transforms the bounds are negative infinity to infinity, so in this case shouldn't it just be the Dirac delta function? Also, how do you get the 2 in front of the sin? When I rewrite using Euler's identity, there is no 2.

$$\int e^{ik(x-x')}dk=\frac{e^{ik(x-x')}}{i(x-x')}\approx \frac{2sin[k(x-x')]}{x-x'}$$

He then substituted x' with the deBroglie wave length at the Fermi level. I'm not sure what the physical meaning is...

 

[NateD]

The unitary transformation is a Fourier transform that has been applied to the exponential function. The 2 in front of the sine comes from Euler's identity, which states that $e^{i\theta}=cos \theta + i sin \theta$. When you substitute the exponential for its components, the 2 appears. The periodic bounds of integration for a Fourier transform are indeed negative infinity to infinity, though in this case you have a shifted exponential which means that you can use the Dirac delta function as an approximation for the integral. The physical meaning of substituting x' with the deBroglie wave length at the Fermi level is that it allows you to calculate the energy levels of a system. The deBroglie wave length is related to the momentum of a particle, which influences its energy.