Hello,
In the derivation of the NFEM in the lecture notes, the following equation is given:
W = \langle k | V | k' \rangle = \frac{1}{a} \int_0^a e^{-ikx} V(x) e^{ik'x}\,\mathrm{d}x
\tag{1}
The way I understand it, the following are defined as follows:
- | k \rangle is a plane-wave with the following wave function: \psi(x) = e^{ikx}
- V(x) is a periodic potential: V(x + a) = V(x)
- The inner product \langle \psi(x) | V(x) |\phi(x) \rangle is defined as
\langle \psi(x) | V(x) |\phi(x) \rangle = \int_{-\infty}^\infty \psi^*(x) V(x) \phi(x) \, \mathrm{d}x
I would therefore assume that
W = \langle k | V | k' \rangle = \int_{-\infty}^\infty e^{-ikx} V(x) e^{ik'x}\,\mathrm{d}x
but this is apparently wrong.
Why are the limits of the integral not \pm \infty in equation (1)? Does it have something to do with the periodicity of V(x)? If so, could you show me how we derive (1) from the more general equation for the inner product?