[PDF] fourier transform periodic boundary conditions

Does forcing periodicity affect the Fourier transform of a wavefunction?

Figure 2.9.1 2.9. 1: Given periodic boundary conditions, only certain k values are allowed in the "unconfined? direction of a quantum wire. But forcing periodicity in real space affects the Fourier transform of the wavefunction. In k -space, the periodic wavefunction is discrete.

Is the Fourier transform bounded?

The Fourier transform F : L1(Rn) ? L?(Rn) is a bounded operator. This follows from the observation that which shows that its operator norm is bounded by 1. Indeed, it equals 1, which can be seen, for example, from the transform of the rect function.

What if a periodic function has a convergent Fourier series?

If is a periodic function, with period , that has a convergent Fourier series, then: where are the Fourier series coefficients of , and is the Dirac delta function. In other words the Fourier transform is a Dirac comb function whose teeth are multiplied by the Fourier series coefficients.

How do you find a Fourier transform pair?

Now consider the Fourier domain function as a shfted impulse at ?=? 0, X (?)=? (?-?0). In this case we find (again using the sifting property) So, X (?)=? (?-?0) and x(t) =ej?0t/2? x ( t) = e j ? 0 t / 2 ? form a Fourier Transform pair.