limit of function $\lim_{x\to\infty} e^{ikx}e^{-ik'x}$

Question

I am studying Principles of Quantum Mechanics Shankar.R and on page 66 he says that there is a way to define the limit of function $\lim_{x\to\infty} e^{ikx}e^{-ik'x}$ "to be the average over a large interval".

what would be the intermediate steps ? I fail to understand it completely. My attempts are that he uses the average value of function over an interval [a,b] (as in calculus) and the definition of the dirac delta function using Fourier transform. 1/Delta becomes zero as it delta goes to infinity.

Moreover i cannot find over the internet where in mathematics does someone define the limit of such functions by using average value over a large interval? Why do we do that? Can someone point me a link.

Answer 1

I have no insight into why he's taking an average. However, here are two important pieces of information:

• The function we are considering here is $f(x) = e^{ikx}e^{-ik'x} = e^{i(k-k')x}$.
• To take the average value of a function $f$over $[a,b]$, we compute $$ \frac{1}{b - a}\int_a^b f(x)\,dx $$

Thus, the average of our function over the interval $[L, L + \Delta]$ is the integral $$ \frac{1}{\Delta} \int_L^{L + \Delta} e^{i(k-k')x}\,dx $$ It is this value whose limit we evaluate as $L,\Delta \to \infty$.