Help in showing that this inner product is zero

[user1139]

Homework Statement

I want to show that the inner product between the plane wave solution and its conjugate is zero.

Relevant Equations

The inner product is defined as $(u_{\vec{k}},u_{\vec{k}'})=-i\int u_{\vec{k}}\partial_{t}u^{*}_{\vec{k}'}-u^{*}_{\vec{k}'}\partial_{t} u_{\vec{k}}\,\mathrm{d}^3 x$

The unormalised plane wave solution is given as $u_{\vec{k}}=e^{i\vec{k}\cdot\vec{x}-i\omega t}$. I want to show that $(u_{\vec{k}},u^{*}_{\vec{k}'})=0$. However, I don't seem to be able to get the answer through direct calculation. Any hints on how to obtain the answer?

 

[vela]

Show your work please.

 

[vanhees71]

Your question is also not well posed. Which Hilbert space are we supposed to work in? Is it $\mathrm{L}^2(\mathbb{R}^3)$ or is it $\mathrm{L}^2(D)$ with some finite region $D$?

 

[Haorong Wu]

It seems that the inner product is propotional to $$\int \exp [i(\vec{k}+\vec{k}')\cdot\vec{x}]\mathrm{d}^3 x.$$

Shouldn't it be a Dirac-delta function?