Help in showing that this inner product is zero

AI Thread Summary
The discussion revolves around demonstrating that the inner product of two plane wave solutions, ##(u_{\vec{k}},u^{*}_{\vec{k}'})##, equals zero. Participants note that the calculation leads to an integral of the form $$\int \exp [i(\vec{k}+\vec{k}')\cdot\vec{x}]\mathrm{d}^3 x$$, which raises questions about the appropriate Hilbert space context, specifically whether it is ##\mathrm{L}^2(\mathbb{R}^3)## or a finite region ##D##. The expectation is that the result should yield a Dirac-delta function, indicating orthogonality when ##\vec{k} \neq \vec{k}'##. Clarification on the setup of the problem is necessary for accurate resolution. The conversation emphasizes the importance of defining the space in which the inner product is calculated.
user1139
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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?
 
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Show your work please.
 
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##?
 
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?
 
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