Help in showing that this inner product is zero

user1139
Messages
71
Reaction score
8
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?
 
Physics news on Phys.org
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?
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
Back
Top