# Confusion on wave packet and creation operator in Mark Srednicki's book

In Mark Srednicki's QFT book, section $$5$$, he mentions following things:

$$a^{\dagger}({\bf k})$$ creates a particle with momentum $$k$$ and is given by $$\begin{equation} a^{\dagger}(k)=-i\int d^3x [e^{ikx}\partial_{0}\phi(x)-\phi(x)\partial_0(e^{ikx})].\tag{5.2} \end{equation}$$ In the next, he defines another operator $$a_1^{\dagger}$$ (see equation 5.6) near momentum $$k_1$$ by $$\begin{equation} a_1^{\dagger}\equiv\int d^3k f_1({\bf k})a^{\dagger}({\bf k}),\tag{5.6} \end{equation}$$ where $$\begin{equation} f_1({\bf k})\propto \exp{[-({\bf k}-{\bf k}_1)^2/4\sigma]}\tag{5.7} \end{equation}$$ is an appropriate wave packet. My confusion is: what is the physical meaning of $$a_1^{\dagger}$$? And what does the "wave packet" mean here? I guess $$a_1^{\dagger}$$ is some operator that creates one-particle state of momentum "near" the given $$k_1$$, but why is the integral defined in whole momentum space?

• Nov 19, 2020 at 20:39
• Feb 11 at 11:00

$$a^{\dagger}_k$$ creates a particle with a definite momentum $$k$$; definite in the sense of Dirac delta function.
While $$a^{\dagger}_1$$ creates a state in which the momentum is not definite but almost smeared over $$3\sigma$$ range about $$k_1$$. Wave packet is proper term to refer such kind of state since this is how we create wave packet in QM.