 by haranaza
Published: 14 janvier 2023 (2 semaines ago)
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Find all the latest versions of the PTC Creo Complete family atInvoicewith industry-leading technical solutions for document management, project management, communication and collaboration, and business intelligence.Q:

How does the textbook QFT associate wavefunctions with particle wave functions?

In Peskin & Schroeder, the textbook to which I’m referring, say the following about how it associates wave functions with the scattering of a particle:

The second interpretation of the wave function is that the wave function \$psi\$ is an amplitude of finding a particle in a small space \$V\$ with momentum \$p\$ and spin \$sigma\$. To see that this is the correct interpretation, we consider the general solution of a wave equation in the form \$e^{iS}\$, which can be separated into a part depending on \$p\$ and a part depending on \$sigma\$: \$\$ psi_{p,sigma}(x) =
intlimits_{ -infty}^{infty}dpleft[intlimits_{V}frac{d^3k}{(2pi)^3}
e^{ -ikx}e^{isigmamathbf{n}cdotmathbf{k}}right]e^{ipx} \$\$
This is just a plane wave, and the amplitude of finding a particle with the wave \$e^{isigmamathbf{n}cdotmathbf{k}}\$ is the \$sigma\$ component of the spin projection of the particle. Note that the amplitude \$psi_{p,sigma}\$ depends on \$x\$ only through \$xpmmathbf{n}cdotmathbf{k}\$ and is therefore independent of \$p\$; this means that it is the amplitude for the particle with spin \$sigma\$ to be located at the point \$x\$ of \$V\$. We therefore call \$psi_{p,sigma}\$ the wave function of the particle with spin \$sigma\$ and momentum \$p\$.

I’m having a hard time seeing why this is the correct interpretation. Can you help me with it?

A:

Consider a wave packet centered around \$mathbf{k}\$ and let it propagate along a straight line through the center of mass frame. Over the course of this time, the wave packet will

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