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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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