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![The Klein-Gordon equation! Here is the simplest field theory: a scalar field \( \phi(t, x) \) that obeys the equation of motion
\[
\partial_t^2 \psi - \nabla^2 \psi + m^2 \psi = 0,
\]
where \( \nabla \) is our usual gradient operator, and \( \partial_t = \frac{\partial}{\partial t} \) is a nice short-hand. Plug the Ansatz
\[
\psi(t, x) = A(k) e^{-iEt + ik \cdot x}
\]
into the equation and find \( E(k) \) such that the Ansatz solves the equation! Surprise!](https://content.bartleby.com/qna-images/question/5a563a2d-4477-421c-8c33-a788a13e3dd2/8554ce92-d1db-4d1e-9429-f2f6a88be51d/psbo5y_processed.png)
Transcribed Image Text:The Klein-Gordon equation! Here is the simplest field theory: a scalar field \( \phi(t, x) \) that obeys the equation of motion
\[
\partial_t^2 \psi - \nabla^2 \psi + m^2 \psi = 0,
\]
where \( \nabla \) is our usual gradient operator, and \( \partial_t = \frac{\partial}{\partial t} \) is a nice short-hand. Plug the Ansatz
\[
\psi(t, x) = A(k) e^{-iEt + ik \cdot x}
\]
into the equation and find \( E(k) \) such that the Ansatz solves the equation! Surprise!
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