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Is it possible to make two separate atoms or electrons entangled on a quantum level? Not photons, those I know can be entangled already

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As soon as two atoms or electrons interact with each other, they become entangled.

Say you shoot two atoms against each other in a collision experiment. At $t=0$ the atoms have a definite momentum $\hbar k$, $|\Psi \rangle = |k\rangle |-k \rangle$. After the collision, they will be in a superposition of different momenta, but the interaction is such that the total momentum must be conserved. Hence the resulting state will look roughly like this

$$ |\Psi \rangle \approx c_1 |k_1\rangle |-k_1 \rangle + c_2|k_2\rangle |-k_2 \rangle +...$$

which is an entangled state. If you want to understand why the result looks like this you should study quantum scattering theory.

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