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My understanding is that when two hydrogen atoms bond (for example) the wavefunctions describing each atom combine into a single molecular wavefunction. Assuming the Copenhagen interpretation, does the wavefunction collapse into two particles whenever we look at the molecule, or is it possible for one electron to collapse while the other remains "fuzzy"? I assume this has something to do with entanglement, and electrons within a single molecule are probably all entangled, right?

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The states of the hydrogen molecule come from the molecular Hamiltonian, not a combination of the atomic states. Molecular orbital theory uses the Born-Oppenheimer approximation and the atomic orbitals to get approximate electronic states (for H2).

The states are not necessarily entangled, except in the trivial antisymmetric sense. But that probably isn't what you mean. And for identical particles, "unentangled states" are usually broadened to include the antisymmetrized product states.

Measuring one electron's position doesn't necessarily collapse the other's, but will affect it. This recent measurement of the H2 wave function may interest you.

These results are true in all interpretations.

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