Double Slit Experiment - Measurement Problem

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Spherical Cow
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Question
Suppose molecules are placed next to the slits that fluoresce when electron passes, emitting a photon that can be imaged. Explain why this is no better than direct illumination, and explain how the two-slit interference pattern is destroyed.

My Answer
In direct illumination, a narrow beam of light is passed through one slit. Each time an electron passes through the slit, it will scatter a few photons, which can be detected. Collision with the photons will change the electron's momentum, thus destroying the interference pattern.

My Question
How is the interference pattern destroyed with the molecular fluorescence experiment? Is it due to the emitted photons interacting with the electrons? I.e. same reasoning as the direct illumination experiment?
 
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Well, what happens when matter fluoresce?
 
The electron excites the molecule. When the molecule relaxes to its ground state, it releases photons.

Is it the transfer of energy from the electron to the molecule that disrupts the interference pattern? Or is it the interaction between the electron and photons (i.e. electron losing momentum)?
 
When the electron transfer energy to the molecule, it transfer its kinetic energy. Specifically, it transfers kinetic energy by colliding with the electrons of the molecule, which makes it lose momentum.