The question you've asked relates to the interpretation of quantum mechanics and the concept of wave function collapse. Different interpretations offer various perspectives on when and how wave function collapse occurs. I'll explain a commonly accepted view, known as the Copenhagen interpretation, which is widely taught and used in the scientific community.
According to the Copenhagen interpretation, the wave function of a quantum system undergoes collapse when it interacts with a measurement apparatus. In the case of the double-slit experiment, where particles (such as electrons or photons) pass through two slits and form an interference pattern on a screen, the wave function collapse occurs when a measurement is made to determine which path the particle took.
When a detector is added to observe which slit the particle passes through, it interacts with the particle and causes the wave function to collapse. The act of detection disturbs the particle's quantum state, destroying the interference pattern and leading to a particle-like behavior with a localized position. The addition of the detector eliminates the wave-like behavior associated with the interference pattern.
If a recorder is connected to the detector to capture the measurement information, it does not have any direct effect on the wave function collapse. The collapse occurs due to the interaction between the particle and the measurement apparatus (in this case, the detector). The recorder merely records the measurement results but does not directly influence the collapse process.
The collapse of the wave function is considered to be an instantaneous and non-reversible process. Once the collapse occurs, the particle is in a definite state (e.g., passing through one specific slit), and subsequent observations will no longer exhibit interference patterns.
It's worth noting that there are alternative interpretations of quantum mechanics, such as the many-worlds interpretation and the pilot-wave theory, which propose different explanations for the behavior of the wave function. These interpretations offer alternative viewpoints on the concept of wave function collapse.