You are correct that in certain experiments, such as electron diffraction, we observe interference patterns that indicate the superposition of states. The wave-like behavior of particles, including electrons, is indeed described by the Schrödinger equation and does not require invoking the collapse of the wave function.
The concept of wave function collapse arises when we consider the measurement process and the interaction of quantum systems with their surroundings. According to the Copenhagen interpretation of quantum mechanics, when a measurement is made on a quantum system, the wave function collapses to one of the eigenstates of the measured observable, and we obtain a definite outcome.
In the case of electron diffraction, the interference pattern is observed when the electrons are not being measured or disturbed by their environment during the experiment. The electrons are in a superposition of states, and their wave functions interfere with each other, leading to the observed diffraction pattern on the detector screen.
The wave function collapse is not necessary in this scenario because we are not measuring any specific property of the electrons during the diffraction experiment. The interference pattern arises due to the inherent wave-like nature of the electron and the interaction with the diffraction apparatus. The Schrödinger equation can accurately describe this behavior without the need for collapse.
It's important to note that different interpretations of quantum mechanics offer alternative explanations and viewpoints on the nature of measurement and wave function collapse. The measurement problem and the precise interpretation of quantum mechanics are still subjects of ongoing debate and research.