Wave function collapse
In the language of Bohmian mechanics or when simply looking at conditional wave functions the collapse of the wave function during measurement processes arises naturally. Measurement here means a coupling of two quantum systems that evntually leads to strong entanglement. As a very basic example take $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$ with $\mathcal{H}_i = \mathbb{C}^2$ and the basis sets $(\sigma_+,\sigma_-)$ and $(\pi_+,\pi_-)$ respectively. Let the initial state be $\Psi_0 = (c_+\sigma_+ + c_-\sigma_-)\otimes \pi_-$ which means the first subsystem is in some superposition and the second one (the pointer device) is in a fixed state (neutral pointer position). Now for a useful pointer device the Hamitonian must couple the systems in such a way that after some time the state evolves more or less into \[ \Psi = c_+\sigma_+\otimes\pi_+ + c_-\sigma_-\otimes\pi_-. \] So obviously $\pi_-$ measures $\sigma_-$ with probability $|c_-|^2$ and $\pi_+$ measures $\sigma_+$ with probability $|c_+|^2$. If the measurement was performed and we have empiricial evidence of either pointer position $\pi_-$ or $\pi_+$. Setting this outcome as the fixed coordinate in the conditional wave function we automatically collapse to $c_+\sigma_+$ or $c_-\sigma_-$ respectively.
A derivation which uses continuous coordinates instead of discrete ones can be done analogously, see for example [1] (§2.4).