In the language of Bohmian mechanics or simply looking at conditonal 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 set $(\chi_+,\chi_-)$. Let the initial state be $\Psi_0 = (c_+\chi_+ + c_-\chi_-)\otimes \chi_-$ which means the first subsystem is in some superposition and the second one (the pointer device) is in a fixed state.