====== Wave function collapse ======

In the language of Bohmian mechanics or when simply looking at [[conditional wave function|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 [(:cite:cwf:norsen2015)] (§2.4).