State vector of a quantum system after a measurement has been performed is given by the measurement postulate. According to this postulate the measuring device is described classicaly, thus the world is divided into quantum and classical parts. Let us investigate a quantum system interacting with a measuring device in more detail. The first steps in a measurement interaction occur at the level of the microscopic components of the macroscopic device. We will describe the microscopic component of the measuring device using quantum mechanics and the rest of the device classically, thus shifting the quantum-classical divide.
The measurement is performed by coupling the quantum system with the detector. Let us consider the interaction of the quantum system with the microscopic part of the detector described by the Hamiltonian [1] \[ H=\gamma p_q x\,, \] where \(x\) is the observable of the quantum system that is being measured, \(p_q\) is the momenum of the detector conjugate to the coordinate of the detector \(q\). A large parameter \(\gamma\) describes the strength of the interaction. The interaction Hamiltonian results in a shift of the detector's position \(q\). If the initial state of the detector is such that \(q\) is well defined, we can obtain the value of \(x\) by the final value of \(q\) after the interaction.
If the wave function of the quantum system before the measurement is \(\psi(x)\), the detector is in the pure state \(\phi(q)\) then the initial state of system and the detector is \[ \Psi(x,q)=\psi(x)\phi(q)\,. \] Assuming that the duration of the interaction is \(\tau\), after the interaction the state becomes \[ \Psi'(x,q)=\exp\left(-\frac{\mathrm{i}}{\hbar}\gamma\tau p_{q}x\right)\Psi(x,q)=\psi(x)\phi(q-\gamma\tau x)\,. \] Applying the projective measurement of the postion \(q\) by the rest of the detector with the result \(q'\) yields the post-measurement wave fuction of the quantum system \[ \psi(x;q')=\frac{1}{\sqrt{P(q')}}\psi(x)\phi(q'-\gamma\tau x)\,, \] where \[ P(q')=\int|\psi(x)|^2|\phi(q'-\gamma\tau x)|^2\,\mathrm{d}x \] is the probability to obtain the outcome \(q'\).
If the initial wave function of the detector \(\phi\) is narrowly centered around \(q_0\), then the final wave function of the quantum system becomes centered around \[ x'=\frac{1}{\gamma\tau}(q'-q_0) \] and the probability to obtain the measured quantity \(x'\) becomes \(P(x')=|\psi(x')|^2\). Thus after the shift of the quantum-classical boundary the changes of the wave function of the quantum system are the same as described by the measurement postulate. This result shows that the precise placement of quantum-classical divide is not important.
For example, let us take the wave function of the detector as \[ \phi(q)=\frac{1}{\sqrt{\epsilon}}f\left(\frac{1}{\epsilon}(q-q_0)\right)\,, \] where \(\epsilon\) is a small quantity describing the width of the detector's wave function and the function $f$ satisfies the condition \[ \int|f(u)|^2\,\mathrm{d}u=1\,. \] In this situation the probability for outcome \(q'\) can be approximated as \[ P(q')\approx\frac{1}{\gamma\tau}|\psi(x')|^2 \] and the probability to obtain the measured quantity \(x'\) is \(P(x')=\gamma\tau P(q')=|\psi(x')|^2\). The wave function of the quantum system after the measurement becomes \[ \psi(x;x')=\frac{\psi(x)}{|\psi(x')|}\sqrt{\frac{\gamma\tau}{\epsilon}}f\left(\frac{\gamma\tau}{\epsilon}(x'-x)\right)\,. \] This is a function with a sharp peak around \(x=x'\). The finite width \(\epsilon\) of the detector's quantum state \(\phi\) results in an measurement error \[ \Delta x=\frac{\epsilon}{\gamma\tau}\,. \]
[1] J. von Neumann, Mathematical Foundations of Quantum Mechanics: New Edition, Translated by R. T. Beyer, Edited by N. A. Wheeler (Princeton Univ. Press, 2018), ISBN 9780691178578, page 287.