The complete description of a closed quantum system consisting of \(N\) particles is provided by its configuration \(Q\), defined by the positions \(\boldsymbol{Q}_1,\ldots,\boldsymbol{Q}_N\) of the particles, together with the wave function \(\Psi=\Psi(\boldsymbol{q}_1,\ldots,\boldsymbol{q}_N)\). Particles with spin are described by spinor-valued wave functions.
Configuration \(Q\) of a quantum system is random, the probability density for the configuration satisfies the quantum equilibrium distribution \[ \rho(Q)=|\Psi(Q)|^2\,. \]
The change in time of the state of a closed quantum system is given by the guiding equation \[ \frac{\mathrm{d}\boldsymbol{Q}_k}{\mathrm{d}t}=\mathbf{v}_k(Q) \] and the Schrödinger's equation \[ \mathrm{i}\hbar\frac{\partial}{\partial t}\Psi=H\Psi\,, \] were \(H\) is the Hamiltonian and \[ \mathbf{v}_k=\frac{1}{\rho}\boldsymbol{J}_k(\boldsymbol{Q}_1,\ldots\boldsymbol{Q}_N) =\frac{\hbar}{M_k}\mathrm{Im}\frac{\nabla_k\Psi}{\Psi} \] is the velocity field on configuration space determined by the wave function. Here \(\rho\) is the probability density, \(\boldsymbol{J}_k\) is the probability current and \[ \nabla_k=\frac{\partial}{\partial\boldsymbol{q}_k}\,. \]
The properties that are genuinely measured in experiments are the positions of particles given by the configuration \(Q\), particularly the position of the apparatus pointer.
The result \(Z\) of an experiment is determined by the initial configuration \(Q=(X,Y)\) of the system \(X\) and apparatus \(Y\). The initial complete state \((Q,\Psi)\) of the system and apparatus evolves deterministically and uniquely determines the outcome of the experiment. However, as the initial configuration \(Q\) is in quantum equilibrium, the outcome of the experiment is random. The role of the apparatus in many real-world experiments is to provide suitable background fields, which introduce no randomness, as well as a final detection, a measurement of the actual positions of the particles of the system. After the measurement the effective wave function of the system is the conditional wave function \(\psi(x)=\Psi(x,Y)\).