de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent Hilbert space based formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schr\(\ddot{o}\)dinger wave function \(\psi \). Using a fixed time wave function \(\psi _{0}\) to make the system configuration space \( M (= \mathbb {R}^n)\) a probability space \(\mathcal {M}_{0} = (\mathbb {R}^n, |\psi _{0}|^2 dx)\), we introduce a stochastic process \(\xi (t)\) such that its expectation value in \(\mathcal {M}_{0}\) equals that of the Heisenberg position operator \(X_{H}(t)\) in the Heisenberg state \(|\psi _{H}\rangle \) corresponding to \(\psi _{0}\). This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process \(\xi (t)\) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the \(\psi _{0}\)-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli’s equation (the Schr\(\ddot{o}\)dinger equation for a nonrelativistic charged spin half particle) is treated as an example. A straightforward derivation of von Neumann’s projection rule employing the Schr\(\ddot{o}\)dinger - Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
