Not quite. But it necessarily has to be described by a different quantum model than unitary dynamics if it is an open system and the rest of the universe is not explicitly modeled.Are you saying that quantum dynamics cannot describe this "jump", but that it necessarily have to be described by classical physics or something outside of any model/theory?
For convenience, physicists often want to describe a small quantum system in terms of only its Hilbert space, when it is in reality not isolated but coupled to a detector (and hence should be described by a unitary deterministic dynamics in a much bigger Hilbert space). This necessarily leads to an effective description of the dynamics of the state of a a small quantum system alone.
Even if the full dynamics of the state of system+detector is deterministic and unitary, the effective dynamics of the state of the system alone is stochastic and nonunitary (dissipative). It can be given by a classical stochastic process for the state vector of the small system. The form of this stochastic process can be derived by refinements of traditional techniques in quantum statistical mechanics. Depending on the kind of coupling to the detector, the effective dynamics is in certain cases a classical jump process described by a master equation, and in other cases a classical diffusion process described by a Fokker-Planck equation, and in general by a combination of both and a deterministic drift term. Here classical refers to the form of the stochastic description - it is still quantum in the sense that the dynamics of a state vector (ray in Hilbert space) is described.
This means that at any fixed time the system is described by a state vector which changes stochastically with time. In a jump process, the state vector changes at random times to another state vector (generalizing the classical dynamics of a Markov chain), and the trajectories are formed by piecewise constant state vectors. In a diffusion process it satisfies instead a Fokker Planck equation (generalizing the classical dynamics of Brownian motion), and the trajectories form Hoelder continuous paths of state vectors, with exponent 1/2.
If a drift term is present, the diffusion process changes to a process described by a stochastic differential equation with a noise term, and the jump process changes to the von Neumann picture of quantum dynamics - namely continuous unitary dynamics interrupted by discontinuous jumps. The jumps are in general governed by POVM probabilities and states.
In special cases the jumps are governed by Born probabilities and eigenstates. Thus von Neumann's dynamics with unitary dynamics interrupted by jumps defined by nonunitary collapse is the correct effective description of the dynamics of certain open systems.
In particular, the stochastic process assigns a trajectory of state vectors to each particular realization of the process, and hence to each single system.
See Section 7 of http://arxiv.org/abs/1511.01069 for a summary how the state of single atoms can be continuously monitored and shows jump of diffusion properties depending on the kind of measurement it is subjected to.
A formal description of the technical side of the reduction process that produces the reduced quantum jump description from the unitary dynamics is given in the references discussed in post #28 below.
These references justify the collapse as an instantaneous approximation on the system-only level to what happens in an interaction with an appropriate measurement device on the system+detector level.