Final answer:
In quantum mechanics, preserving the Gaussianity of states implies a simplified system evolution. The mean field approximation, by transitioning to a quadratic Hamiltonian, approximates the system with non-interacting quasiparticles, but may not capture complex quantum correlations. The probability density of a wave function underpins the location predictions for particles within the framework of the Copenhagen interpretation.
Step-by-step explanation:
In the context of quantum mechanics, when discussing the evolution of theory that preserves the Gaussianity of states, we are considering scenarios where the physical system remains in states that can be fully described by their mean values and variances. This implies that more complex phenomena such as entanglement might not be prominent in the given evolution.
Referring to the mean field approximation, it simplifies the system to a more tractable form by reducing the effect of particle-particle interactions to an average effect exerted by all the other particles. This approximation can indeed often lead to effective quadratic Hamiltonians, which imply the system can be described in terms of non-interacting quasiparticles. However, the recovery of the notion of particles is fundamentally an approximation and doesn't always capture all the dynamics of the original many-body system, especially regarding quantum fluctuations and correlations that are beyond the mean field approach.
Wave packets are an example of how particles are modeled in quantum mechanics—a superposition of plane-wave states that describes a particle localized within a certain region of space. Moreover, due to the Heisenberg uncertainty principle, the more localized a particle is in space, the less certain we are of its momentum. The probability density represented by the square of a wave function is instrumental in determining where a particle might be found upon measurement, a concept that is at the core of the Copenhagen interpretation of quantum mechanics.
The nature and behavior of quantum systems can significantly vary when considering different models and approximations, like the mean field approach versus more exact methodologies that include interactions and fluctuations at a microscopic level. Insights into quantum oscillators and the role of entropy in physical systems, as discussed by Ludwig Boltzmann, highlight the profound relationship between microscopic states and macroscopic properties such as temperature and order.