Structure of the MQCB equations initial conditions, reversibility and observables

STRUCTURE OF THE MQCB EQUATIONS INITIAL CONDITIONS, REVERSIBILITY AND OBSERVABLES 

The mathematical structure of the four MQCB equations (22-24) is such that the time evolution of the combined quantities F t) is uniquely determined by the MQCB equations, and reversibility is given in a strict mathematical sense. Upon propagation r(0)- F(t), changing d/dt— -9/9t and X(t) — X t) propagates the state F(t) backwards to jrield the same initial state F(t) — r(0). 

This simple mathematical structure of the quantities F(t) together with the MQCB equations also need to be connected to physical quantities. For the initial conditions, we have shown above that F(0) can be chosen in a consistent and physically sound way if the full-dimensional initial wavefunction is known. 

However, due to the approximate nature of the MQCB equations, the fulldimensional wave packet is in general not known at later times. In this sense, the time zero is a special time at which, due to the physical situation considered, the full-dimensional wave packet must be known. However, this is not a severe restriction, since in many physically relevant situations the initial state is either a known asymptotic state like in all coUisional processes or an eigenfunction (often the ground state). 

The second point is to use the mathematical objects F(t) to calculate physically measurable observables. Even though the definition of observables is not unique within the MQCB scheme, in all cases considered so far, we use 

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Structure of the MQCB equations initial conditions, reversibility and observables... 

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