From classical mechanics to quantum mechanics Some of these properties will have a value that is a real number,e.g.the position of a particle,others an integer value,e.g.the number of particles that constitute a compound system It is also assumed that all properties can be in principle perfectly known,e.g.they can be perfectly measured.In other terms,the measurement errors can be-at least in principle- always reduced below an arbitrarily small quantity.This is not in contrast with the everyday experimental evidence that any measurement is affected by a finite resolution.Hence.this assumption can be called the postulate of reduction to zero of the measurement error.We should emphasize that this postulate is not a direct consequence of the principle of perfect determination because we could imagine the case of a system that is obiectively determined but cannot be perfectly known Moreover.the variables associated to a system S are in general supposed to be contin- uous,e.g.given two arbitrary values of a physical variable,all intermediate possible real values are also allowed.This assumption is known as the principle of continuity. At this point we can state the first consequence of the three a umptions above:If th state of a system S is perfectly determined at a certain time to and its dynamical variables are continuous and known,then,knowing also the conditions(i.e.the forces that act on the system).it should be possible(at least in principle)to predict with certainty(i.e.with prob- ability equal to one)the future evolution of sfor all times to.This in m me ns that (as we shall see below)are invariant under time reversal (the operation which transforms into -t)also the past behavior of the system for all times t<to is perfectly determined and knowable once its present state is known.Such a con sequence is ally called determin ism.Determinism is implemented by assuming that the system satisfies a set of first-orde differential equations of the form 品s=Fo 1.1) whereSis a vector describing the state of the system.It is also assumed that these equations (called equations of motion)have one and only one solution,and this situation is usual if the functional transformation f is not too nasty Another very important principle,implicitly assumed since the early days of classical mechanics but brought into the scientific debate only in the 1930s.is the principle of sepa rabiliry:given two non-interacting physical systems Si and S2,all their physical properties are separately determined.Stated in other terms.the outcome of a measurement on S cannot depend on a measurement performed on S2. the position todefine mechanics is Let usfirs consider for the sake of simplicity a particle moving in one dimension.Its initial state is well defined by the position xo and momentum po of the particle at time to.The knowledge of the equations of motion of the particle would then allow us to infer the positionx()and the momentum p(r)of the particle at all times It is straightforward to generalize this definition to systems with n degrees of freedom For such a system we distinguish a coordinate configuration space (q.q2.....qn)R and a momentum configuration space(,. pn)E R",where the gi's(j=1.....n) are the generalized coordinates and the pi's(=1...)the generalized momenta.Or 8 From classical mechanics to quantum mechanics Some of these properties will have a value that is a real number, e.g. the position of a particle, others an integer value, e.g. the number of particles that constitute a compound system. It is also assumed that all properties can be in principle perfectly known, e.g. they can be perfectly measured. In other terms, the measurement errors can be – at least in principle – always reduced below an arbitrarily small quantity. This is not in contrast with the everyday experimental evidence that any measurement is affected by a finite resolution. Hence, this assumption can be called the postulate of reduction to zero of the measurement error. We should emphasize that this postulate is not a direct consequence of the principle of perfect determination because we could imagine the case of a system that is objectively determined but cannot be perfectly known. Moreover, the variables associated to a system S are in general supposed to be contin￾uous, e.g. given two arbitrary values of a physical variable, all intermediate possible real values are also allowed. This assumption is known as the principle of continuity. At this point we can state the first consequence of the three assumptions above: If the state of a system S is perfectly determined at a certain time t0 and its dynamical variables are continuous and known, then, knowing also the conditions (i.e. the forces that act on the system), it should be possible (at least in principle) to predict with certainty (i.e. with prob￾ability equal to one) the future evolution of S for all times t > t0. This in turn means that the future of a classical system is unique. Similarly, since the classical equations of motion (as we shall see below) are invariant under time reversal (the operation which transforms t into −t) also the past behavior of the system for all times t < t0 is perfectly determined and knowable once its present state is known. Such a consequence is usually called determin￾ism. Determinism is implemented by assuming that the system satisfies a set of first-order differential equations of the form d dt S = F[S(t)], (1.1) where S is a vector describing the state of the system. It is also assumed that these equations (called equations of motion) have one and only one solution, and this situation is usual if the functional transformation F is not too nasty. Another very important principle, implicitly assumed since the early days of classical mechanics but brought into the scientific debate only in the 1930s, is the principle of sepa￾rability: given two non-interacting physical systems S1 and S2, all their physical properties are separately determined. Stated in other terms, the outcome of a measurement on S1 cannot depend on a measurement performed on S2. We are now in the position to define what a state in classical mechanics is. Let us first consider for the sake of simplicity a particle moving in one dimension. Its initial state is well defined by the position x0 and momentum p0 of the particle at time t0. The knowledge of the equations of motion of the particle would then allow us to infer the position x(t) and the momentum p(t) of the particle at all times t. It is straightforward to generalize this definition to systems with n degrees of freedom. For such a system we distinguish a coordinate configuration space {q1, q2, ... , qn} ∈ IRn and a momentum configuration space {p1, p2, ... , pn} ∈ IRn, where the q j’s (j = 1, ... , n) are the generalized coordinates and the p j’s (j = 1, ... , n) the generalized momenta. On