From classical mechanics to quantum mechanics In this chapter we shall first summarize some conceptual and formal features of classical mechanics(Sec.1.1).Modern physics started with the works of Galileo Galilei and Isaac Newton from which classical mechanics,one of the most beautiful and solid intellectua buildings of the human history,came out.The architecture of classical mechanics was developed between the end of the eighteenth century and the second half of the nineteenth century,and its present form is largely due to Lagrange,Jacobi,and Hamilton.As we shall tof determinism,a rather complex assumption which is far from being obvious.In Sec.1.2 we shall present the two main conceptual features of quantum mechanics on the basis of an ideal inter- ferometry experiment:the superposition principle and the principle of complementarity. In Sec.1.a first formal treatment of quantum-mechanica states is developed: quantu states are represented by vectors in a space that turns out to be a Hilbert space.In Sec.1.4 the significance of probability for quantum mechanics is explained briefly:we will show that probability is o just an ofu mechanics.but an sic feature of the theory.Fur thermore,we shall see m probability is not ruled by Kolmogorov axioms of classical probability.Finally,we discuss the main evidences which have historically revealed the necessity of a departure from classical mechanics. Our task then is to briefly present the principles upon which quantum mechanics is built (in Secs.1.2-1.4)and to summarize in Sec.1.5 the main evidences for this new mechanics 1.1 Review of the foundations of classical mechanics Classical mechanics is founded upon several principles and postulates.sometime only implicitly assumed.In the following we summarize and critically review such assumptions. First of all,in classical mechanics a principle of perfect determination is assumed:all properties of a pysical systemare determined at any time.Here.we define physical system as an object or a collection of objects (somehow interrelated)that can be (directly or indirectly)experienced through human senses,and a property as the value that can be assigned to a physical variable or observable describing S.Perfectly determined means then that each observable)variable describing S has at all times a definite value 1 See [Auletta 20041
1 From classical mechanics to quantum mechanics In this chapter we shall first summarize some conceptual and formal features of classical mechanics (Sec. 1.1). Modern physics started with the works of Galileo Galilei and Isaac Newton from which classical mechanics, one of the most beautiful and solid intellectual buildings of the human history, came out. The architecture of classical mechanics was developed between the end of the eighteenth century and the second half of the nineteenth century, and its present form is largely due to Lagrange, Jacobi, and Hamilton. As we shall see in this chapter, classical mechanics is built upon the requirement of determinism, a rather complex assumption which is far from being obvious. In Sec. 1.2 we shall present the two main conceptual features of quantum mechanics on the basis of an ideal interferometry experiment: the superposition principle and the principle of complementarity. In Sec. 1.3 a first formal treatment of quantum-mechanical states is developed: quantum states are represented by vectors in a space that turns out to be a Hilbert space. In Sec. 1.4 the significance of probability for quantum mechanics is explained briefly: we will show that probability is not just an ingredient of quantum mechanics, but is rather an intrinsic feature of the theory. Furthermore, we shall see that quantum probability is not ruled by Kolmogorov axioms of classical probability. Finally, we discuss the main evidences which have historically revealed the necessity of a departure from classical mechanics. Our task then is to briefly present the principles upon which quantum mechanics is built (in Secs. 1.2–1.4) and to summarize in Sec. 1.5 the main evidences for this new mechanics. 1.1 Review o f the founda tions o f classical mechanics Classical mechanics is founded upon several principles and postulates, sometimes only implicitly assumed. In the following we summarize and critically review such assumptions.1 First of all, in classical mechanics a principle of perfect determination is assumed: all properties of a physical system S are perfectly determined at any time. Here, we define a physical system as an object or a collection of objects (somehow interrelated) that can be (directly or indirectly) experienced through human senses, and a property as the value that can be assigned to a physical variable or observable describing S. Perfectly determined means then that each (observable) variable describing S has at all times a definite value. 1 See [Auletta 2004]
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 continuous, 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 probability 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 determinism. 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 separability: 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
9 1.1 Review of the foundations of classical mechanics the other hand,the phase space r is the setp.p2.....n)R2n.The state of a system with n degrees of freedom is then represented by a point in the 2n- dimensional phase space r. Let us consider what happens by making use of the Lagrangian approach.Here,the equations of motion can be derived from the knowledge of a Lagrangian function.Given a generalized coordinate gi,we define its canonically conjugate variable or generalized momentum p:as the quantity P时=3gg19a…9a以 (1.2) where theare the generalized velocities.In the simplest case(position-independent kinetic energy and velocity-independent potential)we have L(q1,,,91..:gn)=T(g1,,9n)-V(q1,9n) 13) where L is the Lag fmction andTand Vare the kineticand potential enery. respectively.The kinetic energy isafunction of the generalized velocities=1....) and may also be written as (1.4 i.e.as a function of the generalized momenta pj(j=1.....n).where mj is the mass associated with the j-th degree of freedom. In an altemative e approach,a classical system is defined by the function H=T(p1,p2,,pm)+V(q1,q2,9m (1.5) which is known as the Hamiltonian or the energy function,simply given by the sum of kinetic and potential energy.Differently fron he Lagrangian fun n.H is directly because it the nergy of the ystem.The relationship between Lagrangian and Hamiltonian functions is given by H=∑9jPj-L(q1,9n,g1…:4n) (1.6 in coniunction with (12). For the sake of simplicity we have assumed that the Lagrangian and the Hamiltoniar functions are not explicitly time-dependent.The coordinate qe and momentum p&,together with their time derivatives qs,p&,are linked-through the Hamiltonian-by the Hamilton canonical equations of motion =n=- (1.7) which can also be written in terms of the Poisson brackets as 张=qk,H,庆={Pk,H. (1.8)
9 1.1 Review o f the founda tions o f classical mechanics the other hand, the phase space is the set {q1, q2, ... , qn; p1, p2, ... , pn} ∈ IR2n. The state of a system with n degrees of freedom is then represented by a point in the 2ndimensional phase space . Let us consider what happens by making use of the Lagrangian approach. Here, the equations of motion can be derived from the knowledge of a Lagrangian function. Given a generalized coordinate q j , we define its canonically conjugate variable or generalized momentum p j as the quantity p j = ∂ ∂q˙j L(q1, ... , qn; q˙1, ... , q˙n), (1.2) where the q˙k are the generalized velocities. In the simplest case (position-independent kinetic energy and velocity-independent potential) we have L(q1, ... , qn, q˙1, ... : q˙n) = T (q˙1, ... , q˙n) − V(q1, ... , qn), (1.3) where L is the Lagrangian function and T and V are the kinetic and potential energy, respectively. The kinetic energy is a function of the generalized velocities q˙j (j = 1, ... , n) and may also be written as T = j p2 j 2m j , (1.4) i.e. as a function of the generalized momenta p j ( j = 1, ... , n), where m j is the mass associated with the j-th degree of freedom. In an alternative approach, a classical system is defined by the function H = T (p1, p2, ... , pn) + V(q1, q2, ... , qn), (1.5) which is known as the Hamiltonian or the energy function, simply given by the sum of kinetic and potential energy. Differently from the Lagrangian function, H is directly observable because it represents the energy of the system. The relationship between Lagrangian and Hamiltonian functions is given by H = j q˙j p j − L(q1, ... , qn, q˙1, ... : q˙n) (1.6) in conjunction with (1.2). For the sake of simplicity we have assumed that the Lagrangian and the Hamiltonian functions are not explicitly time-dependent. The coordinate qk and momentum pk , together with their time derivatives q˙k , p˙k , are linked – through the Hamiltonian – by the Hamilton canonical equations of motion q˙k = ∂ H ∂pk , p˙k = −∂ H ∂qk , (1.7) which can also be written in terms of the Poisson brackets as q˙k = {qk , H}, p˙k = {pk , H}. (1.8)
10 From classical mechanics to quantum mechanics The Poisson brackets for two arbitrary functions f and g are defined as -(器-影) (1.9 and have the following properties: U,g1=-{g,f. (1.10a) f,C1=0, (1.10b (Cf +C'g.h)=C(f.h)+C'(g,h). (1.10e 0=f,g,h}+{g,h,f》+h,{f,g, (1.10d 品.e={影8+{} (1.10e) where C.C'are constants and h is a third function.Equation(1.10d)is known as the Jacobi identity.The advantage of this notation is that,for any function f of g and p,we can write d f=(f.H (1.11 It is easy to see that Newton's second law can be derived from Hamilton's equations.In fact,from Eq.(1.8)we have =你，=器 (1.12a A=n训=兴 (1.12b) From Eq.(1.12a)one obtains p&=m(the definition of generalized momentum). which,substituted into Eq.(1.12b).gives av (1.13) Since F=-av/aqk is the generalized force relative to the k-th degree of freedom Eq.(1.13)can be regarded as Newton's second law.As a consequence,Newton's second law can be written in terms of a first-order differential equation (as anticipated above). However,in this case we need both the knowledge of position and of momentum for describing a system In classical mechanics the equations of motion may also be determined by imposing that the action (1.140 has an extreme value.This is known as the Principle of least action or Maupertuis- Hamilton principle. The application of this principle yields the Lagrange equations L=0. (1.15)
10 From classical mechanics to quantum mechanics The Poisson brackets for two arbitrary functions f and g are defined as { f , g} = j ∂ f ∂q j ∂g ∂p j − ∂ f ∂p j ∂g ∂q j , (1.9) and have the following properties: { f , g} = −{g, f }, (1.10a) { f ,C} = 0, (1.10b) {C f + C g, h} = C{ f , h} + C {g, h}, (1.10c) 0 = { f ,{g, h}} + {g,{h, f }} + {h,{ f , g}}, (1.10d) ∂ ∂t { f , g} = ∂ f ∂t , g + f , ∂g ∂t , (1.10e) where C,C are constants and h is a third function. Equation (1.10d) is known as the Jacobi identity. The advantage of this notation is that, for any function f of q and p, we can write d dt f = { f , H}. (1.11) It is easy to see that Newton’s second law can be derived from Hamilton’s equations. In fact, from Eq. (1.8) we have q˙k = {qk , H} = pk mk , (1.12a) p˙k = {pk , H}=− ∂V ∂qk . (1.12b) From Eq. (1.12a) one obtains pk = mkq˙k (the definition of generalized momentum), which, substituted into Eq. (1.12b), gives mkq¨k = − ∂V ∂qk . (1.13) Since Fk = −∂V/∂qk is the generalized force relative to the k-th degree of freedom, Eq. (1.13) can be regarded as Newton’s second law. As a consequence, Newton’s second law can be written in terms of a first-order differential equation (as anticipated above). However, in this case we need both the knowledge of position and of momentum for describing a system. In classical mechanics the equations of motion may also be determined by imposing that the action S = t2 t1 dtL(q1, ... , qn, q˙1, ... , q˙n) (1.14) has an extreme value. This is known as the Principle of least action or Maupertuis– Hamilton principle. The application of this principle yields the Lagrange equations d dt ∂L ∂q˙k − ∂L ∂qk = 0, (1.15)
11 1.1 Review of the foundations of classical mechanics which,as Hamilton equations,are equivalent to Newton's second law.In fact,we have aL 城=ms9kand av (1.16 from which it again follows that d av m=-8g (1.17 For this reason,the Lagrange and Hamilton equations are equivalent.The main difference is that the former is a system of n second-order equations in the generalized coordi- nates.whereas Hamilton equations constitute a system of 2n first-order equations in the From the discussion above it turns out that any state in classical mechanics can be repre sented by a point in the phase space,i.e.it is fully determined given the values of position and momentum (see also Subsec.2.3.3)-when these values cannot be determined with ence,in a probabilistic approach the system is described by a distribution of points in the phase space.whose density p at a certain point(,....:p.....p)measures the probability of finding the system in the state defined by that point It follows that o is a real and positive quantity for which d严q'pp(q1qPi…pm)=l (1.18) i.e.the probability of finding the system in the entire phase space is equal to one.The density p allows us to calculate,at any given time,the mean value of any given physical amjo("dd:ub...b)d uo F(q)(p))=da d"pp(lq)(pDF(lq)(p). (1.19) where (q)and (p))stand for(q1.....n)and (p1.....Pn).respectively. The dynamics of a statistical ensemble of classical systems is subjected to the Liouville equation(orcontinuity equation).Let us denote withp(p:)the density of representative points tha at at timet are contained in the infinitesimal volume elementp q and p.Then it is possible to show that we have 安=a+=0 (1.20) 器=H (1.21)
11 1.1 Review o f the founda tions o f classical mechanics which, as Hamilton equations, are equivalent to Newton’s second law. In fact, we have ∂L ∂q˙k = mkq˙k and ∂L ∂q j = − ∂V ∂qk , (1.16) from which it again follows that d dt (mkq˙k ) = − ∂V ∂qk . (1.17) For this reason, the Lagrange and Hamilton equations are equivalent. The main difference is that the former is a system of n second-order equations in the generalized coordinates, whereas Hamilton equations constitute a system of 2n first-order equations in the generalized coordinates qk and momenta pk . From the discussion above it turns out that any state in classical mechanics can be represented by a point in the phase space, i.e. it is fully determined given the values of position and momentum (see also Subsec. 2.3.3) – when these values cannot be determined with arbitrary precision we have to turn to probabilities. As a consequence, in a probabilistic approach the system is described by a distribution of points in the phase space, whose density ρ at a certain point (q1, ... , qn; p1, ... , pn) measures the probability of finding the system in the state defined by that point. It follows that ρ is a real and positive quantity for which dnq dn pρ(q1, ... , qn; p1, ... , pn) = 1, (1.18) i.e. the probability of finding the system in the entire phase space is equal to one. The density ρ allows us to calculate, at any given time, the mean value of any given physical quantity F, i.e. of a function F(q1, ... , qn; p1, ... , pn) of the canonical variables thanks to the relation F¯({q},{p}) = dnq dn pρ({q},{p}))F({q},{p})), (1.19) where {q} and {p}) stand for (q1, ... , qn) and (p1, ... , pn), respectively. The dynamics of a statistical ensemble of classical systems is subjected to the Liouville equation (or continuity equation). Let us denote with ρ(q, p; t) the density of representative points that at time t are contained in the infinitesimal volume element dn p dnq in around q and p. Then it is possible to show that we have dρ dt = {ρ, H} + ∂ρ ∂t = 0 (1.20) or ∂ρ ∂t = {H, ρ}. (1.21)