LECTURE 2—09/23/2020CLASSICALV.S.QUANTUM1.CLASSICALMECHANICS INT*RnConsider the simplest system in classical physics:a freeparticle moving in aforce field in Rn. Here, "free" means there is no other outer force acting on theparticle. Denote by V(r) the potential energy function of the force field, which willbe assumed to be time-independent. For simplicity we take the particle mass m = 1.Denote byr(t) = (ri(t),..,an(t))the position vector of the particle at time t, so that the movement of the particle isdescribed by a curvet-r(t)ERnin the configuration space Rn. The only force acting on the particle isF=-VVwhich depends only on the position of the particle. According to the famous New-ton's rule,thepositionvector of theparticle satisfiesthe Newton's second lau(1)(t) = F(r(t)) =-(VV)(r(t),Here and in the future, dot (or dots) means taking derivatives with respect to t. Aremarkablefact of the system (which is of course a trivial application of the Newton'srule)is the conservation law, which claims that the energy function of the system,(2)E=2l(0)P + V(t),is a conserved quantity, i.e. is a constant independent of t.In many situation, when describing such a system, it is very important to de-scribe not only the position vector of the particle, but also the momentum vector(t) =r(t) = (si(t),...,Sn(t))of the particle.By using the vector S,one can rewrite the Newton's equation as(t) =s(t),s(t) = -VV.This is of course trivial in the theory of ordinary differential equations: one canalways rewrite a higher order differential equation as a system of several first orderdifferential equations by introducing new variables. However, it opens a new door to1
LECTURE 2 — 09/23/2020 CLASSICAL V.S. QUANTUM 1. Classical Mechanics in T ∗R n Consider the simplest system in classical physics: a free particle moving in a force field in R n . Here, “free” means there is no other outer force acting on the particle. Denote by V (x) the potential energy function of the force field, which will be assumed to be time-independent. For simplicity we take the particle mass m = 1. Denote by x(t) = (x1(t), · · · , xn(t)) the position vector of the particle at time t, so that the movement of the particle is described by a curve t 7→ x(t) ∈ R n in the configuration space R n . The only force acting on the particle is F = −∇V which depends only on the position of the particle. According to the famous Newton’s rule, the position vector of the particle satisfies the Newton’s second law (1) ¨x(t) = F(x(t)) = −(∇V )(x(t)), Here and in the future, dot (or dots) means taking derivatives with respect to t. A remarkable fact of the system (which is of course a trivial application of the Newton’s rule) is the conservation law, which claims that the energy function of the system, (2) E = 1 2 |x˙(t)| 2 + V (x(t)), is a conserved quantity, i.e. is a constant independent of t. In many situation, when describing such a system, it is very important to describe not only the position vector of the particle, but also the momentum vector ξ(t) = ˙x(t) = (ξ1(t), · · · , ξn(t)) of the particle. By using the vector ξ, one can rewrite the Newton’s equation as ➝ x˙(t) = ξ(t), ˙ξ(t) = −∇V. This is of course trivial in the theory of ordinary differential equations: one can always rewrite a higher order differential equation as a system of several first order differential equations by introducing new variables. However, it opens a new door to 1
2LECTURE2—09/23/2020CLASSICALV.S.QUANTUMboth physicists and mathematicians: inside this door it is Hamiltonian mechanics,or in the language of mathematics, symplectic geometry.In Hamiltonian mechanics, instead of using the configuration space Rn as thebackground space,peopleusethe phase spaceT*R"=R" x R"={(c,S)rR",sER")as the background space. The movement of the particle is then described by a curvet -(t) = (r(t),s(t) E R" × Rnin the phase space. We will call this curve the classical trajectory of the system. Inthis language any point in the phase space (usually called a classical state) describesa possible situation of the system. Moreover, the total energy of a given systemrepresented by a state (c,)is the value of the energy function (always called theenergy observable, or the Hamiltonian)(3)ISP + V(m) e C(T"R")H =atthepoint (r,).UsingtheHamiltonian H,one can rewritethe system ofequationsabove as[(t) =%(r(t),(t),(4)( () - -((), (),This is usually referred to as the system of Hamilton's equations. If we denote EHbe the vector field (called the Hamiltonian vector field associated to H)Ha(Oa(5)EH=OskOkOrkOEthen the system of Hamilton's equations can be rewritten as(6)(t) = 三H((t).In other words, the solution curve = (t) with initial value (O) = % = (ro, So) isthe integral curve of the vector field 三H starting at the point o-Now given any initial data (0) = (ro, Eo), one can solve the system (6), at leastlocally near t = 0. For simplicity we will assume that the solution exists for all t,in otherwords,we assumethevectorfield Eto becompletel.The advantageofintroducing theHamiltonian vectorfield is thefollowing:itgenerates a flow?(calledthe Hamiltonian flow associated with H),(7)Pt = et=H : R2n → R2n, (0) - pt((0) :=(t),1The solution may“blow up at finite time"even if V is smooth.But 三 is complete if we havea nicecontrol of V,e.g.if Vhas a polynomial growth at infinity.2A flow is a family of maps pt : X → X, depending continuously or smoothly in t E IR, suchthatpt oPs=Pt+s:
2 LECTURE 2 — 09/23/2020 CLASSICAL V.S. QUANTUM both physicists and mathematicians: inside this door it is Hamiltonian mechanics, or in the language of mathematics, symplectic geometry. In Hamiltonian mechanics, instead of using the configuration space R n as the background space, people use the phase space T ∗R n = R n × R n = {(x, ξ)|x ∈ R n , ξ ∈ R n } as the background space. The movement of the particle is then described by a curve t 7→ γ(t) = (x(t), ξ(t)) ∈ R n × R n in the phase space. We will call this curve the classical trajectory of the system. In this language any point in the phase space (usually called a classical state) describes a possible situation of the system. Moreover, the total energy of a given system represented by a state (x, ξ) is the value of the energy function (always called the energy observable, or the Hamiltonian) (3) H = 1 2 |ξ| 2 + V (x) ∈ C ∞(T ∗R n ) at the point (x, ξ). Using the Hamiltonian H, one can rewrite the system of equations above as (4) ✭ x˙ k(t) = ∂H ∂ξk (x(t), ξ(t)), ˙ξk(t) = − ∂H ∂xk (x(t), ξ(t)). This is usually referred to as the system of Hamilton’s equations. If we denote ΞH be the vector field (called the Hamiltonian vector field associated to H) (5) ΞH = ❳ k ❶ ∂H ∂ξk ∂ ∂xk − ∂H ∂xk ∂ ∂ξk ➀ , then the system of Hamilton’s equations can be rewritten as (6) ˙γ(t) = ΞH(γ(t)). In other words, the solution curve γ = γ(t) with initial value γ(0) = γ0 = (x0, ξ0) is the integral curve of the vector field ΞH starting at the point γ0. Now given any initial data γ(0) = (x0, ξ0), one can solve the system (6), at least locally near t = 0. For simplicity we will assume that the solution exists for all t, in other words, we assume the vector field ΞH to be complete1 . The advantage of introducing the Hamiltonian vector field is the following: it generates a flow2 (called the Hamiltonian flow associated with H), (7) ρt = e tΞH : R 2n → R 2n , γ(0) 7→ ρt(γ(0)) := γ(t), 1The solution may “blow up at finite time” even if V is smooth. But ΞH is complete if we have a nice control of V , e.g. if V has a polynomial growth at infinity. 2A flow is a family of maps ρt : X → X, depending continuously or smoothly in t ∈ R, such that ρt ◦ ρs = ρt+s
3LECTURE2—09/23/2020CLASSICALV.S.QUANTUMwhich tells us how the system evolutes in time, given any initial state. In particular,the conservation of energy has the following form in this context:Proposition 1.1. The Hamiltonian H is invariant under the Hamiltonian flow pt:H(pt(%) = H(%).Proof.This follows from a simple direct computation:OH.d.OH=0.是H(r(t),(t) =H(pt(%) =i+dtOrOdt口Remark. In particular, we see that each energy surfaceH-1(E) =((c,s) ER2n [H(r,) =E)is invariant under the flow p(t). In applications we will assume E to be a reg-ular value of H, and assume H is proper, so that H-1(E) is a compact smoothsubmanifold.More generally,one calls any real valued smooth function a = a(r,s) defined onthe phase space a classical obseruable. Any physical experiment concerning the sys-tem should lead to quantities which can be described by the values of some classicalobservables.It is important to study therate of change of a classical observablea(r,)of a classical system as t changes, which can be viewed as a generalization of"the law of conservation of the energy observable" to any observable:Proposition 1.2. If we denote a(t) = a(r(t), s(t)) for any classical observable a,thena H)DaoH(8)a(t) =ZOrkaEkaEKOrkkwhere H is the Hamiltonian (3) described above.Proof.Thecomputation isalmostthesame as above:da aHda aHOa..daEka(t) =/SaEKOrKOEkOEROrkKorkk口It is thus natural to introduce the following notion:Definition 1.3. The Poisson bracket of two smooth functions f,g E C(R" × Rn)isthesmoothfunctionofogafog(9)>(f,g] =EC(Rn × R")OEROCkOxkEkk=1
LECTURE 2 — 09/23/2020 CLASSICAL V.S. QUANTUM 3 which tells us how the system evolutes in time, given any initial state. In particular, the conservation of energy has the following form in this context: Proposition 1.1. The Hamiltonian H is invariant under the Hamiltonian flow ρt : H(ρt(γ0)) = H(γ0). Proof. This follows from a simple direct computation: d dtH(ρt(γ0)) = d dtH(x(t), ξ(t)) = ∂H ∂x x˙ + ∂H ∂ξ ˙ξ = 0. Remark. In particular, we see that each energy surface H −1 (E) = {(x, ξ) ∈ R 2n | H(x, ξ) = E} is invariant under the flow ρ(t). In applications we will assume E to be a regular value of H, and assume H is proper, so that H−1 (E) is a compact smooth submanifold. More generally, one calls any real valued smooth function a = a(x, ξ) defined on the phase space a classical observable. Any physical experiment concerning the system should lead to quantities which can be described by the values of some classical observables. It is important to study the rate of change of a classical observable a(x, ξ) of a classical system as t changes, which can be viewed as a generalization of “the law of conservation of the energy observable” to any observable: Proposition 1.2. If we denote a(t) = a(x(t), ξ(t)) for any classical observable a, then (8) ˙a(t) = ❳ k ❶ ∂a ∂xk ∂H ∂ξk − ∂a ∂ξk ∂H ∂xk ➀ , where H is the Hamiltonian (3) described above. Proof. The computation is almost the same as above: a˙(t) = ❳ k ∂a ∂xk x˙ k + ∂a ∂ξk ˙ξk = ❳ k ❶ ∂a ∂xk ∂H ∂ξk − ∂a ∂ξk ∂H ∂xk ➀ . It is thus natural to introduce the following notion: Definition 1.3. The Poisson bracket of two smooth functions f, g ∈ C ∞(R n × R n ) is the smooth function (9) {f, g} = ❳n k=1 ❶ ∂f ∂ξk ∂g ∂xk − ∂f ∂xk ∂g ∂ξk ➀ ∈ C ∞(R n × R n ).
4LECTURE2-09/23/2020CLASSICALV.S.QUANTUMSo we can rewrite the evolution equation (8) of a classical observable a simply as(10)a(t) =H,al.But, where is symplectic geometry? Well, for a more complicated system, thephase space is a symplectic manifold, i.e.a smooth manifold M together with anon-degenerate closed 2-form w. For example in the case M = T*Rn one just takew=dakΛdEk.Note that the Hamiltonian vector field E associated to H defined by the formula(5) is related to H by the symplectic form via the following equation (which can becalled Hamilton's equations because the system of Hamilton's equations alluded toabove is just the integral curves of 三)(11)1W=dH.Everything above generalize to symplectic manifolds: A smooth function H (calledthe Hamiltonian) will play the role of the energy observable above. The HamiltonianH is preserved along any classical trajectory, which is the trajectory of the Hamil-tonianflow associated to theHamiltonianvectorfield.Onestillhasthenotion ofPoisson bracket in this abstract setting, and the equation of motion is againa ={H,a].Although it is very interesting to consider quite general symplectic manifolds,in this course we will mainly focus on a special class of symplectic manifolds: thecotangent bundleM =T*X ofa smooth manifoldX,which,as wewill seein thefuture, admits a natural symplectic structure.2.QUANTUM MECHANICSIN L?(Rn)Quantum mechanics is much more difficult to understand. The mathematicaltheory to describe a quantum mechanical system is functional analysis. The space ofstates isusually a complexHilbert space(H,(,),ormore precisely,theprojectifiedHilbert space PH which consists of the unit vectors in H. (But we will always workon H itself instead of on PH because the results for H implies the results for PH,while H is much easier to handle.)In the case that the phase space of a classical system is the cotangent bundleT*Rn as we just described, one usually takeH = L(R"),with the usual L?-inner product<f,g)=/f(c)g(a)dr
4 LECTURE 2 — 09/23/2020 CLASSICAL V.S. QUANTUM So we can rewrite the evolution equation (8) of a classical observable a simply as (10) ˙a(t) = {H, a}. But, where is symplectic geometry? Well, for a more complicated system, the phase space is a symplectic manifold, i.e. a smooth manifold M together with a non-degenerate closed 2-form ω. For example in the case M = T ∗R n one just take ω = ❳dxk ∧ dξk. Note that the Hamiltonian vector field ΞH associated to H defined by the formula (5) is related to H by the symplectic form via the following equation (which can be called Hamilton’s equations because the system of Hamilton’s equations alluded to above is just the integral curves of ΞH) (11) ιΞH ω = dH. Everything above generalize to symplectic manifolds: A smooth function H (called the Hamiltonian) will play the role of the energy observable above. The Hamiltonian H is preserved along any classical trajectory, which is the trajectory of the Hamiltonian flow associated to the Hamiltonian vector field. One still has the notion of Poisson bracket in this abstract setting, and the equation of motion is again a˙ = {H, a}. Although it is very interesting to consider quite general symplectic manifolds, in this course we will mainly focus on a special class of symplectic manifolds: the cotangent bundle M = T ∗X of a smooth manifold X, which, as we will see in the future, admits a natural symplectic structure. 2. Quantum Mechanics in L 2 (R n ) Quantum mechanics is much more difficult to understand. The mathematical theory to describe a quantum mechanical system is functional analysis. The space of states is usually a complex Hilbert space (H,h·, ·i), or more precisely, the projectified Hilbert space PH which consists of the unit vectors in H. (But we will always work on H itself instead of on PH because the results for H implies the results for PH, while H is much easier to handle.) In the case that the phase space of a classical system is the cotangent bundle T ∗R n as we just described, one usually take H = L 2 (R n ), with the usual L 2 -inner product hf, gi = ❩ Rn f(x)g(x)dx
LECTURE2—09/23/2020CLASSICAL V.S. QUANTUM5A (t-independent) quantum state (or a wave function) is a function b E L?(R")satisfyinglb(r)da = 1.Note that in quantum mechanics, the position of a particle is no longer a de-termined quantity: one can only compute the probability of finding a particle in acertain region.In the language of Hilbert space above, the probability of finding aquantum particle with wave function in ameasurableregion U c Rn is I(r)Pdr.[This explains why mustbenormalized, i.e.has norm1 in L?(Rn).]What are the guantum observables? They are linear operators A acting on Hwhichare self-adjoint sothat the eigenvalues of A arereal numbers.Wewill assumethat the eigenvalues of A are discrete and the corresponding eigenfunctions canbe chose to form an orthonormal basis of L?(Rn).The quantum mechanics ariomstates the set of eigenvalues (usually called the spectrum) of a quantum observableis exactly the set ofpossible values that can be obtained in ameasurement.Moreprecisely, suppose the eigenvalues of A areA,'s and the corresponding orthonormalset of eigenfunctions are j's, i.e.Api=Ajpi(pk,i) = OkjWe can express any normalized state eL?(Rn) as a summationX=cjpjj=0Note that [|2=1 impliesZlc,/2 = 1.Then the quantum mechanics axiom claims thatthe result of a quantum observation, A, to a quantum system in thestate b, is an eigenvalue of A, in the following sense: the result is(again!)not a determined quantity, any eigenvalue might be theresult, and the probability that one gets the eigenvalue , is Ic,j?As a consequence, we can proveProposition 2.1. The erpected value of a measurement of the quantum observableAtoaquantum systeminthestatew is(12)(A)= (Ab, ).Proof. The expected value isA,lc,l=(Acjpi,Zckk) =(Ab,)S
LECTURE 2 — 09/23/2020 CLASSICAL V.S. QUANTUM 5 A (t-independent) quantum state (or a wave function) is a function ψ ∈ L 2 (R n ) satisfying ❩ Rn |ψ(x)| 2 dx = 1. Note that in quantum mechanics, the position of a particle is no longer a determined quantity: one can only compute the probability of finding a particle in a certain region. In the language of Hilbert space above, the probability of finding a quantum particle with wave function ψ in a measurable region U ⊂ R n is ❩ U |ψ(x)| 2 dx. [This explains why ψ must be normalized, i.e. has norm 1 in L 2 (R n ). ] What are the quantum observables? They are linear operators A acting on H which are self-adjoint so that the eigenvalues of A are real numbers. We will assume that the eigenvalues of A are discrete and the corresponding eigenfunctions can be chose to form an orthonormal basis of L 2 (R n ). The quantum mechanics axiom states the set of eigenvalues (usually called the spectrum) of a quantum observable is exactly the set of possible values that can be obtained in a measurement. More precisely, suppose the eigenvalues of A are λj ’s and the corresponding orthonormal set of eigenfunctions are ϕj ’s, i.e. Aϕj = λjϕj , hϕk, ϕj i = δkj . We can express any normalized state ψ ∈ L 2 (R n ) as a summation ψ = ❳∞ j=0 cjϕj . Note that |ψ|L2 = 1 implies ❳ j |cj | 2 = 1. Then the quantum mechanics axiom claims that the result of a quantum observation, A, to a quantum system in the state ψ, is an eigenvalue of A, in the following sense: the result is (again!) not a determined quantity, any eigenvalue might be the result, and the probability that one gets the eigenvalue λj is |cj | 2 . As a consequence, we can prove Proposition 2.1. The expected value of a measurement of the quantum observable A to a quantum system in the state ψ is (12) hAiψ = hAψ, ψi. Proof. The expected value is ❳ j λj |c 2 j | = hA ❳cjϕj , ❳ckϕki = hAψ, ψi
6LECTURE2—09/23/2020CLASSICALV.S.QUANTUM口Remark. Note that if we multiply by a “"constant phase" ei/h, then (A)(A)ete/ny. In other words, we can't distinguish b and eie/hab. This, together with thefact lllz2 = 1, explains why the“true"model should be the projectified HilbertspacePH instead of HWe have just seen that the classical Hamiltonian H determines the behavior ofthe classical system. What is the quantum Hamiltonian that plays the same rolefora quantum system? In Schrodinger's picture of quantum mechanics,the quantummechanical analogueof the Hamiltonian (3)is thetime-independentSchrodingeroperatorh2H:(13)△+V(r),2and the quantum mechanical analogue of the system of Hamilton's equations (4) isthe Schrodinger equationo=,(14)ihatwhere h = 6.62607015× 10-34 is the Planck's constant and h = h/2元 is the reducedPlanck'sthaeradtentialfV(r) E Co(Rn) is a real-valued function which acts on L?(Rn) by multiplication,and = (t, ) is the quantum mechanical time-evolution of a quantum state attime t. Note that the quantum Hamiltonian H arises from the classical Hamilton-ian H by replacing the momentum variable Ee with the operator 1.Itisveryimportant to notice that H is a self-adjoint operator acting on L?(Rn).What is the quantum analogue of the Hamiltonian flow pt = et=n? Since pt mapsan initial classical state o to its time-t classical state (t), the quantum analoguemustmaps the initial quantum state (O)=oto its time-tquantum statet,whichisthesolutiontotheproblem=0(0)=0oIn other words, the quantum analogue of pt has to be solution operator of theSchrodingerequation, which we can formally denoted byU(t) = e-tH/n : L2(R") → L?(R").Note that U(t) maps an L?-normalized function to an L?-normalized one (which iswhat weneed),since we haveob）=（，a(U(t)bo,U(t)bo) = <(b, H)= 0.）idt
6 LECTURE 2 — 09/23/2020 CLASSICAL V.S. QUANTUM Remark. Note that if we multiply ψ by a “constant phase” e iθ/~ , then hAiψ = hAie iθ/~ψ. In other words, we can’t distinguish ψ and e iθ/~ψ. This, together with the fact kψkL2 = 1, explains why the “true” model should be the projectified Hilbert space PH instead of H. We have just seen that the classical Hamiltonian H determines the behavior of the classical system. What is the quantum Hamiltonian that plays the same role for a quantum system? In Schr¨odinger’s picture of quantum mechanics, the quantum mechanical analogue of the Hamiltonian (3) is the time-independent Schr¨odinger operator (13) Hˆ = − ~ 2 2 ∆ + V (x), and the quantum mechanical analogue of the system of Hamilton’s equations (4) is the Schr¨odinger equation (14) i~ ∂ψ ∂t = Hψ, ˆ where h = 6.62607015 × 10−34 is the Planck’s constant and ~ = h/2π is the reduced Planck’s constant, ∆ = P ∂ 2 ∂x2 k is the Laplace operator, and the potential function V (x) ∈ C ∞(R n ) is a real-valued function which acts on L 2 (R n ) by multiplication, and ψ = ψ(t, ·) is the quantum mechanical time-evolution of a quantum state ψ at time t. Note that the quantum Hamiltonian Hˆ arises from the classical Hamiltonian H by replacing the momentum variable ξk with the operator ~ i ∂ ∂xk . It is very important to notice that Hˆ is a self-adjoint operator acting on L 2 (R n ). What is the quantum analogue of the Hamiltonian flow ρt = e tΞH ? Since ρt maps an initial classical state γ0 to its time-t classical state γ(t), the quantum analogue must maps the initial quantum state ψ(0) = ψ0 to its time-t quantum state ψt , which is the solution to the problem ➝ i~ ∂ψ ∂t = Hψ, ˆ ψ(0) = ψ0. In other words, the quantum analogue of ρt has to be solution operator of the Schr¨odinger equation, which we can formally denoted by U(t) = e −itH/ ˆ ~ : L 2 (R n ) → L 2 (R n ). Note that U(t) maps an L 2 -normalized function to an L 2 -normalized one (which is what we need), since we have d dthU(t)ψ0, U(t)ψ0i = h ∂ψ ∂t , ψi + hψ, ∂ψ ∂t i = 1 i~ hHψ, ψ ˆ i − 1 i~ hψ, Hψˆ i = 0.
LECTURE2—09/23/2020CLASSICAL V.S. QUANTUM7In the language of mathematics, U(t) is a unitary operator3 acting on L?(Rn).Asp(t), the operator U(t) form a Schrodinger (semi-)groupU(t)U(s) =U(t+s).It is usually called the propagator associated with the quantum Hamiltonian H.What is the quantum analogue of the conservation of energy, which tells us thatthe classical Hamiltonian H is unchanged under the flow pt? It should tell us thatthe expected value (H)(t) is unchanged. This can be verified by almost the samecomputation as above:(H))=《自(),(0)=《b,)+()dt2万(H,)=0.(HH,)-Finally let's study the equation of motion in a quantum system. Let A be aquantum observable (which is time independent).As we have seen, at time t theresult of A to a quantum state (t) = (t,-) is (A)+(t). By repeating the samecomputation the third time, we getddH(4)(4(),(0)=的)+(41(A，0）-(HA,)i[A,H],),访where(15)[A, H] = AH- HAis the Lie bracket between the operators A and H, and we have used the facts thatH is self-adjoint and that the conjugate of i is -i. So we end up withProposition 2.2. The equation of motion of a quantum system isd(16)显(4) =([H, A):Comparing this with proposition 1.2, we are led to thefollowing principle:The normalized Lie bracket [l,J is the quantum analogue of thePoisson bracket I,}.Remark. As in the classical case where we can replace R2n by a cotangent bundleT*X, here in the quantum case we can replace Rn by a (Riemannian) manifold X.3Being a unitary operator means that U(t) : L?(Rn) → L?(R") always preserves the innerproduct structure. This is the quantum analogue of the fact that pt : T*Rn → T*R" alwayspreservesthe symplectic structure
LECTURE 2 — 09/23/2020 CLASSICAL V.S. QUANTUM 7 In the language of mathematics, U(t) is a unitary operator3 acting on L 2 (R n ). As ρ(t), the operator U(t) form a Schr¨odinger (semi-)group U(t)U(s) = U(t + s). It is usually called the propagator associated with the quantum Hamiltonian Hˆ . What is the quantum analogue of the conservation of energy, which tells us that the classical Hamiltonian H is unchanged under the flow ρt? It should tell us that the expected value hHˆ iψ(t) is unchanged. This can be verified by almost the same computation as above: d dthHˆ iψ(t) = d dthHψˆ (t), ψ(t)i = hHˆ 1 i~ Hψ, ψ ˆ i + hHψ, ˆ 1 i~ Hψˆ i = 1 i~ hHˆ Hψ, ψ ˆ i − 1 i~ hHˆ Hψ, ψ ˆ i = 0. Finally let’s study the equation of motion in a quantum system. Let A be a quantum observable (which is time independent). As we have seen, at time t the result of A to a quantum state ψ(t) = ψ(t, ·) is hAiψ(t) . By repeating the same computation the third time, we get d dthAiψ(t) = d dthAψ(t), ψ(t)i = hA 1 i~ Hψ, ψ ˆ i + hAψ, 1 i~ Hψˆ i = 1 i~ hAHψ, ψ ˆ i − 1 i~ hHAψ, ψ ˆ i = 1 i~ h[A, Hˆ ]ψ, ψi, where (15) [A, Hˆ ] = AHˆ − HAˆ is the Lie bracket between the operators A and Hˆ , and we have used the facts that Hˆ is self-adjoint and that the conjugate of i is −i. So we end up with Proposition 2.2. The equation of motion of a quantum system is (16) d dthAiψ = i ~ h[H, A ˆ ]iψ. Comparing this with proposition 1.2, we are led to the following principle: The normalized Lie bracket i ~ [·, ·] is the quantum analogue of the Poisson bracket {·, ·}. Remark. As in the classical case where we can replace R 2n by a cotangent bundle T ∗X, here in the quantum case we can replace R n by a (Riemannian) manifold X. 3Being a unitary operator means that U(t) : L 2 (R n) → L 2 (R n) always preserves the inner product structure. This is the quantum analogue of the fact that ρt : T ∗R n → T ∗R n always preserves the symplectic structure