LECTURE309/27/2020QUANTIZATION V.S.SEMI-CLASSICAL LIMITLet's summarizewhatwedidlasttime:Classical worldQuantum worldThe cotangent bundle T*RnThefunction space L?(Rn)Space of stateswith symplectic structurewith Hilbert structurew=Eda,Adei(f,g) = Jen f(r)g(c)daStateapoint (r,s)afunction bwith lll=1HamiltonianThe energy functionThe Schrodinger operatorH(r,E) = +V(r)H=-A+VObservablesReal-valued functionsSelf-adjoint operatorsaE C(T*R")A : L?(R") → L?(Rn)The value a(r, $)Result of an observablean eigenvalue of Awexpected value 《A)=(Ab, )Schrodinger's equationEvolution of a stateHamilton's equations≤=%--器=bEvolution of the systemThe propagator (unitary)The flow (symplectomorphism)U(t) =e-碧: L2(R") -→ L2(R")Pt =etH : T*Rn → T*Rn《H)u(t)do =(H)oConservation of energyH(pt(to, So)) = H(ro, So)Evolution of observablesa= [H,a]%《A)(t) = 《[H, A)(t)In some sense, one of the major task in this courseis to add more classicalquantumcorrespondencetothislist
LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT Let’s summarize what we did last time: Classical world Quantum world Space of states The cotangent bundle T ∗R n The function space L 2 (R n ) with symplectic structure with Hilbert structure ω = P dxi ∧ dξi hf, gi = ❘ Rn f(x)g(x)dx State a point (x, ξ) a function ψ with kψk = 1 Hamiltonian The energy function The Schr¨odinger operator H(x, ξ) = kξk 2 2 + V (x) Hˆ = − ~ 2 2 ∆ + V Observables Real-valued functions Self-adjoint operators a ∈ C ∞(T ∗R n ) A : L 2 (R n ) → L 2 (R n ) Result of an observable The value a(x, ξ) an eigenvalue of A expected value hAiψ =hAψ, ψi Evolution of a state Hamilton’s equations Schr¨odinger’s equation x˙ = ∂H ∂ξ , ˙ξ = − ∂H ∂x i~ ∂ψ ∂t = Hψˆ Evolution of the system The flow (symplectomorphism) The propagator (unitary) ρt = e tΞH : T ∗R n → T ∗R n U(t) = e − itHˆ ~ : L 2 (R n ) → L 2 (R n ) Conservation of energy H(ρt(x0, ξ0)) = H(x0, ξ0) hHˆ iU(t)ψ0 = hHˆ iψ0 Evolution of observables d dta = {H, a} d dthAiψ(t) = h i ~ [H, A ˆ ]iψ(t) In some sense, one of the major task in this course is to add more classicalquantum correspondence to this list. 1
2LECTURE 3-09/27/2020 QUANTIZATION V.S.SEMI-CLASSICAL LIMIT1.QUANTIZATIONThe word “quantization"refers to any systematical way of constructing a quan-tum mechanical description of a system from its classical mechanical description. Inthe language of mathematics,it should be a“functor"that givesSymplectic space (M,w)Hilbert space (H,(, ),(C(M, R), {, ) (self-adjoint operators on H, At theverybeginning of quantummechanics,Dirac suggested several natural axiomsthat a quantization procedure should satisfy:Dirac's Axioms. A quantization procedure assigns to any classical observable a ECo(M, R) a self-adjoint operator Q(a) on some Hilbert space H, such that(D1) (linearity) Q(μa+vb) =μQ(a)+vQ(b) for any μ,VER.(D2)(normalization) Q(1)=Id.(quantizaiton condition) Q({a,b)) = [Q(a), Q(b)].(D3)(minimality) Any complete family of functions on M passes to a complete(D4)family of self-adjoint operators on H.-A family of functions on M is complete if they separate points almosteverywhere on M.-A family of operators on H is complete if they act irreducibly on H, i.e.no nonzero proper closed subspace of H is invariant under the family ofoperators.Unfortunately, such a quantization does not exists!Theorem 1.1 (Groenewold-van Hove no-go theorem). A quantization procedurethat satisfies (D1)-(D4) for all f E Co(M) does not erist.In fact, one can not even quantize all polynomials of degree ≤ 4 in Coo(T*R),:To see this one need to observe the following facts:. The first key ingredient is a Schur-type lemma:Lemma 1.2. Suppose Y is an self-adjoint operator on H such that[Y,Q(r)] = [Y,Q()] = 0,thenY =c·Id for some constant c.Proof. We will need some facts from functional analysis:-By a theorem in functional analysis (c.f. K. Yosida, Functional analysis,6th ed., page 339), if A is a self-adjoint operator and B is a boundedself-adjoint operator, and A, B commutes, then for any Baire functionf, f(A) commutes with B.Inparticular,B commutes with any spectralprojection xE(A) of A onto an interval finite E
2 LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 1. Quantization The word “quantization” refers to any systematical way of constructing a quantum mechanical description of a system from its classical mechanical description. In the language of mathematics, it should be a “functor” that gives Symplectic space (M, ω) Hilbert space (H,h·, ·i), (C ∞(M, R), {·, ·}) (self-adjoint operators on H, i ~ [·, ·]). At the very beginning of quantum mechanics, Dirac suggested several natural axioms that a quantization procedure should satisfy: Dirac’s Axioms. A quantization procedure assigns to any classical observable a ∈ C ∞(M, R) a self-adjoint operator Q(a) on some Hilbert space H, such that (D1) (linearity) Q(µa + νb) = µQ(a) + νQ(b) for any µ, ν ∈ R. (D2) (normalization) Q(1) = Id. (D3) (quantizaiton condition) Q({a, b}) = i ~ [Q(a), Q(b)]. (D4) (minimality) Any complete family of functions on M passes to a complete family of self-adjoint operators on H. – A family of functions on M is complete if they separate points almost everywhere on M. – A family of operators on H is complete if they act irreducibly on H, i.e. no nonzero proper closed subspace of H is invariant under the family of operators. Unfortunately, such a quantization does not exists! Theorem 1.1 (Groenewold-van Hove no-go theorem). A quantization procedure that satisfies (D1)-(D4) for all f ∈ C ∞(M) does not exist. In fact, one can not even quantize all polynomials of degree ≤ 4 in C ∞(T ∗R),. To see this one need to observe the following facts: • The first key ingredient is a Schur-type lemma: Lemma 1.2. Suppose Y is an self-adjoint operator on H such that [Y, Q(x)] = [Y, Q(ξ)] = 0, then Y = c · Id for some constant c. Proof. We will need some facts from functional analysis: – By a theorem in functional analysis (c.f. K. Yosida, Functional analysis, 6th ed., page 339), if A is a self-adjoint operator and B is a bounded self-adjoint operator, and A, B commutes, then for any Baire function f, f(A) commutes with B. In particular, B commutes with any spectral projection χE(A) of A onto an interval finite E
3LECTURE3—09/27/2020QUANTIZATION V.S.SEMI-CLASSICAL LIMIT-Conversely, if B commutes with any spectral projection Xe(A) of A,then by spectral theorem, B commutes with A.Note that as a consequence of thefirst fact,for any finite interval F. Xe(B)commutes with Xe(A).Then as a consequence of the second fact, XF(B)commutes with A.Now we start to prove the theorem. First assume Y is a bounded selfadjoint operator. Then as we just argued, for any finite interval F C R,the spectral projection XF(Y) commutes with Q(r) and Q(),i.e.the rangeof the operator XF(Y) is a subspace of H which is invariant under bothQ(r) and Q(s). But since the two functions and from a complete set offunctions on T*R, by (D4) the two operatorsQ(r)and Q($)act irreduciblyon H. As a result, the range of XF(Y) is either O or H for any F. So thespectrum ofY containsonly onepoint.The case where Y is unbounded is more subtle, because all operatorsinvolved areonlydenselydefined.Under some extraassumptionsthat allcomputations involved make sense in a dense subspace,we can concludethat y?, and thus the operator y2+Id, commutes with Q(r) and Q()Since Y? + Id is positive and self-adjoint, it is invertible.Moreover, itsinverse (y2 +Id)-1 is a bounded self-adjoint operator which commutes withbothQ(r)and Q(s).Nowwecan applywhat we justprovedto claim that口(y2 + Id)-1 has the form cld, and the conclusion follows.. Using the above lemma one can show Q(r) = Q(r)Q() + cId for someconstant c,and then inductivelyproveQ(rm) =Q(r)m and Q(sm) =Q(s)mThe second key observation is that we have the following two different ways towrite a degree 4 polynomial as Poisson brackets of two degree 3polynomials:[3,53] =-9r22=3[r25,r2]A contradiction will appearif calculate Q(r2) and Q(r2) explicitly.Wewill leave the details as an exercise.This is both a bad news and a good news for mathematicians: on one hand, thereis no “perfect theory"describing every aspect of the classical-quantum correspon-dence. On the other hand side, by focusing on different aspects of the correspon-dence, mathematicians and physicists has developed several different theories underthe title "quantization".Among these different quantization schemes, three of themare most popular and are widely studied by (different groups of) mathematicians:(A)Weyl quantization.This is one of the first quantization theory in mathemat-ics (proposed by H.Weyl in 1927).It can be viewed as analysts'quantizationmethod sinceitmainlyuses thelanguage of (Fourier)analysis,butit isalsoveryclosely related to representation theory. In Weyl quantization we only quantizecotangent bundles. One start point to understand the theory is the canonical
LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 3 – Conversely, if B commutes with any spectral projection χE(A) of A, then by spectral theorem, B commutes with A. Note that as a consequence of the first fact, for any finite interval F, χF (B) commutes with χE(A). Then as a consequence of the second fact, χF (B) commutes with A. Now we start to prove the theorem. First assume Y is a bounded selfadjoint operator. Then as we just argued, for any finite interval F ⊂ R, the spectral projection χF (Y ) commutes with Q(x) and Q(ξ), i.e. the range of the operator χF (Y ) is a subspace of H which is invariant under both Q(x) and Q(ξ). But since the two functions x and ξ from a complete set of functions on T ∗R, by (D4) the two operators Q(x) and Q(ξ) act irreducibly on H. As a result, the range of χF (Y ) is either 0 or H for any F. So the spectrum of Y contains only one point. The case where Y is unbounded is more subtle, because all operators involved are only densely defined. Under some extra assumptions that all computations involved make sense in a dense subspace, we can conclude that Y 2 , and thus the operator Y 2 + Id, commutes with Q(x) and Q(ξ). Since Y 2 + Id is positive and self-adjoint, it is invertible. Moreover, its inverse (Y 2 + Id)−1 is a bounded self-adjoint operator which commutes with both Q(x) and Q(ξ). Now we can apply what we just proved to claim that (Y 2 + Id)−1 has the form cId, and the conclusion follows. • Using the above lemma one can show Q(xξ) = Q(x)Q(ξ) + cId for some constant c, and then inductively prove Q(x m) = Q(x) m and Q(ξ m) = Q(ξ) m. The second key observation is that we have the following two different ways to write a degree 4 polynomial as Poisson brackets of two degree 3 polynomials: {x 3 , ξ3 } = −9x 2 ξ 2 = 3{x 2 ξ, xξ2 }. A contradiction will appear if calculate Q(x 2 ξ) and Q(xξ2 ) explicitly. We will leave the details as an exercise. This is both a bad news and a good news for mathematicians: on one hand, there is no “perfect theory” describing every aspect of the classical-quantum correspondence. On the other hand side, by focusing on different aspects of the correspondence, mathematicians and physicists has developed several different theories under the title “quantization”. Among these different quantization schemes, three of them are most popular and are widely studied by (different groups of) mathematicians: (A) Weyl quantization. This is one of the first quantization theory in mathematics (proposed by H. Weyl in 1927). It can be viewed as analysts’ quantization method since it mainly uses the language of (Fourier) analysis, but it is also very closely related to representation theory. In Weyl quantization we only quantize cotangent bundles. One start point to understand the theory is the canonical
NLECTURE3-09/27/2020QUANTIZATIONV.S.SEMI-CLASSICALLIMITquantization introduced by P.Dirac in his doctoral thesis in 1926,in whichthe position functions rk's and the momentum functions Ek's are quantized tothe position operators Q's and the momentum operators P's. The centralrelation between these operators is the canonical commutative relationsh[Qk,Qk] = 0, [Pk, P] = 0, [Qk, P] =oik-Id.which is of course the quantum analogue of the classical facts[k,c] =0,[Sk,S]=0, [Ck,S] = OkjAs we mentioned in lecture 1, in Weyl quantization we takeQ=multiplicaton by khaEx~ Pe=TOrkand as a result we can quantize[5/22+V(r)wH-H-A +V(r)22However,it isnotclearhow to quantize functionsin mired variables likeiSi=i+i, since the multiplications of functions are commutative, whileEi1 =thecompositionsofoperatorsarenotcommutative.[Insomesenseitisthenon.commutativity in the quantum side that makes the whole theory interesting.jH.Weylhad theidea of usingFourier transform to solvethisproblem:for anynicefunctiona(r,E)ECoo(T*Rn),wecan quantizeittotheoperator A:L?(Rn)L?(Rn)defined viaFouriertransform,by[ ei-p-a(“",s)()dyde.(Af)(r) = (awf)(r) :=(2元)nSuch operators are called semi-classical pseudodifferential operators. Wewillsee that the Weyl quantization does not satisfy the quantization condition[aW,6W]+(a,bywbut it does satisfy a weakened version of quantization condition[aw,bW] -(a,b) W + O(h?).We will also quantize canonical relations (like the Hamiltonian flow) betweencotangent bundles.The resulting operator are called Fourier integral opera-tors. Pseudodifferential operators and Fourier integral operators are the majorobjectsofthiscourse.lin the language of representation, we really get a representation of the Heisenberg Lie group
4 LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT quantization introduced by P. Dirac in his doctoral thesis in 1926, in which the position functions xk’s and the momentum functions ξk’s are quantized to the position operators Qk’s and the momentum operators Pk’s. The central relation between these operators is the canonical commutative relations1 [Qk, Qk] = 0, [Pk, Pj ] = 0, [Qk, Pj ] = ~ i δjk · Id, which is of course the quantum analogue of the classical facts {xk, xj} = 0, {ξk, ξj} = 0, {xk, ξj} = δkj . As we mentioned in lecture 1, in Weyl quantization we take xk Qk = multiplicaton by xk ξk Pk = ~ i ∂ ∂xk , and as a result we can quantize H = |ξ| 2 2 + V (x) Hˆ = − ~ 2 2 ∆ + V (x). However, it is not clear how to quantize functions in mixed variables like x1ξ1 = ξ1x1 = x1ξ1+ξ1x1 2 , since the multiplications of functions are commutative, while the compositions of operators are not commutative. [In some sense it is the noncommutativity in the quantum side that makes the whole theory interesting.]H. Weyl had the idea of using Fourier transform to solve this problem: for any nice function a(x, ξ) ∈ C ∞(T ∗R n ), we can quantize it to the operator A : L 2 (R n ) → L 2 (R n ) defined via Fourier transform, by (Af)(x) = (a W f)(x) := 1 (2π) n ❩ e i (x−y)]·ξ ~ a( x + y 2 , ξ)ψ(y)dydξ. Such operators are called semi-classical pseudodifferential operators. We will see that the Weyl quantization does not satisfy the quantization condition [a W , bW ] 6= ~ i {a, b} W , but it does satisfy a weakened version of quantization condition: [a W , bW ] = ~ i {a, b} W + O(~ 2 ). We will also quantize canonical relations (like the Hamiltonian flow) between cotangent bundles. The resulting operator are called Fourier integral operators. Pseudodifferential operators and Fourier integral operators are the major objects of this course. 1 In the language of representation, we really get a representation of the Heisenberg Lie group
LECTURE3—09/27/2020QUANTIZATIONV.S.SEMI-CLASSICAL LIMIT5(B) Geometric quantization. The theory was introduced by Kostant and Souriauin 1970s,and ismainly studied bygeometers.In afuture course on symplecticgeometry, I will describe in detail how to quantize in the general symplecticsetting. In geometric quantization, the Hilbert space will be taken to be the setof smooth sections of some “pre-quantum line bundle" satisfying some extraconditions.(C) Deformation quantization. The theory was introduced by Flato, Lichnerowiczand Sternheimer in 1976, and has been studied extensively since then mainly byalgebraists. In this theory, the main focus is trying to understand quantizationas a deformation of the (Poisson) structure of the algebra of classical observ-ables.For example,people want to construct a new noncommutativeproduct* on Co(M)[] (whose elements are formal power series in h with coefficientsin C(M), where M is a symplectic or Poisson manifold) such that(f,g) + O(h2).f*g-g*f=We will see that in Weyl quantization, such a star product can be defined forcotangent bundles.2. SEMICLASSICAL ANALYSISRoughly speaking, semiclassical analysis is the reverse of quantization: We wouldlike to get the information about the classical system from the information aboutthe corresponding quantum system. As we mentioned in Lecture 1, the guideline ofthe subject is the so-called Bohr's correspondence principle:Bohr's Correspondence Principle. The classical mechanics can be realized astheformal h-→o limit of the corresponding quantum mechanics.Remark. When we say h -→ O limit, we don't really evaluate the limit, but insteadstudy the asymptotic behavior of our objects as h -→ 0For example, the three theorems we mentioned in Lecture I are all in this spirit:(1) The Egorov's theoremeitQ/naWe-itQ/n = (pta)W+O(h)is a rigorous mathematical theorem justifying our claim “the propagatorU(t) = e-itQ/n is the quantum analogue of the classical geodesic flow ptassociated to the Hamiltonian q(r,s).(2) The Weyl law(2h) (Vol(r,5) | H(z,5) ≤) + (h)#tells us that asymptotically, the number of quantum energy below coincideswith the measure of the classical region with classical energy below >
LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 5 (B) Geometric quantization. The theory was introduced by Kostant and Souriau in 1970s, and is mainly studied by geometers. In a future course on symplectic geometry, I will describe in detail how to quantize in the general symplectic setting. In geometric quantization, the Hilbert space will be taken to be the set of smooth sections of some “pre-quantum line bundle” satisfying some extra conditions. (C) Deformation quantization. The theory was introduced by Flato, Lichnerowicz and Sternheimer in 1976, and has been studied extensively since then mainly by algebraists. In this theory, the main focus is trying to understand quantization as a deformation of the (Poisson) structure of the algebra of classical observables. For example, people want to construct a new noncommutative product ? on C ∞(M)[~] (whose elements are formal power series in ~ with coefficients in C ∞(M), where M is a symplectic or Poisson manifold) such that f ? g − g ? f = ~ i {f, g} + O(~ 2 ). We will see that in Weyl quantization, such a star product can be defined for cotangent bundles. 2. Semiclassical Analysis Roughly speaking, semiclassical analysis is the reverse of quantization: We would like to get the information about the classical system from the information about the corresponding quantum system. As we mentioned in Lecture 1, the guideline of the subject is the so-called Bohr’s correspondence principle: Bohr’s Correspondence Principle. The classical mechanics can be realized as the formal ~ → 0 limit of the corresponding quantum mechanics. Remark. When we say ~ → 0 limit, we don’t really evaluate the limit, but instead study the asymptotic behavior of our objects as ~ → 0. For example, the three theorems we mentioned in Lecture 1 are all in this spirit: (1) The Egorov’s theorem e itQ/~ a W e −itQ/~ = (ρ ∗ t a) W + O(~) is a rigorous mathematical theorem justifying our claim “the propagator U(t) = e −itQ/~ is the quantum analogue of the classical geodesic flow ρt associated to the Hamiltonian q(x, ξ)”. (2) The Weyl law #{j | λj ≤ λ} = 1 (2π~) n (Vol({(x, ξ) | H(x, ξ) ≤ λ}) + O(~)) tells us that asymptotically, the number of quantum energy below λ coincides with the measure of the classical region with classical energy below λ
6LECTURE 3-09/27/2020 QUANTIZATION V.S.SEMI-CLASSICAL LIMIT(3) The quantum ergodicity theorem set up a correspondence between classi-cal ergodicity of the geodesic flow and quantum ergodicity of theLaplacianeigenfunctions, which we will explain later.As the course goes, you will see many other theorems of this flavor.We end today's lecture by a classical example for which we can calculate all theeigenvalues explicitly, and verify Weyl's law in this special case. Example: the harmonic oscillator.Considerthen-dimensionalHarmonicoscillator2台d+2We want to compute all the eigenvalues of H.To do so we start with the creationoperators1ha+V-lrkCk=V2V-10rkand the annihilation operatorah1Ak=Ck:-1rV2(V-1arkYou may feel strange where do the creation and annihilation oper-ators comefrom.To see this let'sfirst turnto the correspondingclassical system of harmonic oscillator, i.e.the system with Hamil-tonianH(r,$)=(ls/2+r/2)/2.ThesystemofHamiltonianequationscanbewrittenasddt9This isa coupled system.Oneway (maybe not the simplest way inthis example though)to solve such a coupled system is to decouplingthe system via eigenvectors of the coefficient matrix. It is easy to findout the unit eigenvectors, which are11-i,1)T(i.1)1andU1=02V2V2Itturns outthatif weset1and(SK+iCE)(-)Ck=akV2:then the system of Hamiltonian equations is decoupled into two sim-pleequationsCk=ick,ak=-iak
6 LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT (3) The quantum ergodicity theorem set up a correspondence between classical ergodicity of the geodesic flow and quantum ergodicity of the Laplacian eigenfunctions, which we will explain later. As the course goes, you will see many other theorems of this flavor. We end today’s lecture by a classical example for which we can calculate all the eigenvalues explicitly, and verify Weyl’s law in this special case. ♠ Example: the harmonic oscillator. Consider the n-dimensional Harmonic oscillator Hˆ = − ~ 2 2 ❳n j=1 d 2 dx2 j + |x| 2 2 . We want to compute all the eigenvalues of Hˆ . To do so we start with the creation operators Ck = 1 √ 2 ❶ ~ √ −1 ∂ ∂xk + √ −1xk ➀ and the annihilation operator Ak = C ∗ k = 1 √ 2 ❶ ~ √ −1 ∂ ∂xk − √ −1xk ➀ . You may feel strange where do the creation and annihilation operators come from. To see this let’s first turn to the corresponding classical system of harmonic oscillator, i.e. the system with Hamiltonian H(x, ξ) = (|ξ| 2+|x| 2 )/2. The system of Hamiltonian equations can be written as d dt ❺ xk ξk ➄ = ❺ 0 1 −1 0➄❺xk ξk ➄ This is a coupled system. One way (maybe not the simplest way in this example though) to solve such a coupled system is to decoupling the system via eigenvectors of the coefficient matrix. It is easy to find out the unit eigenvectors, which are v1 = 1 √ 2 (i, 1)T and v2 = 1 √ 2 (−i, 1)T . It turns out that if we set ck = 1 √ 2 (ξk + ixk) and ak = 1 √ 2 (ξk − ixk), then the system of Hamiltonian equations is decoupled into two simple equations c˙k = ick, a˙ k = −iak
LECTURE3—09/27/2020QUANTIZATION V.S.SEMI-CLASSICAL LIMIT7which can be solved easily.Now youimmediatelysee that the cre-ationoperator CkandtheannihilationoperatorAkaremerelythe"quantizations"of the"decoupled variables"c and ax!One can easily check by direct computations that[Cj,Ch] = 0 =[Aj, Al] and[A, Ch] = hojk· IdMoreover,sinceha品-V-1e+CiAjV-1or-1or,2 82h20r22we getH=(1)“h·IdECiA, -2j=1and thus[Ci,H] = C,C,Aj - C,A,Ci= C,[Cj,A] = -hCjand by a similar computation one has[A, H] = hA,It follows that if X is an eigenvalue of H, i.e.Hu = Au,thenHCju=C;(Hu+hu)=(Λ+h)CjuandHA,u=A,(Hu-hu) =(Λ-h)Aju.In other words, the creation operator C, maps an eigenfunction u of H associatedto eigenvalue ^ to an eigenfunction Cu of H associated to the eigenvalue ^+ h,while the annihilation operator A, maps u to an eigenfunction Au (if it is nonzero)associated to the eigenvalue ^- h.Tofinish the computation ofeigenvalues,we observe from (1)that《u, u) =ZIAjuP +l2,(2)jfrom which we see that(1)if入isan eigenvalueofH,then入≥h.(2)a function u is an eigenfunction associated to ^=nh if and only if A,u= 0holds for all 1 ≤j ≤ n
LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 7 which can be solved easily. Now you immediately see that the creation operator Ck and the annihilation operator Ak are merely the “quantizations” of the “decoupled variables” ck and ak! One can easily check by direct computations that [Cj , Ck] = 0 = [Aj , Ak] and [Aj , Ck] = ~δjk · Id. Moreover, since CjAj = 1 2 ❶ ~ √ −1 ∂ ∂xj + √ −1xj ➀ ❶ ~ √ −1 ∂ ∂xj − √ −1xj ➀ = − ~ 2 2 ∂ 2 ∂x2 j + x 2 j 2 − ~ 2 we get (1) Hˆ = ❳n j=1 CjAj + n 2 ~ · Id and thus [Cj , Hˆ ] = CjCjAj − CjAjCj = Cj [Cj , Aj ] = −~Cj , and by a similar computation one has [Aj , Hˆ ] = ~Aj . It follows that if λ is an eigenvalue of Hˆ , i.e. Huˆ = λu, then HCˆ ju = Cj (Huˆ + ~u) = (λ + ~)Cju and HAˆ ju = Aj (Huˆ − ~u) = (λ − ~)Aju. In other words, the creation operator Cj maps an eigenfunction u of Hˆ associated to eigenvalue λ to an eigenfunction Cju of Hˆ associated to the eigenvalue λ + ~, while the annihilation operator Aj maps u to an eigenfunction Aju (if it is nonzero) associated to the eigenvalue λ − ~. To finish the computation of eigenvalues, we observe from (1) that (2) hHu, u ˆ i = ❳ j kAjuk 2 + n 2 ~kuk 2 , from which we see that (1) if λ is an eigenvalue of Hˆ , then λ ≥ n 2 ~. (2) a function u is an eigenfunction associated to λ = n 2 ~ if and only if Aju = 0 holds for all 1 ≤ j ≤ n
8LECTURE 3-09/27/2020 QUANTIZATION V.S.SEMI-CLASSICAL LIMITAdirectcomputationshowsthatAju(r) =0 u(rj) = ce-g/2hAs a consequence, we see that h is indeed the smallest eigenvalue of H and theassociated eigenfunction isuo(r) = e-(rP /2hwhich is often called the ground state.Starting from theground state, one canfind more eigenvalues using the creationoperator (this iswhythe operator called creationoperator):Foranymulti-indexQ= (α1,*.,,an) N"o, the function?Ua= C....Ca"uois an eigenfunction of H associated with the eigenvalue"h+[a|hAa2where as usual we denote [a = Qi +..+ an.Finally we claim that these eigenvalues are all the eigenvalues of H:If >>his an eigenvalue of H with eigenfunction u, then by (2), there is some j such thatA,u O, and thus -h is also an eigenvalue of H with eigenfunction A,u. As aconsequence,we get. any eigenvalue of H with 入>h must has the form = h +kh for somekEN,. if u is the corresponding eigenfunction, then there exists Q1, ., Qn E Nzowith [a|=k such thatA"u:=A1...Aanu= uo.Now we are ready to proveProposition 2.1. The eigenvalues of H arenh."h+h,h+2h...h+kh....2h, 2Moreover, for each k ≥O, the associated eigenspace has a basisua=Cauo := Ca1..Canuo,Jal=k.Proof. We have already proved the first half of the proposition. We have alreadyseen that us are eigenfunctions associated to the eigenvalues h+aα h.Notethatif[a|=[Bbut αβ,then uaug,sinceABCauo=0forIB|=a|butβαwhileA°Cαuo = Q!uo.So these ua's are linearly independent eigenfunctions.2Note that Cju = O has no nontrivial L? solution. So these u&'s are non-zero functions
8 LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT A direct computation shows that Aju(xj ) = 0 ⇐⇒ u(xj ) = ce−x 2 j /2~ , As a consequence, we see that n 2 ~ is indeed the smallest eigenvalue of Hˆ and the associated eigenfunction is u0(x) = e −|x| 2/2~ , which is often called the ground state. Starting from the ground state, one can find more eigenvalues using the creation operator (this is why the operator called creation operator): For any multi-index α = (α1, · · · , αn) ∈ N n ≥0 , the function2 uα = C α1 1 · · · C αn n u0 is an eigenfunction of Hˆ associated with the eigenvalue λα = n 2 ~ + |α|~, where as usual we denote |α| = α1 + · · · + αn. Finally we claim that these eigenvalues are all the eigenvalues of Hˆ : If λ > n 2 ~ is an eigenvalue of Hˆ with eigenfunction u, then by (2), there is some j such that Aju 6= 0, and thus λ − ~ is also an eigenvalue of Hˆ with eigenfunction Aju. As a consequence, we get • any eigenvalue of Hˆ with λ > n 2 ~ must has the form λ = n 2 ~ + k~ for some k ∈ N, • if u is the corresponding eigenfunction, then there exists α1, · · · , αn ∈ N≥0 with |α| = k such that Aαu := A α1 1 · · · Aαn n u = u0. Now we are ready to prove Proposition 2.1. The eigenvalues of Hˆ are n 2 ~, n 2 ~ + ~, n 2 ~ + 2~, · · · , n 2 ~ + k~, · · · . Moreover, for each k ≥ 0, the associated eigenspace has a basis uα = C αu0 := C α1 1 · · · C αn n u0, |α| = k. Proof. We have already proved the first half of the proposition. We have already seen that uα’s are eigenfunctions associated to the eigenvalues n 2 ~ + |α|~. Note that if |α| = |β| but α 6= β, then uα 6= uβ, since A βC αu0 = 0 for |β| = |α| but β 6= α while A αC αu0 = α!u0. So these uα’s are linearly independent eigenfunctions. 2Note that Cju = 0 has no nontrivial L 2 solution. So these uα’s are non-zero functions
9LECTURE3—09/27/2020QUANTIZATION V.S.SEMI-CLASSICAL LIMITTo show that there is no other element in the same eigenspace, we suppose v isan eigenfunction with eigenvalue nh+kh,and yis L?-perpendicular to all ugs with[al = k. We just observed that there exists Q with [a| = k such that Aau = uo. Itfollows(uo, uo) = (Aav,uo)= (u,Cuo) =0,口a contradiction.Remark.There is an explicit formula for these eigenfunctions. In the case of dimen-sion l, the normalized eigenfunctions associated to the eigenvalueh +jh are12 (h)e-%H;(gi()= 3whereH;(r) =(-1)e*2 1ois a polynomial in , known as the Hermite polymomial. In the higher dimensionalcase, the eigenfunctions are simply the products of these functions.Although one can deduce from some general theorems that for the Harmonicoscillator, these eigenfunctions span L?(R"),we can give a direct proof using theseexplicit expressions.For simplicity we only consider the case n =1,and we takeh = 1. The key observation is that H, is a degree j polynomial with nonzero leadingterm. It follows that the span of po,..., Pm are exactly p(r)e2/2, where p(a) isany polynomial of degree no more than m.Nowwe notice that for any complexe-r/2converges to ee-/2 in L?.Now suppose (r)number c, the series k=ois orthogonal to all Pm's. Then by taking c = -i we will geti5e-ap/2b()dr = 0.Since e-1}°/2b(r) belongs to L?(R), using basic theory of Fourier transform (whichis our topic of next lecture) we see e-lp/2b(r) = 0, i.e. b = 0.Finally wearereadytocheck Weyl'slawfor the Harmonicoscillator:wehave#=(a+0)= [α: lal ≤(- h)/h)2(/h)" + 0(h=")while1s122Vol((r, ) : ≤ >)) = (V2)2n × the volume of unit ball in IR2n,22and Weyl's law in this case follows since the volume of unit ball in R2n is
LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 9 To show that there is no other element in the same eigenspace, we suppose v is an eigenfunction with eigenvalue n 2 ~ + k~, and v is L 2 -perpendicular to all uαs with |α| = k. We just observed that there exists α with |α| = k such that Aα v = u0. It follows hu0, u0i = hA α v, u0i = hv, Cαu0i = 0, a contradiction. Remark. There is an explicit formula for these eigenfunctions. In the case of dimension 1, the normalized eigenfunctions associated to the eigenvalue 1 2 ~ + j~ are ϕj (x) = 1 2 j j! 1 (π~) 1/4 e − x 2 2~ Hj ( x √ ~ ), where Hj (x) = (−1)j e x 2 d j dxj (e −x 2 ) is a polynomial in x, known as the Hermite polynomial. In the higher dimensional case, the eigenfunctions are simply the products of these functions. Although one can deduce from some general theorems that for the Harmonic oscillator, these eigenfunctions span L 2 (R n ), we can give a direct proof using these explicit expressions. For simplicity we only consider the case n = 1, and we take ~ = 1. The key observation is that Hj is a degree j polynomial with nonzero leading term. It follows that the span of ϕ0, · · · , ϕm are exactly p(x)e −x 2/2 , where p(x) is any polynomial of degree no more than m. Now we notice that for any complex number c, the series P∞ k=0 c kx k k! e −x 2/2 converges to e cxe −x 2/2 in L 2 . Now suppose ψ(x) is orthogonal to all ϕm’s. Then by taking c = −iξ we will get ❩ R e −ixξe −|x| 2/2ψ(x)dx = 0. Since e −|x| 2/2ψ(x) belongs to L 2 (R), using basic theory of Fourier transform (which is our topic of next lecture) we see e −|x| 2/2ψ(x) = 0, i.e. ψ = 0. Finally we are ready to check Weyl’s law for the Harmonic oscillator: we have #{j : λj ≤ λ} = {α : n 2 ~ + |α|~ ≤ λ} = ➜ α : |α| ≤ (λ − n 2 ~)/~ ➟ = 1 n! (λ/~) n + o(~ −n ) while Vol({(x, ξ) : |ξ| 2 2 + |x| 2 2 ≤ λ}) = (√ 2λ) 2n × the volume of unit ball in R 2n , and Weyl’s law in this case follows since the volume of unit ball in R 2n is π n n! .