PROBLEM SET 1SEMICLASSICALMICROLOCALANALYSISDUE:OCT.19, 2020(1)[Poisson bracket]Prove that the Poisson bracket [,} turns Co(T*R") into Poisson algebra, namelyfor any f,g, h E Co(T*Rn),(a) (Anti-commutativity) [f,g) =-{g, f).(b) (Jacobi's identity) (f, (g, h)) + (g, [h, f)}) + [h, (f,g)) = 0.(c) (Leibniz's rule) (f, gh) =[f,g)h +g(f,h).Moreover, prove that(d) (Commutator relation) 三(f.g) = [三,三g].[Ref: Zworski, Semiclassical Analysis, Lemma 2.9](2)[Ehrenfest Theorem]Prove the Ehrenfest Theorem that relates the time derivative of the expectation valuesof the position and momentum operators to the expectation value of the force-vV:最(Qi)(t) =(Ps)(t),量(P)(t)=-(g,V)(t)[Ref:Hall,Quantumtheoryformathematicians,Section3.7.5](3)[No-go theorem]Complete the proof of the no-go theorem as outlined in Lecture 3. Namely, provethat one can't quantize all polynomials in r and of degree ≤ 4 if we assume Dirac'saxioms (D1)-(D4) listed in the Lecture 3. For this purpose we let Q(r) and Q($) be thequantizations of the position function and the momentum function respectively.(a)Prove:There exists constant csuch thatQ(rE)=Q(r)Q()+c.Id(b) Inductively prove that for any m EN, we haveQ(rm)= Q(r)m and Q(sm)= Q(s)m(c) Compute{2,r3 and (?,31, and proveQ(r2) = 2()°Q() + Q()Q(n)Q(t) = Q(c)Q(E) + Q(5)2Q(2)22(d) Compute [r?g, r2} and [r3, 3], and deduce a contradiction.[Ref:Mv2014notes..Lecture11
PROBLEM SET 1 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: OCT. 19, 2020 (1) [Poisson bracket] Prove that the Poisson bracket {·, ·} turns C∞(T ∗R n ) into Poisson algebra, namely for any f, g, h ∈ C∞(T ∗R n ), (a) (Anti-commutativity) {f, g} = −{g, f}. (b) (Jacobi’s identity) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0. (c) (Leibniz’s rule) {f, gh} = {f, g}h + g{f, h}. Moreover, prove that (d) (Commutator relation) Ξ{f,g} = [Ξf , Ξg]. [Ref: Zworski, Semiclassical Analysis, Lemma 2.9] (2) [Ehrenfest Theorem] Prove the Ehrenfest Theorem that relates the time derivative of the expectation values of the position and momentum operators to the expectation value of the force −∇V : d dthQj iψ(t) = hPj iψ(t) , d dthPj iψ(t) = −h∂jV iψ(t) . [ Ref: Hall, Quantum theory for mathematicians, Section 3.7.5] (3) [No-go theorem] Complete the proof of the no-go theorem as outlined in Lecture 3. Namely, prove that one can’t quantize all polynomials in x and ξ of degree ≤ 4 if we assume Dirac’s axioms (D1)-(D4) listed in the Lecture 3. For this purpose we let Q(x) and Q(ξ) be the quantizations of the position function x and the momentum function ξ respectively. (a) Prove: There exists constant c such that Q(xξ) = Q(x)Q(ξ) + c · Id. (b) Inductively prove that for any m ∈ N, we have Q(x m) = Q(x) m and Q(ξ m) = Q(ξ) m. (c) Compute {ξ 2 , x3} and {x 2 , ξ3}, and prove Q(x 2 ξ) = Q(x) 2Q(ξ) + Q(ξ)Q(x) 2 2 , Q(xξ2 ) = Q(x)Q(ξ) 2 + Q(ξ) 2Q(x) 2 (d) Compute {x 2 ξ, xξ2} and {x 3 , ξ3}, and deduce a contradiction. [Ref: My 2014 notes, Lecture 1.] 1
2PROBLEM SET1SEMICLASSICALMICROLOCALANALYSISDUE:OCT.19,2020(4)[Hermitepolynomials)Consider the Harmonic oscillator with n = 1. Denote u;(r) = Ciuo be the eigenfunction of H associated with the eigenvalue (i+)h, where C=(+-Ia)is the creation operator, and uo(r) = e-r2/2h. To get an explicit expression, write）e-5元hH(ui(r) =(a) Calculate Ho, Hi,H2 and H3.(b) Use the facts Cuj = uj+1 to prove Hi+i(r) = 2rH;(r) - H,(r). As a conse-quence, H, is a polynomial of degree j, called Hermite polynomials.(c) First prove Auj = juj-1, then prove H'(r) = 2jHj-i(r). As a consequence, Her-mite polynomials satisfy the recurrencerelation Hj+1=2rH,(r)-2jHj-i(r).(d) Prove the Rodrigues' formula for Hn:H;(a) =(-1)er(兴)(e-r"),(e) Prove that Hn has the following exponential generating functionAtie2rt-t?-H,(r)！j=0[Ref:Hall, Quantum theoryformathematicians, Chapter 11](5)[The uncertainty principle]Let A be a quantum observable, a quantum state, and a e R. The expression(A) := <(A-a-Id)2)can be used as a measure of how much the observable A in the state fails to beconcentrated at a. In particular,if we take a = (A), then we write s(A) := (A), (A)(which is just the standard deviation) and call it the uncertainty of A in a state b.(a) Prove: For any self-adjoint operators A and B, any EDom(AB)nDom(BA)and any a,b e R, we have(A)(B)≥K[A,B)].(b) Let Q = and P - be the canonical quantum observables (acting on L?(R)associated with the position and momentum . Prove: For E Dom(QP) nDom(PQ),(0)(P)≥4[Ref:Folland, Harmonic analysis in phase space, Theorem 1.34][Ref:Hall,Quantumtheoryformathematicians,Chapter12.]
2 PROBLEM SET 1 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: OCT. 19, 2020 (4) [Hermite polynomials] Consider the Harmonic oscillator with n = 1. Denote uj (x) = C ju0 be the eigenfunction of Hˆ associated with the eigenvalue (j + 1 2 )~, where C = √ 1 2 √ ~ −1 ∂ ∂x + √ −1x is the creation operator, and u0(x) = e −x 2/2~ . To get an explicit expression, write uj (x) = √ −1 √ 2 j ~ j 2 Hj ( x √ ~ )e − x 2 2~ . (a) Calculate H0, H1, H2 and H3. (b) Use the facts Cuj = uj+1 to prove Hj+1(x) = 2xHj (x) − H0 j (x). As a consequence, Hj is a polynomial of degree j, called Hermite polynomials. (c) First prove Auj = juj−1, then prove H0 j (x) = 2jHj−1(x). As a consequence, Hermite polynomials satisfy the recurrence relation Hj+1 = 2xHj (x) − 2jHj−1(x). (d) Prove the Rodrigues’ formula for Hn: Hj (x) = (−1)j e x 2 ( d dx) j (e −x 2 ). (e) Prove that Hn has the following exponential generating function e 2xt−t 2 = X∞ j=0 Hj (x) t j j! . [Ref: Hall, Quantum theory for mathematicians, Chapter 11.] (5) [The uncertainty principle] Let A be a quantum observable, ψ a quantum state, and a ∈ R. The expression δ ψ a (A) := q h(A − a · Id)2iψ can be used as a measure of how much the observable A in the state ψ fails to be concentrated at a. In particular, if we take a = hAiψ, then we write δ ψ(A) := δ ψ hAiψ (A) (which is just the standard deviation) and call it the uncertainty of A in a state ψ. (a) Prove: For any self-adjoint operators A and B, any ψ ∈ Dom(AB) ∩ Dom(BA) and any a, b ∈ R, we have δ ψ a (A)δ ψ b (B) ≥ 1 2 |h[A, B]iψ| . (b) Let Q = x and P = ~ i d dx be the canonical quantum observables (acting on L 2 (R)) associated with the position x and momentum ξ. Prove: For ψ ∈ Dom(QP) ∩ Dom(P Q), δ ψ a (Q)δ ψ b (P) ≥ ~ 2 . [Ref: Folland, Harmonic analysis in phase space, Theorem 1.34] [Ref: Hall, Quantum theory for mathematicians, Chapter 12.]
PROBLEM SET 1SEMICLASSICAL MICROLOCAL ANALYSISDUE:OCT.19,20203(6)[SemiclassicalFourier transform]Define the semiclassical Fourier transform of a function E to beFrp(3) := (Fp)(o(r)dr(a) Write down the analogue statements of Proposition 1.2 of Lecture 4for Fh,andprove them directly (namely not as a consequence of Proposition 1.2).(b) Write down the semiclassical versions of Theorem 1.3, Corollary 1.4 and Theorem2.3 (for which we compute the semiclassical Fourier transform of eQa")(c) Let P, = f, be the momentum operator. Prove: For ES,TOr(E)1(Pi)=/(d)Prove thefollowing semiclassical version of uncertainty principle:For any E,I1gollL2 l;Fhll2 ≥llollL2-Fhl/2.(e)Figure out the the condition for the equality in (d)[Ref:Zworski,SemiclassicalAnalysis,$3.3](7)[Poisson summationformula]Let f E g(Rn). Define h(r) := Zuezn f(r +v)(a) Prove: the function h is smooth and periodic with period Zn.(b) Expand h into its Fourier series, and prove:E f(u) = f(2元μ)VEZnHEZn[Ref: Guillemin-Sternberg, Semi-classical Analysis, g15.10.4.][As an application, read g15.10.1, g15.10.3 and g15.11](8)[Almost analytic extension]For any f e (R), we say a function f e Co(C) is an almost analytic ertension off if it is an extension of f which supports near the real axis such that ,f vanishesto infinite order near the real axis, namelyfl = f, suppf c [z : [Im(2)/≤1) and ,f(2) =O(Im(z2)]0).(a)Fix any cut-off function x ECo((-1,1)) with x=1 on [-1/2,1/2]. Prove:F(a):= 2x() / (us)(5)es(+) dsis an almost analytic extension of f.(b) Let f be an almost analytic extension of f.Use Green's formula to prove:1 / aj dz.f(t) =iJct-z[Ref: Zworski, Semiclassical Analysis, Theorem 3.6 and Theorem 14.8.]
PROBLEM SET 1 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: OCT. 19, 2020 3 (6) [Semiclassical Fourier transform] Define the semiclassical Fourier transform of a function ϕ ∈ S to be F~ϕ(ξ) := (Fϕ)( ξ ~ ) = Z Rn e − ix·ξ ~ ϕ(x)dx. (a) Write down the analogue statements of Proposition 1.2 of Lecture 4 for F~, and prove them directly (namely not as a consequence of Proposition 1.2). (b) Write down the semiclassical versions of Theorem 1.3, Corollary 1.4 and Theorem 2.3 (for which we compute the semiclassical Fourier transform of e i 2~ x T QxT ). (c) Let Pj = ~ i ∂ ∂xj be the momentum operator. Prove: For ϕ ∈ S , hPj iϕ = Z Rn ξj |F~ϕ(ξ)| 2 dξ. (d) Prove the following semiclassical version of uncertainty principle: For any ϕ ∈ S , kxjϕkL2 · kξjF~ϕkL2 ≥ ~ 2 kϕkL2 · kF~ϕkL2 . (e) Figure out the the condition for the equality in (d). [Ref: Zworski, Semiclassical Analysis, §3.3] (7) [Poisson summation formula] Let f ∈ S (R n ). Define h(x) := P v∈Zn f(x + v). (a) Prove: the function h is smooth and periodic with period Z n . (b) Expand h into its Fourier series, and prove: X v∈Zn f(v) = X µ∈Zn ˆf(2πµ). [Ref: Guillemin-Sternberg, Semi-classical Analysis, §15.10.4.] [As an application, read §15.10.1, §15.10.3 and §15.11.] (8) [Almost analytic extension] For any f ∈ S (R), we say a function ˜f ∈ C∞(C) is an almost analytic extension of f if it is an extension of f which supports near the real axis such that ∂¯ z ˜f vanishes to infinite order near the real axis, namely ˜f|R = f, supp ˜f ⊂ {z : |Im(z)| ≤ 1} and ∂¯ z ˜f(z) = O(|Im(z)| ∞). (a) Fix any cut-off function χ ∈ C∞ 0 ((−1, 1)) with χ ≡ 1 on [−1/2, 1/2]. Prove: ˜f(z) := 1 2π χ(y) Z R χ(yξ) ˆf(ξ)e iξ(x+iy) dξ is an almost analytic extension of f. (b) Let ˜f be an almost analytic extension of f. Use Green’s formula to prove: f(t) = 1 πi Z C ∂¯ z ˜f(z) t − z dz. [Ref: Zworski, Semiclassical Analysis, Theorem 3.6 and Theorem 14.8.]