Lecture 4 January 14, 2016

Transcription

1 MATH 262/CME 372: Applied Fourier Analysis and Winer 206 Elemens of Modern Signal Processing Lecure 4 January 4, 206 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long; Edied by E. Candes & E. Baes Ouline Agenda: Uncerainy Principle. Weyl-Heisenberg Uncerainy Principle 2. Quanum mechanical inerpreaion Las Time: Moivaed by he boxcar funcion, whose Fourier ransform is no summable, we inroduced he space of square-inegrable funcions L 2 (R). We proved he Parseval-Plancherel heorem, which shows he Fourier ransform preserves he L 2 -inner produc and is herefore an isomery (modulo a facor 2π). Using his resul, we were able o exend he Fourier ransform o L 2 (R) and hus make sense of he inverse Fourier ransform of he boxcar funcion as a squareinegrable funcion. The properies we proved also apply o his exension, wih he remark ha some equaliies hold almos everywhere. We also defined he Fourier ransform in higher dimensions by simply ieraing he single-variable ransform. 2 Weyl-Heisenberg s Uncerainy Principle The uncerainy principle is commonly known in physics as saying ha one canno know simulaneously he posiion and he momenum of a paricular wih infinie precision. In fac, his saemen is an implicaion of ha mahemaical observaion ha f and ˆf canno boh be concenraed. This is somehing we can inuiively see from some examples. For insance, our calculaions involving Gaussians (Lecure ) revealed ha increasing or decreasing he sandard deviaion in he ime (or spaial) domain has he opposie effec in he frequency (or Fourier) domain. Fig. illusraes his phenomenon. Taking his example o he exreme cases of 0 or infinie sandard deviaion gives he dela funcion in one domain and he consan funcion in he oher. The former is compleely localized, while he laer is compleely delocalized. Even wihou hese examples, one can iniially undersand he uncerainy principle from he ideniy F () = af(a) ˆF (ω) = ˆf(ω/a). Tha is, dilaing (compressing) ime is equivalen o compressing (dilaing) frequency. Wha he uncerainy principle does is provide a precise, quaniaive saemen of his noion.

2 Time domain: The blue curve corresponds o a signal f concenraed in ime, whereas he red curve corresponds o a signal f 2 spread in ime. Frequency domain: The Fourier ransform ˆf is spread in frequency, whereas he Fourier ransform ˆf 2 concenraes in frequency. Figure : Visual evidence of he uncerainy principle To sae he resul, we need o make a definiion. For f L 2 (R), define he spread of f o be σ 2 (f) = inf 0 R f 2 ( 0 ) 2 f() 2 d. We can inerpre his quaniy in probabilisic erms by noing ha p() = f() 2 / f 2 is a probabiliy densiy funcion, as (i) p() 0, (ii) p() d =. Therefore, one can recognize σ 2 (f) as he variance of a random variable X wih densiy p(), where he minimizing 0 is exacly he mean 0 = EX = p() d. Then σ(f) is, of course, he sandard deviaion of X. Similarly, q(ω) = ˆf(ω) 2 / ˆf 2 is he densiy of random variable Y having mean ω 0 = EY = ωq(ω) dω and sandard deviaion σ( ˆf). Wih his noaion, we can now sae he uncerainy principle: Theorem (Weyl-Heisenberg Uncerainy Principle). If f L 2 (R) is no idenically 0, hen (i) σ(f)σ( ˆf) /2. 2

3 (ii) Equaliy holds if and only if f is a ranslaion and modulaion of a Gaussian funcion. Thus, in erms of simulaneous ime and frequency concenraion, Gaussians are maximal wih respec o mean squared-deviaion. Proof. Le us begin wih some simplifying reducions. Firs, we may assume ha X from above is cenered (EX = 0), since ranslaions and modulaions do no affec spread: F = f( + 0 ) σ 2 (F ) = σ 2 (f) and σ 2 ( ˆF ) = σ 2 (e iω 0 ˆf) = σ 2 ( ˆf). Second, we may assume ha Y is also cenered, for he same reason: F () = e iω 0 f() σ 2 (F ) = σ 2 (f) and σ 2 ( ˆF ) = σ 2 ( ˆf(ω + ω 0 )) = σ 2 ( ˆf). Third, f by f/ f does no change spread, and he densiy funcions p() and q(ω) are lef unchanged. So we may assume f has norm. Numerically, our assumpions ell us (i) f() 2 d = ω ˆf(ω) 2 dω = 0, and (ii) f() 2 d =. Finally, smooh funcions are dense in L 2 (R), and he Fourier ransform preserves L 2 -norm (modulo a facor of 2π). So i suffices o demonsrae he uncerainy principle for smooh, in paricular differeniable, funcions. Now we will express σ( ˆf) in erms of f. Noe ha f () F iω ˆf(ω), and hen he Parseval-Plancherel heorem yields σ 2 ( ˆf) = ˆf ω 2 ˆf(ω) 2 dω = iω 2 2π ˆf(ω) 2 dω = f () 2 d, where we used ha f 2 = implies ˆf 2 = 2π. Consequenly, he uncerainy principle is equivalen o 2 f() 2 d f () 2 d 4, () meaning i is also a saemen abou a rade-off beween concenraion and regulariy. Namely, a funcion canno simulaneously be very concenraed and have small derivaives (in he L 2 sense). To complee he proof by checking (), we will employ he wo mos ubiquious ools of analysis: he Cauchy-Schwarz inequaliy and inegraion by pars. Recall he saemen of Cauchy-Schwarz: For g, h L 2 (R), g()h() d = g, h g h = g() 2 d h() 2 d. Following Weyl s proof (see [2], p. 393), we assume ha f() 2 decays faser han / a infiniy: f() 2 0 as. (2) 3

4 When f L 2 (R), (2) holds; he verificaion is echnical, and he ineresed reader can find i a he end of he proof. When f / L 2 (R), we have σ( ˆf) =, and so he uncerainy inequaliy cerainly holds. Apply Cauchy-Schwarz o g() = f() and h() = f (). Since g = σ(f) and h = σ( ˆf), we have f()f () d σ(f)σ( ˆf). We also have f()f () d σ(f)σ( ˆf) and so f()f () + f()f () d 2σ(f)σ( ˆf) Now inegraion by pars gives f()f () d + f()f () d = d d f() 2 d = f() 2 f() 2 d =. (from (2) and (ii)) We have hus shown σ(f)σ( ˆf) /2. Equaliy holds in Cauchy-Schwarz if and only if g h. So he uncerainy principle is saisfied wih equaliy if and only if f() f (). In his case, one can solve he differenial equaion o see f () = f() (3) σ2 f() = f(0)e 2 /2σ 2. Since we allow ranslaions, modulaions, and rescalings (i.e. undoing our reducions from he beginning), in general we have f() e iω0 e ( 0 ) 2 2σ 2. Tha is, f is a modulaion and ranslaion of a Gaussian funcion, as claimed. Remark : Assume f L 2 (R) is differeniable, wih f L 2 (R). Verificaion of (2) proceeds as follows: Suppose agains he claim ha here exiss ϵ > 0 such ha f() 2 > ϵ for arbirarily large. By possibly replacing f() wih f( ), we may assume his holds for arbirarily large posiive. Since f L 2 (R), we may choose so large ha f (u) 2 du 4. Applying Cauchy-Schwarz, we find ha for any, 2 ( 2 f( ) f() 2 = f (u) du f (u) du) ( ) f (u) 2 du. 4

5 So for [, + ϵ/], Hence for such. I follows ha Bu f L 2 (R), and so u 2 f(u) 2 du Therefore, (4) is a conradicion. f( ) f() 2 ϵ 4. f( ) f() 2 + ϵ lim u 2 f(u) 2 du > ϵ > ϵ 2 + ϵ u 2 f(u) 2 du = ϵ d = ϵ2 4. (4) Remark 2: If he RHS of (3) were given posiive sign, f() would be an exponenial wih posiive exponen and hus no in L 2 (R). Remark 3: We can verify direcly ha Gaussians saisfy he uncerainy principle wih equaliy: and Indeed, f() = e 2 /2 ˆf(ω) = 2πe ω2 /2 = f() 2 = e 2 = σ 2 (f) = 2, = ˆf(ω) 2 = 2πe ω2 = σ 2 ( ˆf) = 2. σ 2 (f)σ 2 ( ˆf) = 4. 3 Quanum mechanical inerpreaion Le us consider he descripion of a simple physical sysem, such as an elecron. From a classical poin of view, he physical sae of he elecron a any insan is characerized by six quaniies, ha is, is hree posiion coordinaes, and is hree momenum coordinaes. In oher words, o each ime insan we associae a poin in he phase space. Physical quaniies of ineres, such as he energy, can be measured from hese quaniies. However, in quanum mechanics he physical descripion is characerized by a sae vecor, Sae vecor: ψ H, where H is an infinie-dimensional Hilber space. The quaniies of ineres are represened by observables, which are symmeric operaors acing on he Hilber space Observable: A L(H), A = A. For insance, he posiion X and momenum P are observables. In classical mechanics, hese only exrac he posiion coordinaes or momenum coordinaes from a poin in he phase space, bu in quanum mechanics hey are symmeric linear operaors acing on H. 5

6 Since hey are symmeric, hey can be diagonalized and have a complee se of eigenvecors and real eigenvalues. We can define x eigenvecor of X wih eigenvalue x, ha is, X x = x x, and p eigenvecor of P wih eigenvalue p, ha is, P p = p p. As usual, we le eigenvecors be normalized. Since { x } x and { p } p are boh complee orhonormal ses, we can decompose he ideniy operaor as I = dx x x = dp p p. Fundamenal posulae of Quanum Mechanics: For an observable A wih eigensaes a and a sae ψ, he oucome of measuring A is a wih probabiliy a ψ 2 and he sysem ransiions o he sae a, ha is, ψ measure A a wih probabiliy a ψ 2. An experimen ha shows his behavior is Sern-Gerlach experimen. For a descripion of his experimen and is significance, you can see hp://en.wikipedia.org/wiki/sern-gerlach_ experimen. The original paper by Oo Sern and Walher Gerlach (in German) can be found here hp://link.springer.com/aricle/0.007%2fbf Also he fundamenal posulae is par of wha is known as he Copenhagen inerpreaion of quanum mechanics (see hp://en.wikipedia.org/wiki/copenhagen_inerpreaion#principles). 3. Uncerainy relaions Since he oucomes of measuring an obsevable are random variables, we can define he expecaion of an observable. Le a denoe he se of eigenvecors of he observable A and assume he sysem under consideraion is on a sae ψ. Then is expeced value is ā = A = da a a, ψ 2 and is variance ( A) 2 = da (a ā) 2 a, ψ 2. Heisenberg s uncerainy relaion saes ha: ( X) 2 ( P ) 2 (ħ/2) 2 (5) wih ( X) 2 = ( P ) 2 = dx (x x) 2 x, ψ 2 dp (p p) 2 p, ψ 2. 6

7 In oher words, he produc of sandard deviaions is a leas ħ/2, where ħ is he reduced Planck consan h/2π. As a remark, x, ψ is called he he wave funcion and ofen denoed by Ψ(x). These are he coefficiens of our sae ψ in he basis of eigenvecors for X. Clearly Ψ(x) 2 dx = and Ψ(x) 2 is usually inerpreed as he probabiliy finding he sysem (say, an elecron) on an (infiniesimal) neighborhood of x. To see why Heisenberg s uncerainy relaion holds, we make use of his: Claim 2. If x is an eigensae of X wih eigenvalue x and, likewise, p is an eigensae of P wih eigenvalue p, hen he do produc obeys x, p = 2πħ e i ħ px. Define he wave funcion Φ(p) in momemum space via Φ(p) = p, ψ. Since ψ = x x, ψ dx, Φ and Ψ are relaed by Φ(p) }{{} wave funcion in momenum space = p ψ = = x ψ p x dx 2πħ Ψ(x) e i ħ px dx. }{{} wave funcion in posiion space Similarly, we can wrie Ψ(x) }{{} wave funcion in posiion space = x ψ = = p ψ x p dx 2πħ Φ(p) e i ħ px dp. }{{} wave funcion in momenum space The Fourier ransform arises naurally, and can be hough as a change of variables beween posiion space and momenum space. Hence, we can wrie uncerainies abou posiion and momemum as ( X) 2 = (x x) Ψ(x) 2 dx ( P ) 2 = (p p) ˆΨ(p/ħ) 2 dp; from here, Theorem proved earlier gives Heisenberg s uncerainy relaion (5). 7

8 3.2 Momemum We need o argue in favor of Claim 2, and in order o do his, we inroduce he momemum operaor. We begin by considering a special class of maps acing on saes, he ranslaion map U(a) for a R defined as U(a) x x + a. This family of maps defines a group. We can consider is behavior for small δa, ha is U(δa) = I i ħ P δa + o(δa2 ). The operaor appearing on he firs-order erm of he above expansion is by definiion he momenum operaor. Roughly speaking, he momenum operaor causes an infiniesimal displacemen on a physical sae. Informally, U(δa) ψ Ψ(x) = Ψ(x) Ψ (x)δa + o(δa 2 ), and consequenly he momenum operaor is such ha Ψ(x) iħ d dxψ(x). Using our noaion, is acion on a sae ψ is given by ( ) d P ψ = iħ dx x ψ x dx. Wih ψ = p and using P p = p p, his gives ( ) d p p = iħ dx x p x dx. Taking he do produc wih he bra x gives ( ) d p x p = iħ dx x p x x dx = iħ d x p. dx This is a differenial equaion wih soluion x p = 2πħ e i ħ px, and consequenly he eigensaes of he momenum represened in he posiion basis are complex exponenials wih (spaial) frequency given by he momenum eigenvalue p. 3.3 Heisenberg s original formulaion In his 927 paper [], Heisenberg gave a slighly differen formulaion of all of his. Firs noe ha he mean oucome when applying A is given by ( ) A = ψ A ψ = ψ A da a a ψ = da ψ A a a ψ = da a a, ψ 2. In linear algebra, we would wrie ψ A ψ as ψ Aψ. 8

9 Se A = A A I, so ha ( A) 2 = A 2 A 2, is jus he variance of he oucome as before. Las, he commuaor or Lie bracke is [A, B] = AB BA, which is a measure, in some sense, of how incoheren he observables A and B are. Theorem 3 (Heisenberg (927)). For a sae ψ and any observables A and B we have ( A) 2 ( B) 2 4 [A, B] 2. This mahemaical inequaliy is also a sor of Cauchy-Schwarz inequaliy and I will omi he proof. Claim 4. We have Hence, [X, P ] 2 = ħ 2, from where we obain [X, P ] = iħ I. ( X) 2 ( P ) 2 ħ2 4. Proof. We compue he acion of XP in he basis of eigensaes of X, and recall ha X x = x x and P ψ = iħ d dx x, ψ x dx. On he one hand, XP ψ = iħ x d ( x, ψ ) x dx dx while on he oher I hus follows ha [X, P ] ψ = iħ P X ψ = iħ d (x x, ψ ) x dx. dx ( x d ) d x, ψ + (x x, ψ ) x dx = iħ dx dx x, ψ x dx = iħ ψ. References [] Werner Heisenberg. Über den anschaulichen inhal der quanenheoreischen kinemaik und mechanik. Zeischrif für Physik, 43(3-4):72 98, 927. [2] Hermann Weyl. The heory of groups and quanum mechanics. Courier Corporaion,