MA4A7 Quantum Mechanics: Basic Principles and Probabilistic Methods

Transcription

1 MA4A7 Quaum Mechaics: Basic Priciples ad Probabilisic Mehods Course held by Volker Bez i Warwick,. Lecure oes ake ad yped by Sephe Harris. Course held agai ad lecure oes revised by Volker Bez i Warwick,.

2 Coes Iroducio. Basic Axioms Cosisecy of he Axioms Soluio o he Free Schrödiger equaio Ehrefes equaios Quaum Mechaics via Operaors. Hilber spaces ad operaors Hilber spaces Operaors Commuaors Exisece of soluio o he Schrödiger equaio Observables ad Eigevalues Heiseberg Picure of Quaum Dyamics Uceraiy Priciple The Sabiliy of he Hydroge Aom The Harmoic Oscillaor 4 3. Classical Harmoic Oscillaor Quaum Harmoic Oscillaor Eigevalues ad Eigevecors Dyamics Periodiciy Dyamics of mea values Dyamics of observables The Feyma-Kac Formula 3 4. Feyma iegral Troer Produc Formula Feyma s igeious ierpreaio Feyma iegrals for imagiary ime Browia Moio Markov Propery Feyma-Kac Formula Some applicaios of he Feyma-Kac formula

3 Chaper Iroducio. Basic Axioms The Quaum Mechaics descripio of a poi paricle is give by hree hypohesis: H The sae of a paricle a ime is described by a fucio ψ(, : R 3 C (he wavefucio. The ime evoluio is give by he Schrödiger equaio i ψ(x, = ψ(x, + V (xψ(x, (. m where = h π, where h is Plack s cosa. m is he mass of he paricle ad V : R3 R is he poeial i which he paricle moves. is he Laplace operaor. H The probabiliy o fid he paricle a ime i a regio A R 3 is give by ψ(x, dx ψ(x, = ψ(x, ψ(x, is ierpreed as he probabiliy desiy, hus R 3 ψ(x, dx = We ca compare his o Newo s law: A mẋ = p ṗ = V (x( } mẍ = V (x( Classical mechaics gives us a pah alog which he paricle ravels, whereas Quaum mechaics gives us a pah of probabiliy desiies which correspod o he paricles posiio. V is he same i quaum ad classical mechaics H3 The probabiliy o fid he value of he paricle momeum i he se B R 3, a ime is give by ˆψ( p (π 3, dp (. B

4 CHAPTER. INTRODUCTION 3 where ˆψ(k, = R 3 e ik x ψ(x, dx is he Fourier rasform of ψ wih respec o x. Thus he momeum is relaed o he oscillaio of frequecies of ψ. We will see laer ha H3 makes sese as he aalogue of momeum Axioms H-H3 oly give probabiliies, o rajecories. I classical mechaics iiial posiio ad momeum deermie he rajecory, whereas i quaum mechaics, iiial wave fucio deermies ψ(x, for every.. Cosisecy of he Axioms Sice ψ(x, ad ˆψ( p (π 3, are ierpreed as probabiliy desiies, hey have o have oal iegral, for all imes. We ow prove his uder somewha resricive codiios. I holds for greaer geeraliy. Defiiio.. Assume V : R 3 R is coiuous, a fucio ψ : R 3 R C is called a classical soluio o (. if ψ(x, C (R for every x ad ψ(, C (R 3 for every ad if (. holds for every x ad. (Laer we shall sudy weak soluios which are o C A classical soluio is o good eough. We eed ψ(x, dx < for H ad H3, i.e. we eed ψ(x, as x fas eough. (i.e. we eed boudary codiios. A coveie assumpio is: Assume ha for all ime iervals [a, b] here exiss C a,b < ad α > 3 wih for every x R 3, [a, b]. ψ(x, C a,b x α ψ(x, C a,b x α (.3 We remark ha α = 3 is he borderlie iegrabiliy of ψ(x, ψ(x, dx ψ(r, r dr π i.e. so eed ψ(r, o have a leas r 3 decay a. So as a decay rae, (.3 is opimal. Lemma.. Assume ha V is coiuous, ψ is a classical soluio o (. ad saisfies (.3 he ψ(x, dx is cosa. Proof. Firs we rewrie (. as ψ(x, = iβ ψ(x, + iγv (xψ(x, (.4

5 CHAPTER. INTRODUCTION 4 wih β = m ad γ =. Now for B R = {x R 3 : x < R} d ψ(x, d dx = ψ(x, ψ(x, dx = ( d B R d ψ(x, ψ(x, + ψ(x, ψ(x, dx B R B R (. = = IBP = iβ Now iegraig his over gives ψ(x, dx ψ(x, dx = β B R B R The sice where α > 3. The [iβ ψ + iγv ψ]ψ + ψ[( iβ ψ + ( iγv ψ]dx B R (iβ( ψψ iβψ ψdx sice V = V B [ R ( ψ ψ ψ ψdx + B R Area of B R = 4πR ψ ψ C, R α (.5 βc, 8π R α 3 B R [( ψψ ψ ψ] x x ds [ ] ( ψψ ψ( ψ x x ds dτ (.5 B R R The limi of he LHS of (.5 is ψ(x, dx ψ(x, dx ad so i follows ha ψ(x, dx = ψ(x, dx for every ] Thus if ψ(x, dx = he ψ(x, dx = for all. Theorem.3 (Uiqueess. If wo soluios o (. ad (.3 agree a ime =, he hey are agree for all ime. Proof. If ψ, φ solves (., he ψ φ solves (., ad ψ(x, φ(x, = for every x by assumpio. The ψ(x, φ(x, dx = for all ime ad so ψ(x, φ(x, = for all ad x. We remark ha his heorem implies causaliy..3 Soluio o he Free Schrödiger equaio The free refers o he fac ha we have liear moio, o exeral forces. I Newoia mechaics, eiher he paricle says sill or moves i a sraigh lie. We cosider (. wih V =, ad i aomic uis, we ca assume = m =. i (x, = ψ(x, (.6

6 CHAPTER. INTRODUCTION 5 A simple soluio ca be guessed: ψ(x, = e i(k x+ k called de Broglie s wave. Bu ψ(x, = ad so ψ(x, dx =. To fid square iegrable soluios, we eed a more sysemaic approach. Recall he Fourier rasform: (Fψ(k, = ˆψ(k, = e ik x ψ(x, dx R 3 ad is iverse Moreover, F x j F = ik j : (F ψ(x, = ˇψ(k, = (π 3 e ik x ˆψ(k, dk R 3 [F x j F ψ](k = = [F x (π 3 j (π 3 F ] e i k x ψ( kd k (k e i k x [i k j ψ( k] d k }{{} (k g( k = FF g(k = g(k = ik j ψ(k ad hus i follows F F = k, where we hik of F,, x j, F as operaors of fucios. Oe ca say ha is diagoal i Fourier space. By viewig f(x as a vecor wih idex x (hik specrum!, he F F f(k = k f(k So, we ca solve (.6 by lookig a i i Fourier space: The soluio o his is rivial for fixed k: i ψ(x, = ψ(x, i ˆψ(k, = F F Fψ = k ˆψ(k, ˆψ(k, = e i k ˆψ(k, ad is well defied if ˆψ(k, L (equivalely ψ(x, L. Recall ha he Fourier represeaio is good for sudyig momeum from (.. Noe ha we have immediaely Newo s firs law - k i coservaio of momeum, usig e = ˆψ( k (π 3, dk = B This also meas ha wavepackes are spread over ime. B ˆψ( k (π 3, dk We are ieresed i posiio space, so we eed o do a iverse Fourier rasform. ψ(x, = F ˆψ(k, = (π d e ik x k i e ˆψ(k, dk = (π d e ik x e i k [ ] e ik y ψ(y, ody dk

7 CHAPTER. INTRODUCTION 6 Now do he iegral over k: e ik (x y e i k dk = e i (x y (π d = e i (x y (πi d e i (k (x y dk Where for he las iequaliy we have used ha he iegral is i Gaussia form wih mea (x y ad variace. Ad hus i ψ(x, = (πi d e i (x y ψ(y, dy (.7 We defie a operaor P : ψ(, ψ(,, called he propagaor. I is give by iegraig agais he kerel K(x y, = (πi d e i (x y (.8 A impora special case is he Gaussia wavepacke (i d dimesios. I is give by he iiial codiio: ψ(x, = eiv (x x x x πσ 4σ 4 The iegral i (.7 ca ow be doe explicily o give [ ] ψ(x, = ( σ π d σ v + i x x 4 (σ + i exp 3 σ + i σ v Wha does ψ(x, look like? ψ(x, is a Gaussia, wih oal probabiliy, mea posiio x ad sadard deviaio (or uceraiy σ. Furhermore i ca be show ha ψ(x, has mea x + v ad sadard deviaio σ( = σ + ( σ. Noe ha (i The mea solves mẍ = x( = x ẋ( = v (ii uceraiy icreases over ime (iii speed is ecoded by oscillaios. Noe ha Newo s equaios - i.e. behaves classically Re(ψ(x, = d 4 cos(v πσ (x x e x x 4σ i.e. faser oscillaios = bigger v = faser wavefucio..4 Ehrefes equaios We ve see ha he mea posiio of a Gaussia wavepacke solves free Newo s equaio (for he free Schrodiger equaio. A similar hig holds fore more geeral wavepackes ad poeials.

8 CHAPTER. INTRODUCTION 7 Defiiio.4. For a wavefucio ψ(x, wih ψ(x, = he expeced posiio ad expeced momeum are vecors give by X( = x ψ(x, dx R d P ( = k ˆψ( k R d (π d, dk respecively. The idea is ha ψ(x, is he posiio probabiliy desiy ad (π d ˆψ( k, he momeum probabiliy desiy. Takig derivaives of X( ad P ( we migh expec ha hey saisfy Ẍ( = X( =? ( V X( m similar o mẍ( = V (x(. Bu his is o rue. However, we do have he followig Proposiio.5. Proof. d d X j( = = = R d d d X j( = m P j( d d [x jψ(x, ψ(x, ]dx R d x j [( ψψ + ψ ψ]dx x j [(iβ ψ + iγv ψψ + ψ(iβ ψ + iγv ψ]dx R d = iβ x j [( ψψ ψ( ψ]dx R d = iβ x j [( ψψ ψ ψ]dx IBP R d = iβ ( xj ψψ ψ( xj ψdx R d ad by iegraio by pars o he secod erm we ge = iβ ( xj ψψdx (.9 R d = ( m R i x j ψψdx d = m (π d R i x j ψ ˆψdk Placherel d = m (π d k j ˆψ ˆψdk R d = p j ˆψ( p m R d (π d dp = m P j(

9 CHAPTER. INTRODUCTION 8 Cosider (.9 agai, akig oe more derivaive ad (.4 usig ad iegraio by pars d d X j( = iβ ψ xj ψ + ψ xj ψdx R d = iβ ( iβ ψ iγv ψ xj ψ + ψ xj [iβ ψ + iγv ψ]dx R d = iβ iγv ψψ + iγ(v ψ ψdx R d = γβ (V ψψ + V ψ ψ ψ V ψdx R d = γβ V ψ dx = ( xj V (x ψ(x, dx R m d Thus, we have show ha Def = m x jv ( d d X( = V ( m where he righ had side is o he same as V ( X(. We ca compare his o Newo s equaio d d x( = m V (x( So he mea fucio fulfils a Newo equaio bu he force is give by he mea value of he poeial wih respec o he wave fucio, o by he poeial a he mea posiio. We have proved Theorem.6 (Ehrefes s Equaios. Le V : R d R be i C, ψ C be a soluio o he (. saisfyig ψ dx = x ψ dx < ( V (x ψ dx < The (i (ii d d X( = m P ( d d P ( = V ( Now he mea value of he eergy P ( E( = + V ( m k = m ˆψ( k (π d, dk + V (x ψ(x, dx Classically E( = p( m + V (x, ad This is also rue i quaum mechaics. E( = p(ṗ( + V (x, ẋ( m = p([ṗ( + V (x, ] = m

10 CHAPTER. INTRODUCTION 9 Theorem.7. Le ψ(x, be a soluio of (. wih ψ + V ψ dx < he E( = Proof. Exercise

11 Chaper Quaum Mechaics via Operaors. Hilber spaces ad operaors.. Hilber spaces If we have he equaio (he Schrödiger equaio wih = m = i ψ = ( + V ψ = Hψ he i is reasoable o suspec ha he soluio will be of he form ψ( = e ih ψ bu we eed o make sese of a expoeial of a operaor. I his secio we sudy operaors from fucioal aalysis ad apply hem o he sudy of he Schrödiger equaio. Defiiio.. A ier produc o a vecor space X is a map, : X X C, ha saisfies he followig for every α, β C ad v, w, z X (i Lieariy i secod erm: v, αw + βz = α v, w + β v, z (ii Cojugae symmery: v, w = w, v (iii Posiiviy: v, v ad v, v = if ad oly if v =. The above implies ha αv +βw, z = α v, w +β v, z. Also he map v v := v, v is a orm. A complex vecor space X wih a ier produc is called Hilber if i is complee wih respec o his ier produc. A Hilber-Schmid basis is a se of muually orhogoal vecors ha is big eough o cosruc X. Formally i is a family {v : N} of muually orhogoal vecors such ha he se of fiie liear combiaios { α j v j : N, α j R, { j } N} j= is dese i X. If {v } is a Hilber-Schmid basis he Parseval relaio: w = N w, v Fourier series : w = N w, v v

12 CHAPTER. QUANTUM MECHANICS VIA OPERATORS.. Operaors Uforuaely may operaors ha we come across i physics are ubouded maps A : H H. These are defied o a a subse D(A H. We look a some commo operaors: (i I,, id : ψ ψ (ii Muliplicaio by a co-ordiae ( posiio operaor is ubouded (iii Muliplicaio by a fucio V : R d C x j : ψ x j ψ M V, V : ψ V ψ; (V ψ(x = V (xψ(x (iv Momeum operaor (ubouded P j : ψ i xj ψ (v Laplace operaor (ubouded : ψ x j ψ j= (vi Iegral Operaors (Kψ(x = K(x, yψ(ydy for some kerel K(x, y : R d R d C (vii Schrödiger operaor (ubouded H : ψ m ψ + V ψ If a operaor is bouded, he he operaor orm is give by A := Aψ sup Aψ sup < ψ H, ψ = ψ H\{} ψ..3 Commuaors Defiiio.. The commuaors of A ad B is [A, B] = AB BA e.g. [ x, x] = x (xf(x x x f(x = f(x + xf (x xf (x = f(x (Noe ha here are some domai issues here as o all fucios are differeiable.

13 CHAPTER. QUANTUM MECHANICS VIA OPERATORS. Exisece of soluio o he Schrödiger equaio We reformulae he Schrödiger equaio (. i laguage of operaors: i ψ = Hψ ψ( = ψ (. wih H = + V (ake = m =. If H were a umber he soluio would be rivial: ψ( = e ih ψ. If H was a marix we could defie defie ( i U = H =: e ih! ad check ha = U ψ = ihu ψ U ψ = ψ So U ψ solves (.. The same works for bouded operaors. A operaor U such ha U ψ solves (. is called a propagaor of (.. Problems arise whe H is ubouded, sice ( i =! H may o make sese. For example if H =, he H is defied oly for ψ C. Thus we eed a leas ψ C. We could ry o defie e ih := lim ( i H bu we have he same problem. Wha does work, however, is e ih := lim ( + i H (. because ( + i H exiss ad is bouded. Before coiuig we make a few more defiiios. Defiiio.3. A operaor is symmeric if Aψ, φ = ψ, Aφ ψ, φ D(A For example he operaor M V is symmeric if ad oly if V is real. Sice M V ψ, φ = V (xψ(xφ(xdx = ψ(xv (xφ(xdx = ψ, M V φ Noe ha usig iegraio by pars, x is ai-symmeric, i x is symmeric ad x is symmeric. I paricular is symmeric, ad hus H is also symmeric. Lemma.4. Le A be a symmeric operaor. For every φ D(A, δ > (iδ + Aφ δ φ (.3 Proof. (iδ + Aφ = (iδ + Aφ, (iδ + Aφ = δ φ + Aφ + iδ φ, Aφ iδ Aφ, φ = δ φ + Aφ δ φ

14 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 3 So i paricular (iδ + H is always ijecive. However, as a operaor (iδ + H : C L i is o oo (he domai is oo small. Defiiio.5. The adjoi A of a operaor A is he uique operaor wih A ψ, φ = ψ, Aφ φ D(A, ψ D(A where D(A = {ψ L : ψ, Aφ C ψ φ } for some cosa C ψ ad for all φ D(A}. A symmeric operaor is called self-adjoi if D(A = D(A The good ews is ha H is self-adjoi o suiable domais for all physically releva poeials V (bu o, for example, for V (x = C x wih C > 4. Proposiio.6. Assume ha A is self-adjoi he for all δ R\{} (iδ + A is bouded ad (iδ + A < δ. So, for example if for φ L pu u = (iδ φ, he u is he soluio o he ihomogeeous equaio. iδu u = φ We ow reur o cosrucig e ih. Theorem.7. Le H be self-adjoi wih D(H dese i L. The for each φ L exiss i L U φ := lim ( + i H φ Proof. Se V ( = ( + i H. By proposiio.6 ad he fac ha + ah = a( a + H he ( + ah = a ( a + H. Thus we have ( + i H = i ( i + H I follows V ( for every N. Furhermore we have To see his, for φ D(H compue = lim V (φ = φ (.4 V (φ φ = ( ( + ih φ = ih + ih φ = ih + ih φ = i( + ih Hφ I follows ha V (φ φ ( + ih Hφ. For geeral φ, by deseess of D(H i L ake φ D(H wih φ φ < ɛ. The V (φ φ = (V ( φ (V ( φ + (V ( (φ φ H φ + φ φ ɛ

15 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 4 for every ɛ. For geeral oe This implies (V ( φ = (V ( φ = (V ( (V ( k φ k= k= (V ( φ V ( V ( k We ow show ha {V (φ} N is a Cauchy sequece for every φ. Tha is V ( V m ( L ad so his implies srog covergece. Noe m, ψ L ɛ d (V ( V m (φ = lim ɛ ɛ ds [V m( sv (sφ]ds = lim ɛ ɛ [ V m( sv (s + V m ( sv m(s]φds (.5 Now [ ] V m( sv (s V m ( sv (s = ih ( + i( s m Hm+ ( + is H ( + i s m Hm ( + is [ H+ + is = ih H i( s m H ] ( + i( s m Hm+ ( + is ( H+ s = H s ( m i( s + ( m m H + is H }{{}}{{} as,m bouded by Bu wha abou H? Assume φ D(H, he.5 ( s s i( s ( + m m H m ( + is H H φ }{{} ( + m H φ m, (.6 So (V (φ is Cauchy for φ D(H. For geeral φ we use deseess of D(H. We claim he followig (proof is lef as a exercise } (i D(H = (H + i D(H (ii rage((h + i D(H dese = D(H Wih his we ge {V (} N is a Cauchy sequece ad so a limi exiss. Laer we shall wrie U = e ih bu for i o warra he oaio we eed o check ha i acually behaves like a expoeial. So far we have show ha he operaor U defied as acually exiss. U φ := lim V (φ

16 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 5 Theorem.8. Defie U as above. The (i U is srogly coiuous i.e. (i For every φ D(H L lim U φ = U φ φ L. U Hφ = HU φ. So i paricular U φ D(H for each ad φ D(H. (iii For φ D(H i (U φ = H(U φ. So φ(x, = U φ (x solves (. wih iiial codiio φ. Proof. (a V (φ V ( φ as he resolve is aalyic (for > or by direc compuaio for = (see above. By (.6 covergece is uiform i o o compac iervals. (b Le φ D(H The Now le ψ = V (φ The we have (i ψ U φ (ii (Hφ N is coverge i L U Hφ = lim V (Hφ = lim HV (φ. Sice H is a closed operaor we have U φ D(H ad HU φ = lim Hφ = U Hφ. (c By par (b he claim is ha Cosider he limi i (U φ = H(U φ = U Hφ lim V ( = lim ( + i H φ i = lim H( ( + i H φ = lim ih( + i H V (φ We will ow show ha U behaves like a expoeial. Defiiio.9. A operaor U is uiary if U U = UU =. If U is uiary he So uiaries are always isomeries. Uφ = Uφ, Uφ = φ, U Uφ = φ. Theorem.. U = lim V ( is uiary for every R.

17 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 6 Proof. Clearly he adjoi of V ( is give by V ( = ( i H Now cosider W ( := V (V ( = ( + H. We will show ha W ( srogly. For φ D(H, Now, sice (W ( φ = (W ( φ = (W ( (W ( k φ (.7 k= ( + H φ φ + Hφ H φ φ we have ha ( + H. Also ( (W ( φ = + H φ = ( H + H This shows he orm of (W ( φ is bouded by H φ. So we ca boud he orm of (.7 by H φ which eds o zero as. For geeral φ, we use a approximaio argume. φ. Corollary. (Coservaio of probabiliy. For all ψ D(H he soluio ψ(x, of (. wih iiial codiio ψ fulfils ψ(, = ψ (.8 Fially we show ha U is a group. Theorem.. For every, s R we have U U s = U +s Proof. Usig he fac HU φ = U Hφ for φ D(H d ds (U su s = U su s + U s U s = HU s U s + U s HU s =. So s U s U s is cosa. The resul follows. So we coclude ha e ih := U is a uiary group of operaors wih (e ih = e ih ad e ih ψ solves i ψ = Hψ ψ( = φ D(H

18 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 7.3 Observables ad Eigevalues A observable is a self-adjoi operaor o he Hilber space L. For example: (i Poisso operaor X (ii Momeum operaor (iii Eergy operaor E (iv Agular momeum L (X j ψ(x = x j ψ(x P j : ψ i xj ψ (Eψ(x = ( m + M V ψ(x L x φ = i (x x3 x 3 x Defiiio.3. The mea of a operaor A wih respec o he wave fucio ψ is A ψ = ψ, Aψ Theorem.4 (Evoluio of he mea. If ψ solves (. he d d A ψ( = ψ, i [H, A]ψ = i [H, A] ψ( Proof. d d ψ, Aψ = d ψ(x, (Aψ(x, dx d = ψ, Aψ + ψ, Aψ = i Hψ, Aψ + ψ, A i Hψ = Aψ, i (HA AHψ = Aψ, i [H, A]ψ Corollary.5. The eergy is coserved. E( := H ψ = P + V ψ If [A, H] = for ay observable, he A ψ( = cosa ad we call A a coserved quaiy. For example, P = i x is coserved i he free Schrödiger equaio.

19 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 8 Defiiio.6. A eleme ψ L \{} is called a eigevecor (or eigefucio of A if here exiss a eigevalue λ such ha Hψ = λψ. Each eigefuio ψ gives rise o a coserved operaor, P ψ defied by P ψ φ = ψ, φ ψ i.e. he oe-dimesioal projecio oo he subspace spaced by φ. The HP ψ φ = H ψ, φ ψ = ψ, φ Hψ = ψ, φ λψ P ψ Hφ = ψ, Hφ ψ = Hψ, φ ψ = λψ, φ ψ = λ ψ, φ ψ. sice λ is a eigevalue of a symmeric operaor H, λ ψ = ψ, Hψ = Hψ, ψ = λ ψ, i.e. λ R. I follows ha for a j R N α j P ψj j= for eigefucios ψ j is a coserved quaiy ad more geerally i ca be show ha all specral projecios are coserved. We ca use eigefucios o sudy dyamics. For ψ a eigefucio wih eigevalue λ we have e ih ψ = e iλ ψ sice i ψ = Hψ = λψ. So ψ is a saioary soluio of (.. As such e iλ ψ produces he same expeced values as ψ. More precisely A e iλ ψ = e iλ ψ, Ae iλ ψ = ψ, e +iλ Ae iλ ψ = ψ, Ae +iλ e iλ ψ = ψ, Aψ = A ψ wave fucios ha differ oly by a cosa (o-x-depede phase describe he same physical sae. Proposiio.7. Assume ha ψ(x, = = α ψ (x wih Hψ = λ ψ. The (a ψ D(A if ad oly if = λ α < (a If eve λ 4 α < = he ψ(, = α e iλ ψ is he uique soluio of (..4 Heiseberg Picure of Quaum Dyamics Recall ha for ay observable we have A ψ( = e i H φ, Ae i H φ = φ, e + i H Ae i }{{ H } φ A( = A( φ.

20 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 9 So isead of solvig (. we ca solve he followig equaio i he space of operaors: This gives d d A( = i [H, A(] i He i H Ae i H + e i H A( i He i H = i [H, A(] (.9.5 Uceraiy Priciple Fudameal properies i probabiliy heory ( Mea value: E(X = xdµ(x Sadard deviaio: σ(x = (x E(X dµ(x I Quaum Mechaics, he posiio mea value is X ψ = x ψ(x dx or more geerally A ψ = ψ, Aψ For posiio i s clear ha (x X ψ ψ(x dx = X ψ X ψ is he variace. So X ψ X ψ is he sadard deviaio. I geeral A := is he uceraiy (i.e. sadard deviaio of A. A ψ A ψ Cosider he Gaussia wave packes wih zero mea value ad zero mea momeum. Tha is ψ(x = πσ e x 4σ. I ca be easily check ha X j ψ = ad P j ψ =. Furhermore X j = x j e x (πσ d σ dx = } {{... } x j e x j (πσ σ dx j = σ d P j = Pj ψ = ψ, Pj ψ = ψ(x( j ψ(xdx IBP = = ( x j σ ψ( x j ψdx = σ 4σ 4 j ψ j ψdx x j ψ(x dx = 4σ

21 CHAPTER. QUANTUM MECHANICS VIA OPERATORS So eiher we make σ small ad ge good iformaio abou posiio or we make σ large ad ge good iformaio abou momeum, bu we cao have boh. I paricular X j ψ P j ψ = The followig heorem shows ha his is he bes we ca hope for: Theorem.8 (Heiseberg s Uceraiy Priciple. For ay ψ D(Xj D(P j we have ψ X j ψ P j ψ Implicaio: i is i priciple impossible o measure boh posiio ad speed of a paricle o arbirary accuracy. To prove heorem.8 we will eed he caoical commuaio relaio (CCR: [P j, X j ] = i which follows from he observaio P j X j f = xj (x i f(x = f(x + x j f (x X j P j f = x j xj (f(x = x j f (x Theorem heorem.8 follows immediaely from Theorem.9 (Absrac Uceraiy Priciple. Le A, B be self-adjoi operaors. The for ay ψ D(A D(A D(B D(B we have Proof. We use Now use A ψ B ψ [A, B] ψ [X, Y ] ψ = ψ, (XY Y Xψ = Xψ, Y ψ Y ψ, Xψ = Xψ, Y ψ Xψ, Y ψ = Im Xψ, Y ψ Xψ, Y ψ Xψ Y ψ The X = A A ψ Y = B B ψ }{{}}{{} α β [X, Y ] = [A α, B β] = [A, B] [A, β] [α, B] + αβ[, ] = [A, B] ad Xψ = (A αiψ, (A αiψ = Aψ, Aψ Aψ, αψ αψ, Aψ + αψ, αψ = ψ, A ψ α ψ, Aψ α ψ, Aψ + α ψ = A ψ A ψ The equaliy ca be used o prove he followig: [X, Y ] φ = i Im Xψ, Y ψ (.

22 CHAPTER. QUANTUM MECHANICS VIA OPERATORS.6 The Sabiliy of he Hydroge Aom I classical mechaics here is a ucleus ad oe elecro. The ucleus is much heavier, so we cosider is posiio fixed ad calculae he moio of he elecro. They ierac vis he Coulomb poeial V (x = This meas ha he classical eergy is give by e x x uc = e x. (. E = ρ x. x This meas ha if he elecro falls io he ucleus, i gais ifiie eergy, which is impossible i he real world. The same problem seems o exis i celesial mechaics, bu here, he plaes ca be show o go o sable orbis aroud he su for a very log ime. The siuaio is differe for a elecro orbiig aroud a ucleus: As i moves o a curved orbi, i will always accelerae, herefore emi radiaio ad be slowed dow. So a classical elecro orbiig a classical ucleus would crash io ha ucleus sooer raher ha laer. I quaum mechaics his does o happe. Wha do you mea? Agai H is he Hamiloia operaor, i his example wih poeial give above, i.e. We eed ha H = m e x H ψ = ψ, Hψ = ψ, m ψ ψ, e x ψ is bouded from below i he variable ψ. More precisely we wa The key is if{ H ψ : ψ L } > Theorem.. For all ψ i a d dese subspace of D(H we have he followig ψ, ψ ψ, 4 x ψ(x dx ψ(x 4 x dx Proof. Firs we prove (oe xj x = xj x 3. ad x = i d x j x x jf(x = x x j x j x f(x = x j x d [ x P j x, X j] (. j= [ ] x j x x j x x jf(x [ xj x 3 f(x + x jf(x ] + x j x jf(x

23 CHAPTER. QUANTUM MECHANICS VIA OPERATORS subracig ad summig over j from o d gives he righ-had side of (.. This equaliy ad (. allows us o express d d ψ, x ψ = Im( x P j x ψ, x jψ Now usig he equaliy P j x = x P j + [P j, x ] = x P j + i xj x 3. This gives us Summig over j gives j= x P j x ψ, x jψ = x ( x P j + i x j x 3 ψ, x jψ = x P jψ, x j ψ i x j x 4 ψ, x jψ = P j ψ, j= x j x ψ i ψ, x j x 4 ψ d d ψ, x ψ = Im x j P j ψ, x ψ + ψ, x ψ We use his ad he he Cauchy-Schwarz iequaliy ψ, x ψ = = C S = = 4 d ( (d Im( x j P j ψ, x ψ j= j= 4 d ( (d x j P j ψ, x ψ 4 ( (d P x ψ, x ψ 4 ( (d P ψ, P ψ x x ψ, x x ψ 4 d ( (d ψ, j= P j }{{} ψ ψ, 4 ( (d ψ, ψ, d j= x ψ x j x 4 ψ I follows ψ, which gives he claim (d = 3. 4 x ψ (d ψ, ψ ψ, x ψ I he above proof we have used Cauchy-Schwarz iequaliys o L (R d, R d wih f, g d = f i, g i L (R d,r j=

24 CHAPTER. QUANTUM MECHANICS VIA OPERATORS 3 For he Hydroge aom i R 3 we fid wih f(x = 8m x ψ, Hψ = ψ, ( m e e ψ = ψ, ψ ψ, x m x ψ e x me4 8m ψ, ψ = ψ, f(xψ x which proves sabiliy.

25 Chaper 3 The Harmoic Oscillaor The quaum harmoic oscillaor is he quaum mechaical aalogue of he classical harmoic oscillaor. I is oe of he few quaum mechaical sysems for which a simple exac soluio is kow. Furhermore i is oe of he mos mos impora model sysems i quaum mechaics because a arbirary poeial ca be approximaed as a harmoic poeial a he viciiy of a sable equilibrium poi. This is because paricles move very lile if hey si close o a poeial eergy miimum ad have low kieic eergy. I his case we ca approximae eergy by is Taylor expasio. V V (x + (x x V (x + (x x V (x The V (x ca be se o zero by a eergy shif ad V (x = as we are a a miimum. Thus we ge he followig V (x = κ x where κ is some cosa. 3. Classical Harmoic Oscillaor Usig he classical relaios ẋ( = m ρ( ρ( = κx( Seig ω = κ m we ge he soluios x( = A si(ω b The eergy of he sysem is give by ρ( = mωa cos(ω b H class = ρ m + κ x 3. Quaum Harmoic Oscillaor I quaum mechaics i oe dimesio H = m x + κ x (3. 4

26 CHAPTER 3. THE HARMONIC OSCILLATOR 5 ad i d dimesios H = m + xax where A is a posiive defiie marix, bu we shall be dealig oly wih oe dimesio. We ca simplify he expressio by scalig. For ψ L (R saisfyig (3. defie Pu y = λx, ad τ = ω he Thus we have ψ(λx, ω = ψ(x, wih λ = mw, ω = κ m xψ(x = x( ψ(λx, ω = λ ψ (y, τ = mω ψ (y, τ x ψ(x, = λ (λx ψ(λx, ω = mω y ψ(y, τ ψ(x, = ψ(λx, ω = ω τ ψ(y, τ Hψ(x, = mω m ψ (y, τ + κ mω x ψ(y, τ = ω ψ (y, τ + ω m = ω ( ψ (y, τ + y ψ( x i ψ(x, = iω τ ψ(y, τ mω y ψ(y, τ This gives us he more simplified quaum harmoic oscillaor equaio of ( ψ (y, τ + y ψ(y, τ = i τ ψ(y, τ (3. I a similar way he above we ca use scalig o pu m = = i he geeral Schrödiger equaio ( Eigevalues ad Eigevecors We wa o solve For his a differe scalig workigs beer: ψ(x = ψ( x λ Hψ = λψ (3.3 wih λ = m, ω = k m. The if ψ solves (3.3, he ψ solves ψ x + ω x ψ = λ ψ We drop he oaio ad sudy xψ + ω x ψ = λψ (3.4 ad we will fid all is soluios. We shall use he followig sraegy:

27 CHAPTER 3. THE HARMONIC OSCILLATOR 6 (i Fid oe eigefucio (ii Fid he operaors A +, A ha map each eigefucio o he ex higher (or ex lower eergy eigefucio. This give us ifiiely may eigefucios. (iii Show his gives hem all We sar wih (ii ad defie Defiiio 3.. The creaio operaor (a frequecy ω is ad he aihilaio operaor (a frequecy ω is A + := ( d dx + ωx A := ( d dx + ωx I fac i ca be easily show ha A = A +. So we wrie A = A, A = A + Lemma 3.. A A = H ω AA = H + ω Proof. The proof of he oher is similar. AA ψ = ( d dx + ωx( d dx + ωxψ = ( d dψ + ωx(ωxψ dx dx = (ωψ + ωxdψ dx d ψ dx ω x ψ ωx dψ dx = ( d ψ dx + ω x ψ + ω ψ = (H + ω ψ Before we impleme sep (ii, we shall firs prove he followig proposiios: Proposiio 3.3. Ay eigevalue of H is greaer ha or equal o ω. Proof. Suppose λ is a eigevalue of H. The λ = λ ψ = ψ, λψ = ψ, Hψ = ψ, (A A + ω ψ = ψ, (A Aψ + ψ, ω ψ = Aψ + w w

28 CHAPTER 3. THE HARMONIC OSCILLATOR 7 Proposiio 3.4. Assume Hψ = λψ for some λ ω. The (a A ψ is a eigefucio of H wih eigevalue of λ + ω (b For λ > ω, Aψ is a eigefucio of H wih eigevalue of λ ω (c λ = ω if ad oly if Aψ = Proof. Sar wih (c. Assume Hψ = λψ. The Aψ = ψ, A Aψ = ψ, (H ω ψ = ψ, (λ ω ψ = (λ ω ψ which is zero if ad oly if λ = ω. For (b, assume agai Hψ = λψ he H(Aψ = (AA ω Aψ = A(A Aψ ω ψ = (H ω ψ ω ψ = (λ ω ψ ω ψ For (a = (λ ωψ H(A ψ = (A A + ω A ψ = A (AA ψ ω ψ = (H + ω ψ + ω ψ = (λ + ω ψ + ω ψ = (λ + ωψ We eed o check ha A ψ. Noe ha A ψ = ψ, AA ψ = ψ, (H + ω ψ = ψ, λψ + ω ψ > So sep (ii is complee. For sep (iii: Proposiio 3.5. If λ is a eigevalue of H he λ { ω + ω : N} Proof. Assume by way of coradicio, λ = ω(γ +, γ (, 3 N. Le ψ be he correspodig eigefucio. The A ψ is a eigefucio wih eigevalue λ ω = ωγ. So H(A ψ = γωa ψ. Now apply A agai o fid H(A + ψ = (γ wa + ψ ad sice γ < his coradics proposiio 3.3. So we have show A is a bijecio from he eigespace wih eigevalues ω +ω o he eigespace wih eigevalues ω + ( + ω. Similarly A is a bijecio from he ( + + ω-eigespace o he ( + ω-eigespace. I ow remais o fid oe eigespace, i.e. eigefucios correspodig o λ = ω (sep. They are give, by par (c of proposiio 3.4 o be exacly he fucios for which Aψ =. Thus Aψ = (ψ (x + ωxψ(x = ψ (x ψ(x = ωx log(ψ(x = ω x + c ψ(x = e ω x e c

29 CHAPTER 3. THE HARMONIC OSCILLATOR 8 We eed ψ =, so we fid = e c e ωx dx = e c ( π ω. Thus φ(x = ( ω π 4 e ω x Theorem 3.6. The eigevalues of he (scaled, oe dimesioal quaum harmoic oscillaor equaio H osc = x + ω x are give by λ = ( + ω for N {}. The correspodig ormalised eigefucios are φ = ω 4 ( d π 4! dx + wx e ω x Proof. For = see above. The by way of iducio assume for, he is a eigefucio, bu o ormalised. We have A ψ = ( d dx + ωxψ A ψ = ψ, AA ψ = (λ + ω ψ = ( + ω. So i follows ha ψ + = + ω A ψ is a eigefucio ad ormalised. A direc calculaio of he above gives ( d dx + ωx e ω x = H e ω x wih H (x =, H = x, H 3 = 4x,... H (x is called he h Hermie polyomial. Noe ha ψ (x > for all x ad is he oly eigefucio wih ha propery. 3.4 Dyamics We wa o solve wih ψ(x, = ψ (x Periodiciy Le us wrie ( x + x ψ(x, = i ψ(x, (3.5 ψ (x = α ψ (x =

30 CHAPTER 3. THE HARMONIC OSCILLATOR 9 wih ψ eigefucios o H = ( x + x Hψ = ( + ψ (where we are cosiderig ω =. I ca be show ha {ψ } N is a orhoormal basis of L (R. The by proposiio.7 ψ(x, = = α e i(+ ψ (x = [ ] α e i c H (x = e x i Thus ψ(x, is π periodic up o a global phase, which is ivisible o all observables Dyamics of mea values Recall he Ehrefes equaios For he harmoic oscillaor d d X( ψ = P ( ψ d d P ( ψ = V ( ψ V (x ψ = ψ, ( x ψ = ψ, xψ = x ψ = V ( x ψ so i his case (bu o i geeral he mea values follow precisely he classical rajecio Dyamics of observables For he mea values we ow kow Solvig his we ge The rick is o oe ha (3.6 Thus defiig: d d X( ψ = X( ψ. X( = cos( X( + si( P ( (3.6 P ( = cos( P ( si( X( ψ(, Xψ( = e ih ψ, Xe ih ψ = cos( ψ(, Xψ( + si( ψ(, P ψ( X( := e ih Xe ih = cos(x + si(p This implies X ( := e ih X e ih = e ih X } e ih {{ e ih } = I is he same for all powers, so his implies e ix( = ψ, e ix ψ = ψ(x, e ix dx Xe ih = = i! X ( = exp(i(cos(x + si(p

31 CHAPTER 3. THE HARMONIC OSCILLATOR 3 This becomes useful ogeher wih he Weyl relaio (proof as exercise: e i(ax+bp = e iax e ibp e i ab Theorem 3.7. [Mehler Kerel] The soluio ψ(x, of (3.5 is give by ψ(x, = K (x, yψ (ydy where is he Mehler Kerel e i [ (e i K (x, y = exp x y (e i y x ] π( e i ( e i Proof. See laer (uses Feyma-Kac formula Noe ha for = π, e i = i, e i =. So [ ] K (x, y = e i π 4 exp π 4 ((ix + y + (iy + x = e i π 4 e ixy π ad so K is he kerel of he Fourier rasform, i.e. urs posiio io momeum ad vice-versa. For = kπ he kerel is udefied. Bu for kπ i coverges o a δ kerel.

32 Chaper 4 The Feyma-Kac Formula 4. Feyma iegral The aim of his secio is o compue he iegral kerel for e ih i geeral, i.e. fid a fucio K (x, y such ha e ih f(x = K (x, yf(ydy. Why? Because he he soluio of (.6 is give by a iegral. 4.. Troer Produc Formula We kow (a H = K (x, y = e i (πi (x y (free moio (b H = + x K (x, y = Mehler Kerel ad ha s all. We do kow he kerel of e i( e iv : K (x, y = (πi e i (x y iv (y e bu uforuaely Bu wha is rue is he followig e i( +V e i( e iv Theorem 4. (Troer Produc Formula. Le A,B be operaors. Assume, aleraively (i eiher A or B are bouded (ii or A, B ad A + B are self-adjoi ad bouded below. The for ay λ C wih Re(λ ad ay ψ D(A D(B D(A + B we have e λ(a+b ψ = lim (exp( λ A exp( λ B ψ 3

33 CHAPTER 4. THE FEYNMAN-KAC FORMULA 3 Proof. Oly for (i. I his case he heorem is called he Lie produc formula. Wihou loss of geeraliy le λ =. Ad le S = e A+B ad T = e A e B he e A+B (e A e B = S T = S T S Thus, Sice, = T k (S T S k k= + T S T S e A+B (e A e B T k (S T S k S e A+B k= (S T (max{ T, S } k= = (S T (max{ T, S } (S T (e A e A B e T = e A e B e A e B Fially sice S ad T are bouded operaors, A e B e B e + TS T So S = + A + B + (A + B! +... T = ( + A + A! +...( + B + B! +... = + A + B + A + AB + B +... S T = c somehig i A, B k!k k= Le us apply his o H = + V ad λ = i (so A =, B = V. exp( λ Aexp( λ B has kerel wih The we kow K (x, y = (πi e i x y e i V (y. I follows, [ Kerel of e ih ](x, y = lim K (x, x...k (x, x K (x, ydx...dx

34 CHAPTER 4. THE FEYNMAN-KAC FORMULA 33 sice, leig H = e ih f(x roer = lim e i H e i V e i H e i V }{{}... = lim Fubii = lim = lim [ = lim dx K (x, x... imes dx K (x, x e i H e i V f(x dyk (x, yf(y dx...dx dyk (x, x K (x, x...k (x, x K (x, yf(y [ dx...dx exp dx...dx K (x, x K (x, x...k (x, x K ( i k= ( ] (x, y f(ydy x k+ x k V (x k+ ( πi d ] f(ydy (4. wih x = x, x = y. 4.. Feyma s igeious ierpreaio We ow cosider he par iside he expoeial ad Feyma s ierpreaio of i. S = ( x k+ x k V (x k+ k= k= Le φ : R R d be piecewise liear wih φ ( k = x k for k =,,..., ad x = x ad x = y. The ( S = φ ( k+ φ ( k V (φ ( k+ ad isead of iegraig over x i we ca iegrae over fucios ha sar a x ad fiish a y. Wha s more, S coverges as a Riema sum o φ (s V (φ(sds =: S(φ,. So by akig he limi i he expoe oly, gives a aswer. S(φ, is he classical acio correspodig o he classical Newo equaio ẍ = V (x. So we ca ge he iegral kerel of e ih by (i evaluaig he classical acio all possible pahs (ii averagig over he resuls. So i a sese, he fial iegral should be over (R d or raher (R d [,] (he space of all fucios [, ] R d. Grea heurisic advaage: Sice S(φ, is saioary a he classical soluio hose pahs ha are close o i shall cou mos ( saioary phase argume. Feyma we oe sep furher ha (4. ad ook he limi also i he measure. The resul: e i H (x, y = e i S(φ, dφ

35 CHAPTER 4. THE FEYNMAN-KAC FORMULA 34 where dφ is he Lebesgue measure o all fucios from x o y. The bad ews is ha i ca be show ha here is o way o make sese of such a measure. This does o sop physiciss! Ad he Feyma iegral is used widely (e.g. for guessig resuls ha have o be proved by oher meas. We wa a rigorous geeral heory, ad have o sele for less Feyma iegrals for imagiary ime Marc Kac oiced ha Feyma iegrals make mahemaical sese for imagiary ime. Pu differely e H = e i( ih has kerel give by a space of fucios. So from ow o we aim o fid he kerel of e H for R. Idea: Icorporae he by φ (s ds erm io he measure. So cosider he measure o fucios µ (φ( A, φ( A,..., φ( ( A = ( π = where d dx...dx e x x A (x e x x A(x... A (x e y x dx...dx g (x, x A (x g (x, x A (x... A (x g (x, y (4. g s (x, y = (πs d e s x y Noe ha his fucio saisfies g s (x, yg (y, zdy = g +s (x, z ad replacig he A (x i (4. wih e V ( x gives back he real ime of (4.. Wha abou he limi? The idea is o view he measures µ as he fiie dimesioal disribuios of some big measure. So he quesio is: Is here a measure µ o fucios φ : [, ] R d such ha for all <... < we have µ (φ( A, φ( A,..., φ( A = g (x, x A (x g (x, x A (x g 3 (x 3, x A3 (x 3...g (y, x dx...dx? The aswer is yes by Kolmogorov s cosisecy (someimes exesio or exisece heorem: Theorem 4. (Kolmogorov cosisecy heorem. Le {µ,..., : < <... < < } be a family of fiie measures (e.g. probabiliy measures. Each u,..., is assumed o be o (R d ad we assume he cosisecy codiio µ,.., k,τ, k+,... (A... A k R d A k+... A = µ,.., k, k+,... (A... A k A k+... A for every j (, ad τ [, ] wih k < τ < k+ ad A j R d measurable. The here exiss a measure µ o (R d [,] (i.e. fucios from [, ] o R d such ha µ(φ( A,..., φ( A = µ,..., (A... A

36 CHAPTER 4. THE FEYNMAN-KAC FORMULA 35 Le us check ha our measures are cosise. µ 4 (φ( 4 3 A, φ( 4 R, φ( 4 B = µ 4, 4, 3 (A R B 4 x x 4 e = dx (π 4 d A (x dx e = [e 4 H A e 4 H R de 4 H h y ](x = [e 4 H A e H h y ](x = µ 4, 3 4 (A B x x 4 (π 4 d R (x dx 3 e x 3 x 4 (π 4 d y x 3 B (x 3 e 4 (π 4 d }{{} =:h y(x 3 The measure µ will ur ou o be he codiioal Browia moio from x o y i ime. We wrie W x,y (dφ for i. The formally e H (x, y = lim e k+ k= V (φ( µ (dφ = e V (φ(sds W x,y (dφ (4.3 Noe ha he erm iside he expoeial coverges o V (φ(sds bu we eed o be careful abou he limi of µ. We will prove his rigorously laer. (4.3 is called he Feyma-Kac formula. We aim o prove i. 4. Browia Moio Classical Cosrucio: Le {X i } i N be idepede radom variables wih probabiliy P(X i = = P(X i = =. Pu S = i= X i. So S ca be viewed as piecewise liear fucio, jus as before. Acually if X i is Gaussia radom variable raher ha Beroulli we ge precisely he picure we had i he imagiary ime Feyma iegrals. P(S A,..., S A = g (, x A (x g (x, x A (x...g (x, x A (x dx...dx Now we wa o make he grid fier. Ideify i N wih i N R ad sed N. Keepig sep size would make he fucio o rough. Try jump size N, bu he N i= N X i = N S law of large umbers E(X = So his sep size is oo small!. The ceral limi heorem implies. N i= N X i disribuio N (, Now formalise his: Defie a map G : {, } N C([, ]; R, (X i i N B (N : [, ] R, where B (N s { N k i= X i liear ierpolaio if s = k oherwise

37 CHAPTER 4. THE FEYNMAN-KAC FORMULA 36 Le W (N be he image measure of he Beroulli measure (B(, N uder G i.e. for a subse of A of C([, ]; R we defie W N o be he pushforward measure W (N (A = (B(, N [G (A] (4.4 Noe ha he σ algebra o C([, T ]; R will always be he oe geeraed by poi evaluaios f f( for some. Facs abou W (N : (i I is coceraed o piecewise liear fucios wih corers a k N ad slopes N (ii Each possible pah has weigh N Theorem 4.3 (Dosker. W (N coverges weakly (wih respec o he opology of local uiform covergece o a probabiliy measure W o C([, T ]; R W is called Browia moio sarig a. The same cosrucio works o C(R +, R or C(R +, R d. Shifig he sarig poi of he radom walk o x R d gives W x - Browia moio sarig a x. Sice W is a measure o C(R +, R d we ca say higs abou almos all pahs q : R + R d. A remark o oaio: a pah is a (radom fucio draw accordig o W x. We will use q(s or q s for a pah (evaluaed a ime s. Remarks (a q( = x for W x -almos all pahs (b q is owhere differeiable W x -almos surely. Sice he fiie approximaios slope N we could expec ha q q s s for small s. This is almos correc. The realiy is a bi more ivolved. lim sup lim sup δ q q = almos surely l(l( q s q = almos surely s l( s Theorem 4.4 (Fiie Dimesioal Disribuios o Browia Moio. For < < <... < ad A, A,..., A B(R d (Borel ses where P(A = W x (A. The = where H = W x (q A,..., q A e x x (π d A (x e ( x x (π( d A(x e ( x x (π( d A(x dx...dx = [e H A e ( H A e (3 H...e ( H A ](x (4.5

38 CHAPTER 4. THE FEYNMAN-KAC FORMULA 37 Skech Proof. Use Dosker s Theorem 4.3 wih Gaussia jumps: Sar wih jumps of size,, 3,.. ad he refie he grid (reaiig origial pois,,... By he semi-group propery of (W x (N (q A,..., q A = righ had side of (4.5 for all N ad he lef had side coverges o W x by Dosker. A equivale defiiio of Browia moio is he uique probabiliy measure W x o C(R +, R d such ha W x (q A,..., q A = [e H A e ( H A...e ( H A ](x for all < <... < R +, A,..., A R d ope where H =. 4.. Markov Propery Iuiively he behaviour of a pah q for imes afer depeds oly o q q(. Formally, cylider ses are ses of fucios for he form C (A = {q C(R +, R d : q( A} for some A B(R d (Borel se. Geerae he σ algebra F {} = σ({c (A : A B(R d } A subse B of C(R +, R d is i F {} if i ca be decided for ay fucio, q B, wheher q belogs o B or o by kowledge of q( aloe. Defie F [a,b] = σ({c (A : A B(R d, [a, b]} Defiiio 4.5. Le F : C(R +, R d R be measurable. The for a b we defie codiioal expecaio, W x (F F [a,b], o be he (almos surely uique radom variable C(R +, R d R such ha (i W x (F F [a,b] is F [a,b] measurable (ii For all F [a,b] measurable fucios G we have W x (F F [a,b] (qg(qdw x (q = F (qg(qdw x (q We ca hik of F [a,b] measurable fucios, F (F mf [a,b], as fucios ha deped oly o q s for a s b. For example: If F (q = F (q = b a q(sds is F [a,b] measurable q(sq( sds is F [,] measurable F (q = q (sds

39 CHAPTER 4. THE FEYNMAN-KAC FORMULA 38 The W ( q sds = = = qsdsdw (q = ( πs ( s ds = 3 3 ( qsdw (q ds x e s x dx ds iegraig over all pahs, lookig oly a q(s. Iuiively W x (q A F [a,b] (q is he probabiliy ha q A if we kow ha q s = q s for s [a, b]. Theorem 4.6. Codiioal expecaio exiss. Theorem 4.7 (Codiioal Expecaio of Browia Moio. For f L, se y R d we have W y (f(q F {s} (q = (e ( sh f(q s = W q s (f(q s he las expressio is he expecaio wih respec o a sadard Browia moio sared a he poi q(s a ime s. Proof. [e ( sh f](q s depeds oly o q s ad so is F {s} measurable. Now for g mf {s} we fid W y ([e ( sh f](q s g(q s Thm. 4.4 = (e sh ge ( sh Thm. 4.4 f(y = W y (g(q s f(q The moral is ha Browia moio sars a fresh a x R d whe codiioed o be a x R d. Theorem 4.8 (Markov Propery. For all F mf [, > we have W y (F F [,] = W y (F F {} Proof. Sice F {} F [,], he clearly W y (F F {} mf [,]. For G mf [,] we firs cosider G(q = g (q g (q...g (q wih < <... < = ad F (q = f (q + f (q +...f m (q +m. The W y (GW y (f F {} Thm. 4.7 = W y (g (q g (q...g (q [e (+ H f e (+ +H f......e (+m +m H f m ](q Thm. 4.4 = [e H g e ( H g.....g e (+ H f e (+ +H f...e (+m +m H f m ](y = W y (F G

40 CHAPTER 4. THE FEYNMAN-KAC FORMULA 39 Defie τ : L (C(R +, R d L (C(R +, R d which raslaes Browia moio. For example: F : g T We will eed he followig cosequece of his (τ h(q = h({q +s : s R} q sds (τ F (q = F (q + = T q+sds Lemma 4.9. Le f, g : C(R +, R R + wih f mf [,], W x ( f < ad sup y W y ( g <. The W x (fτ g W x (f sup W y (g y Proof. By he defiiio of codiioal expecaio (ii: W x (fτ G f mf [,] = W x (fw x (τ g : F [,] Markov = W x (fw x (τ g : F {} Thm4.7 = W x (fw q( (g W x (f sup W y (g y = W x (f sup W y (g y 4.3 Feyma-Kac Formula Theorem 4.. Assume V : R d R is bouded above ad below ad coiuous. The for each f, g L (R d we have f, e (H+V = f(xw (e x V (qsds g(q dx R d ( = f(x e V (qsds g(q dw x dx R d his meas ha i L sese Proof. exercise - usig Troer ad DCT ( e (H+V = W x e V (qsds g(q Bu wha we really wa is difficul poeials like V (x = x i R3. For isaces does he iegral wih respec o Browia moio of e V (qsds g(q make sese? For example V (x = x. Where he iegrad is ifiie we hope ha such pahs have measure zero. A key resul is:

41 CHAPTER 4. THE FEYNMAN-KAC FORMULA 4 Theorem 4. (Kashmiski. Le V : R d R, V be measurable wih ( sup W x x R d V (q s ds = α < he Proof. We will show ha ( sup W x e V (qsds x α ( I := sup W x x R! ( d ( The sup x R d W x e V (qsds = I = α. To show ha oe: ( I =! sup W x ds... ds V (q s...v (q s x R d ( = sup W x ds ds... ds V (q s V (q s...v (q s x R d s s = sup W x V (q x R d s [,] s...v (q s ds...ds s s... s Fubii = sup x R d ds s ds... s ds W x ( V (q s ds α (4.6 V (q s V (q s...v (q s s V (q s ds The secod equaliy follows sice he iegrad is symmeric i s,..., s, so we ca iegrae over he lower riagle, i.e. over he se {(s,..., s : s... s } ad muliply by he umber of differe orderigs!. Tha is, here are! permuaios of s,..., s ad V (s...v (s = V (s π(...v (s π( for all permuaios π : {,..., } {,..., }. Now, usig codiioal expecaio ad he fac ha V (q s V (q s...v (q s is F s measurable he, W (V x (q s V (q s...v (q s V (q s ds s repea imes o ge resul. s W x V (q s...v (q s sup W y V (q r dr y }{{} α αw x ( V (q s...v (q s Defiiio 4.. A measurable fucio V : R d R is Kao class, V K(R d if (i sup x R d { x y } V (y dy < if d = (ii lim r sup x R d { x y r} g(x y V (y dy = if d

42 CHAPTER 4. THE FEYNMAN-KAC FORMULA 4 where { l x d = g(x = d 3 x d V K(R d ca be locally sigular bu o oo much. Check his for V (x = x i R 3 : q lim sup x= { x y <r} dy = lim dy = r x R x y y q q q+ d So x K(R3 bu x / K(R 3. r y r Defiiio 4.3. V is locally Kao-class, V K loc, if K V K for every compac se K. V is Kao-decomposable,V K ±, if V = V + V wih V +, V ad V + K loc ad V K. ( Remember ha we wa W x e V (qsds < so we do care oo much abou wha V + does a ifiiy. Theorem 4.4. A o-egaive fucio V K(R d if ad oly if ( lim sup W x V (q s ds = x R d Proof for d = 4. By Fubii ( W x V (q s ds = Fubii = The by usig he subsiuio x y s e x y s W x (V (q s ds = e s x y V (ydyds (πs d [ ] e s x y ds V (ydy (πs d =: u(s, he x y d ds = (πs d (π d x y e u u d du =: Q( x y, So ( lim sup W x V (q s ds = lim sup Q( x y, rv (ydy x R 4 r x R 4 We show ha he righ had side of he above equaio is equal o lim sup g x y { x y <r} V (ydy r x R 4 where g is as i he defiiio of Kao-class (defiiio 4.. To see his, oe ha Q( x y, r g( x y for x y r while Q( x y, r if x y r. We oly look a d = 4, he x y Q( x y, r = (π (x y e (π 4 e { x y }

43 CHAPTER 4. THE FEYNMAN-KAC FORMULA 4 So 4π e r The firs iequaliy shows ha g(x y { x y r} V (ydy Q( x y, rv (ydy 4π e r g(x y { x y r 4 V (ydy } + 4π e x y r g(x yv (ydy lim sup W x x R 4 ( { x y r 4 } V (q s ds = V K(R 4 For V K(R 4, he firs erm i he las lie vaishes whe akig lim r sup x R 4. The secod erm also vaishes: if V K(R 4, here exiss r such ha g(x, yv (ydy < sup x R 4 x y <r for all r < r. We pariio R d io cubes of size r ad fid o each cube ha he secod erm max{e x y r : y cube } V (yg(x ydy } {{ } summig up over all he cubes gives a fiie resuls (sice he fucio decays very quickly which vaishes as r. Corollary 4.5. Assume V K(R d. The for ay ( sup W x exp V (q s ds < (4.7 x R d Proof. By heorems 4.4, 4., for sufficiely small > we fid sup W (exp x V (q s ds < x α where α <. Now, via codiioal expecaio ( W (exp( x V (q s ds = W (W x x exp( Ierae for arbirary ime iervals. = W x (exp( = W x (exp( α W x ( α. ( exp( V (q s ds exp( V (q s dsw x (exp( V (q s dsw q (exp( V (q s ds V (q s ds F [,] V (q s ds F [,] V (q s ds

44 CHAPTER 4. THE FEYNMAN-KAC FORMULA 43 Corollary 4.6. Assume V K ± (R d. The for ay ( sup W x exp V (q s ds < (4.8 x R d Proof. Proof exercise. Noe we have a egaive expoeial so V + oly makes higs small, bu why is V (q sds fiie for almos all Browia moio pahs? This corollary meas we ca make he followig defiiio: Defiiio 4.7. For V K ± (R d, f L (R d defie ( (P f(x := W x e V (qsds f(q {P } is called he Feyma-Kac semigroup The aim is show ha P f = e ( +V f. To achieve his we will firs show ha {P } is a srogly coiuous symmeric semi-group i L. Srog coiuiy meas ha We fis show ha P is symmeric. Lemma 4.8. If lim P f f L = f L f(xp g(xdx < he f(xp g(xdx = g(xp f(xdx Thus, i L, his meas f, P g = P f, g. For he proof, recall he measure iroduced jus before Kolmogorov s Cosisecy Theorem 4.. W x,y (g A,..., g A = ( + i= π( i+ i dx...dx e + i= j j xj xj A (x... A (x wih = < <... < < + = ad x = x, x + = y. This is called Codiioal Browia Moio from x o y i ime. Proof of lemma 4.8. Noe ha for s < W x,y (f(q s g(ydy = W x (f(q s g(q (we ca prove his usig a deseess argume ( f(xw x e V (qsds g(q dx = f(xg(yw x,y (e V (qsds g(q dxdy (4.9

45 CHAPTER 4. THE FEYNMAN-KAC FORMULA 44 We use he followig fac of Browia Moio (see exercise: ime reversibiliy W x,y (q A,..., q A = W y,x (q A,..., q A which exeds o geeral fucios of pahs. The RHS of 4.9 = f(xg(yw y,x (e V (q sds dxdy usig subsiuio s s ( = g(yw y e V (qsds f(q = g(yp f(ydy Nex we show ha P acs o L as a bouded operaor. I fac, we show more. Theorem 4.9 (Simo-98. Le f L p (R d wih p. The for every q p We say ha P is bouded from L p o L q. P f L q (R d ad sup{ P f L q : f L p } <. Proof. The proof uses he Riesz-Thori ierpolaio heorem: If a operaor A is bouded from L pj L qj for p,..., p ad q,..., q wih p j, q j he i is bouded from L r o L s such ha ( r, s is i he covex hull of So we oly eed o prove i is bouded (i from L o L (ii from L o L (iii from L o L {( p j, q j : j =,..., } ad we have already see (iii. For (i, le f L Now > Lemma 4.8 = = P f dx = ( W x e V (qsds f(q dx ( W x e V (qsds f(q dx ( f(x W x e V (qsds (q dx }{{} sup x...< by Kashmiski s heorem 4. ( W x e V (qsds f(q (P f dx Thus P f L ( P f dx sup x W x e V (qsds (q f L.

46 CHAPTER 4. THE FEYNMAN-KAC FORMULA 45 For (ii, firs show ha P f L for f L. ( P f = sup x ( Cauchy-Schwarz e V (qsds f(q dw x (q ( e V (qsds dw x (q sup x sup e x } {{ } C sice V K ± (f(q dw x (q V (qsds dw x (q sup e x y x (π d (f(y dy }{{} Fially we show ha P f L for each f L. For each g L P f, g = P f(xg(xdx = P g(x f(x < }{{} L which jusifies he use of lemma 4.8, ad so by above, P f, g P g f C g f d f (π A geeral fac: f L sup{ f(xg(xdx : g L, g = } < So P f L, P f L C f L ad fially P f L = P P f L C C f Oce we kow ha P = e H his shows he amazig saeme ha Corollary 4.. Suppose ha u solves u(x, = u + V u u(x, u wih V K ±. The u L p implies u(x, L q for all ad q p So P akes L p fucios i L q for q p. I fac, eve more is rue: Theorem 4.. Le V K ± (R d. The for each f L p, p ad each >, P f is a coiuous fucio. For he proof we eed Lemma 4.. For V K we have lim sup W x x R d ( e V (qsds =

47 CHAPTER 4. THE FEYNMAN-KAC FORMULA 46 Proof. Noe ha W x ( e V (qsds k= ( [ k! W x ] k V (q s ds (4. Pu α( = sup x W x ( V (q s ds. By heorem (4.4 lim α( =. By Kashmiski heorem 4. sup RHS of (4. α( k = α( x α( k= Proof of heorem 4.. Noe ha P = P P, so by heorem 4.9, we oly eed o cosider f L. Assume firs ha V K(R d. Defie g τ (x = W x ( e τ V (qsds f(q The ( ( g τ (x Markov = W x W qτ e τ V (q sds f(q τ = [e τh (P τ ]f(x So x g τ (x is coiuous as i solves he hea equaio Now g τ P f L g τ = g τ g τ ( = P τ f L ( = sup W x ( e τ V (qsds e τ V (qsds f(q x Markov = sup x ( W x ( e τ V (qsds W qτ (e τ V (q sds f(q ( sup W x e τ x V (qsds sup y ( sup W y e r V (qsds f L r O he righ had side, he firs erm eds o zero as τ. The secod erm is bouded by a cosa, by corollary 4.6 ad f. So P f is he uiform limi of coiuous fucios ad herefore coiuous. For V K ± he proof is compleed by showig ha i his case P is he uiform limi o compac ses of coiuous fucios, ad herefore agai coiuous. Thus he proof is compleed by he followig lemma. Lemma 4.3. Le V K ±. Pu V (x = V (x { x } K(R d ad pu ( P, (x = W x e V(qsds f(q The for f L, lim P, f(x = P f(x uiformly o compac subses of R d.

48 CHAPTER 4. THE FEYNMAN-KAC FORMULA 47 Proof. For all pahs q s ha ever leave {x R d : x } clearly V (q s ds = V (q s ds Pu A = {w C(R +, R d : sup s w s }. The by Cauchy-Schwarz iequaliy (( P f(x P, f(x = W x e V (qsds e V(qsds ( A + c A f(q [ ( ( W x e V (qsds e V(qsds ] f(q [W x ( A c ]. Noice ha he secod erm o he righ had side is equal o W ({w x C(R +, R d : sup w s > } s by properies of Browia moio. We ow show ha he supremum over x R d of he firs erm o he righ had side is fiie, idepedely of. This is because ad he secod erm is bouded ( W x e V (qsds + e V(qsds e V(qs+V (qsds ( = W x e V (qsds ( +W x e V(qsds + cross erms }{{} < by Kashmiski s 4. W x ( e V (qsds W x ( e V (q sds < as V K ± (R d. Similarly he cross erms are bouded uiformly i x ad. Takig sup x M for compac ses M R d above proves he lemma. Nex we show Theorem 4.4. Assume ha V is Kao-decomposable. The he semigroup P is srogly coiuous, i.e. lim P f f L = for all f L. Proof. We firs cosider f bouded wih compac suppor, say f(x D ad f(x = whe x > R. We wrie Q = e H, for he propagaor of he hea equaio. Recall ha Q f(x = (π d/ wih H = e x y f(y dy. I is a classical resul of he hea equaio (ad ca be checked usig he above formula ha lim Q f f L = for all f L. The sraegy of his proof is o compare P wih Q. Lemma (4. gives ( Q f(x P f(x = e V (qs ds dw x (q