Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 42
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Kleinert, PATH INTEGRALS1772.17 Time Evolution Amplitude of Freely Falling ParticleFor positive z, the Airy function can be expressed in terms of modified Bessel functions Iν (ξ) and Kν (ξ):33√rz1 zAi(z) =[I−1/3 (2z 3/2 /3) − I1/3 (2z 3/2 /3)] =K1/3 2z 3/2 /3 .(2.601)2π 3For large z, this falls off exponentially:3/21Ai(z) → √ 1/4 e−2z /3 ,2 πzz → ∞.(2.602)− π < argξ ≤ π/2,π/2 < argξ ≤ π,(2.603)For negative z, an analytic continuation34Iν (ξ) = e−πνi/2 J(eπi/2 ξ),Iν (ξ) = e−πνi/2 J(eπi/2 ξ),leads toi1√ hz J−1/3 (2(−z)3/2 /3) + J1/3 (2(−z)3/2 /3) ,(2.604)3where J1/3 (ξ) are ordinary Bessel functions. For large arguments, these oscillate likeAi(z) =Jν (ξ) →s2cos(ξ − πν/2 − π/4) + O(ξ −1 ),πξ(2.605)from which we obtain the oscillating part of the Airy functionAi(z) → √hi13/2sin2(−z)/3+π/4,πz 1/4z → −∞.(2.606)The Airy function has the simple Fourier representationAi(x) =Z∞−∞dk i(xk+k3 /3)e.2π(2.607)In fact, the momentum space wave functions of energy E arehp|Ei =sl −i(pE−p3 /6M )l/εh̄eε(2.608)fulfilling the orthogonality and completeness relationsZdphE 0 |pihp|Ei = δ(E 0 − E),2πh̄ZdE hp0 |EihE|pi = 2πh̄δ(p0 − p).(2.609)The Fourier transform of (2.608) is equal to (2.599), due to (2.607).33A compact description of the properties of Bessel functions is found in M.
Abramowitz and I.Stegun, op. cit., Chapter 10. The Airy function is expressed in Formulas 10.4.14.34I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 8.406.1782.182 Path Integrals — Elementary Properties and Simple SolutionsCharged Particle in Magnetic FieldHaving learned how to solve the path integral of the harmonic oscillator we are readyto study also a more involved harmonic system of physical importance: a chargedparticle in a magnetic field. This problem was first solved by L.D. Landau in 1930in Schrödinger theory.352.18.1ActionThe magnetic interaction of a particle of charge e is given byAmag =ecZtbtadt ẋ(t) · A(x(t)),(2.610)where A(x) is the vector potential of the magnetic field.
The total action isA[x] =ZtbtadteM 2ẋ (t) + ẋ(t) · A(x(t)) .2c(2.611)Suppose now that the particle moves in a homogeneous magnetic field B pointingalong the z-direction. Such a field can be described by a vector potentialA(x) = (0, Bx, 0).(2.612)But there are other possibilities. The magnetic fieldB(x) = ∇ × A(x)(2.613)as well as the magnetic interaction (2.610) are invariant under gauge transformationsA(x) → A(x) + ∇Λ(x),(2.614)where Λ(x) are arbitrary single-valued functions of x. As such they satisfy theSchwarz integrability condition [compare (1.41)–(1.42)](∂i ∂j − ∂j ∂i )Λ(x) = 0.(2.615)For instance, the axially symmetric vector potentialÃ(x) =1B×x2(2.616)gives the same magnetic field; it differs from (2.612) by a gauge transformationÃ(x) = A(x) + ∇Λ(x),(2.617)1Λ(x) = − B xy.2(2.618)with the gauge function35L.D.
Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965.H. Kleinert, PATH INTEGRALS1792.18 Charged Particle in Magnetic FieldIn the canonical form, the action readsA[p, x] =Ztbta(e1p − A(x)dt p · ẋ −2Mc2 ).(2.619)The magnetic interaction of a point particle is thus included in the path integral bythe so-called minimal substitution of the momentum variable:ep−−−→ P ≡ p − A(x).(2.620)cFor the vector potential (2.612), the action (2.619) becomesA[p, x] =Ztbtadt [p · ẋ − H(p, x)] ,(2.621)with the Hamiltonian11p2+ MωL2 x2 − ωL lz (p, x),H(p, x) =2M82(2.622)where x = (x, y) and p = (px , py ) andlz (p, x) = (x × p)z = xpy − ypx(2.623)is the z-component of the orbital angular momentum.
In a Schrödinger equation, thelast term in H(p, x) is diagonal on states with a given angular momentum aroundthe z-axis. We have introduced the field-dependent frequencyωL =eB,Mc(2.624)called Landau frequency or cyclotron frequency. This can also be written in termsof the Bohr magnetonh̄e,(2.625)µB ≡McasωL = µB B/h̄.(2.626)The first two terms in (2.622) describe a harmonic oscillator in the xy-plane witha field-dependent magnetic frequencyωB ≡ωL.2(2.627)Note that in the gauge gauge (2.612), the Hamiltonian would have the rotationally noninvariant formH(p, x) =p21+ MωL2 x2 − ωL xpy2M2(2.628)rather than (2.622), implying oscillations of frequency ωL in the x-direction and afree motion in the y-direction.1802 Path Integrals — Elementary Properties and Simple SolutionsThe time-sliced form of the canonical action (2.619) readsANe=N+1 Xn=1i1 h 222pn (xn − xn−1 ) −p + (py n − Bxn ) + pzn ,2M x n(2.629)and the associated tome-evolution amplitude for the particle to run from xa to xbis given by(xb tb |xa ta ) =N ZY3d xnn=1" NY+1 Zn=1i Nd 3 pnA ,3 exph̄ e(2πh̄)#(2.630)with the time-sliced actionNA =2.18.2N+1 Xn=1i1 h 222pn (xn − xn−1 ) −p + (py n − Bxn ) + pzn .2M x n(2.631)Gauge PropertiesNote that the time evolution amplitude is not gauge-invariant.
If we use the vectorpotential in some other gaugeA0 (x) = A(x) + ∇Λ(x),(2.632)the action changes by a surface termZee tbdt ẋ · ∇Λ(x) = [Λ(xb ) − Λ(xa )].∆A =c tacThe amplitude is therefore multiplied by a phase factor on both ends(2.633)(xb tb |xa ta )A → (xb tb |xa ta )A0 = eieΛ(xb )/ch̄ (xb tb |xa ta )A e−ieΛ(xa )/ch̄ .(2.634)For the observable particle distribution (x tb |x ta ), the phase factors are obviouslyirrelevant.
But all other observables of the system must also be independent of thephases Λ(x) which is assured if they correspond to gauge-invariant operators.2.18.3Time-Sliced Path IntegrationSince the action AN contains the variables yn and zn only in the first termPN +1n=1 ipn xn , we can perform the yn , zn integrations and find a product of N ∆functions in the y- and z-components of the momenta pn . If the projections of p tothe yz-plane are denoted by p0 , the product is(2πh̄)2 δ (2) p0N +1 − p0N · · · (2πh̄)2 δ (2) p02 − p01 .(2.635)These allow performing all py n , pz n -integrals, except for one overall py , pz .
The pathintegral reduces therefore to(xb tb |xa ta ) =×Z∞−∞#" NYN Z ∞+1 Zdpy dpz Ydpx ndx2πh̄(2πh̄)2 n=1 −∞ n n=1(2.636)ip2exppy (yb − ya ) + pz (zb − za ) − (tb − ta ) zh̄2M("#)expi NA ,h̄ xH. Kleinert, PATH INTEGRALS1812.18 Charged Particle in Magnetic Fieldwhere ANx is the time-sliced action involving only a one-dimensional path integralover the x-component of the path, x(t), with the sliced actionANx=p2e1px n (xn − xn−1 ) − x n −py − Bxn2M2McN+1 hXn=12 i.(2.637)This is the action of a one-dimensional harmonic oscillator with field-dependentfrequency ωB whose center of oscillation depends on py and lies atx0 = py /MωL .(2.638)The path integral over x(t) is harmonic and known from (2.168):sMωL2πih̄ sin ωL (tb − ta )inhMωLi× exp(xb − x0 )2 + (xa − x0 )2 cos ωL (tb − ta )h̄ 2 sin ωL (tb − ta )(xb tb |xa ta )x0 =− 2(xb − x0 )(xa − x0 )} .(2.639)Doing the pz -integral in (2.636), we arrive at the formula(z −z )21iM b a⊥(xb tb |xa ta ) = qe 2h̄ tb −ta (x⊥b tb |xa ta ),2πih̄(tb − ta )/M(2.640)with the amplitude orthogonal to the magnetic field⊥(x⊥b tb |xa ta )MωL≡2πh̄Z∞−∞dx0 eiM ωL x0 (yb −ya )/h̄ (xb tb |xa ta )x0 .(2.641)After a quadratic completion in x0 , the total exponent in (2.641) reads"1iMωL−(x2b + x2a ) tan[ωL (tb − ta )/2] + (xb − xa )22h̄sin ωL (tb − ta )MωLx + xayb − ya−itan[ωL (tb − ta )/2] x0 − b−h̄22 tan[ωL (tb − ta )/2]MωL (xb + xa )2(yb − ya )2+itan[ωL (tb − ta )/2] +2h̄22 tan[ωL (tb − ta )/2]MωL(xb + xa )(yb − ya ).+i2h̄"The integration MωLR∞−∞#!2#(2.642)dx0 /2πh̄ removes the second term and results in a factorMωL2πh̄sπh̄.iMωL tan[ωL (tb − ta )/2](2.643)1822 Path Integrals — Elementary Properties and Simple SolutionsBy rearranging the remaining terms, we arrive at the amplitude(xb tb |xa ta ) =s3ωL (tb − ta )/2iMexp (Acl + Asf ) ,2πih̄(tb − ta ) sin[ωL (tb − ta )/2]h̄(2.644)with an actionihM (zb − za )2 ωL+cot[ωL (tb − ta )/2] (xb − xa )2 + (yb − ya )2Acl =2tb − ta2+ωL (xa yb − xb ya )(2.645)and the surface termAsf =2.18.4beMωL(xb yb − xa ya ) = B xy .a22c(2.646)Classical ActionSince the action is harmonic, the amplitude is again a product of a phase eiAcl anda fluctuation factor.
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