Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 45
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Theproper order is achieved by rewriting Ĥ asĤ =M 2x̂ (t) − 2x̂(t)x̂(0) + x̂2 (0) + [x̂(t), x̂(0)] ,2t2(2.746)and calculating the commutator from Eq. (2.744) and the canonical commutation rule [p̂i , x̂j ] =−ih̄δij asih̄(2.747)[x̂(t), x̂(0)] = − Dt,Mso that we find the desired expressionĤ = H(x̂(t), x̂(0); t) =DM 2x̂ (t) − 2x̂(t)x̂(0) + x̂2 (0) − ih̄ .22t2t(2.748)Its matrix elements (2.738) can now immediately be written down:H(x, x0 ; t) =MD2(x − x0 ) − ih̄ .2t22t(2.749)H.
Kleinert, PATH INTEGRALS1952.23 Heisenberg Operator Approach to Time Evolution AmplitudeFrom this we find directly the exponential factor in (2.740)R0Di M2E(x, x0 ; t) = e−i dt H(x,x ;t)/h̄ = exp(x − x0 ) − log t .h̄ 2t2(2.750)Inserting (2.750) into Eq. (2.740), we obtain010hx t|x 0i = C(x, x )tD/2iM0 2exp(x − x ) .h̄ 2t(2.751)A possible constant of integration in (2.750) depending on x, x0 is absorbed in the prefactor C(x, x0 ).This is fixed by differential equations involving x:hi−ih̄∇hx t|x0 0i = hx|p̂e−iĤt/h̄ |x0 i = hx|e−iĤt eiĤt/h̄ p̂e−iĤt/h̄ |x0 i = hx t|p̂(t)|x0 0i.ih̄∇0 hx t|x0 0i = hx|e−iĤt/h̄ p̂|x0 ihx t|p̂(0)|x0 0i.(2.752)Inserting (2.744) and using the momentum conservation (2.743), these become−ih̄∇hx t|x0 0i =ih̄∇0 hx t|x0 0i =M(x − x0 ) hx t|x0 0i,tM(x − x0 ) hx t|x0 0i.t(2.753)Inserting here the previous result (2.751), we obtain the conditionsi∇0 C(x, x0 ) = 0,−i∇C(x, x0 ) = 0,(2.754)which is solved only by a constant C.
The constant, in turn, is fixed by the initial conditionlim hx t|x0 0i = δ (D) (x − x0 ),t→0to beDrM,2πih̄so that we find the correct free-particle amplitude (2.125)C=0hx t|x 0i ≡rM2πih̄tD(2.755)iM0 2(x − x ) .exph̄ 2t(2.756)(2.757)Note that the fluctuation factor 1/tD/2 emerges in this approach as a consequence of the commutation relation (2.747).2.23.2Harmonic OscillatorHere we are dealing with the Hamiltonian operatorĤ = H(p̂, x̂) =p̂2M ω2 2+x ,2M2(2.758)which has to be brought again to the time-ordered form (2.736).
We must now solve the Heisenbergequations of motiondx̂(t)dtdp̂(t)dt==i p̂(t)i h,Ĥ, x̂(t) =h̄Mii hĤ, p̂(t) = −M ω 2 x̂(t).h̄(2.759)(2.760)1962 Path Integrals — Elementary Properties and Simple SolutionsBy solving these equations we obtain [compare (2.151)]p̂(t) = Mω[x̂(t) cos ωt − x̂(0)] .sin ωt(2.761)Inserting this into (2.758), we obtainĤ =which is equal toĤ =oM ω2 n222[x̂(t)cosωt−x̂(0)]+sinωtx̂(t),2 sin2 ωtM ω2 2x̂ (t) + x̂2 (0) − 2 cos ωt x̂(t)x̂(0) + cos ωt [x̂(t), x̂(0)] .22 sin ωt(2.762)(2.763)By commuting Eq.
(2.761) with x̂(t), we find the commutator [compare (2.747)][x̂(t), x̂(0)] = −ih̄ sin ωtD,Mω(2.764)so that we find the matrix elements of the Hamiltonian operator in the form (2.738) [compare(2.749)]H(x, x0 ; t) =M ω2 2D020x+x−2cosωtxx− ih̄ ω cot ωt.22 sin2 ωtThis has the integral [compare (2.750)]ZiDM ω h 2sin ωt2dt H(x, x0 ; t) = −x + x0 cos ωt − 2 x x0 − ih̄ log.2 sin ωt2ω(2.765)(2.766)Inserting this into Eq.
(2.740), we find precisely the harmonic oscillator amplitude (2.170), apartfrom the factor C(x, x0 ). This is again determined by the differential equations (2.752), leavingonly a simple normalization factor fixed by the initial condition (2.755) with the result (2.756).Again, the fluctuation factor has its origin in the commutator (2.764).2.23.3Charged Particle in Magnetic FieldWe now turn to a charged particle in three dimensions in a magnetic field treated in Section 2.18,where the Hamiltonian operator is most conveniently expressed in terms of the operator of thecovariant momentum (2.620),e(2.767)P̂ ≡ p̂ − A(x̂),cas [compare (2.619)]P̂2Ĥ = H(p̂, x̂) =.(2.768)2MIn the presence of a magnetic field, its components do not commute but satisfy the commutationrules:eeeh̄eh̄[P̂i , P̂j ] = − [p̂i , Âj ] − [Âi , p̂j ] = i(∇i Aj − ∇j Ai ) = i Bij ,cccc(2.769)where Bij = ijk BK is the usual antisymmetric tensor representation of the magnetic field.We now have to solve the Heisenberg equations of motiondx̂(t)dtdP̂(t)dt==ii hĤ, x̂ (t) =h̄ii hĤ, P̂(t) =h̄P̂(t)Meh̄eB(x̂(t))P̂(t) + i∇j Bji (x̂(t)),McMc(2.770)(2.771)H.
Kleinert, PATH INTEGRALS2.23 Heisenberg Operator Approach to Time Evolution Amplitude197where B(x̂(t))P̂(t) is understood as the product of the matrix Bij (x̂(t)) with the vector P̂. In aconstant field, where Bij (x̂(t)) is a constant matrix Bij , the last term in the second equation isabsent and we find directly the solutionP̂(t) = eΩL t P̂(0),(2.772)where ΩL is a matrix version of the Landau frequency (2.624)eBij ,McΩL ij ≡(2.773)which can also be rewritten with the help of the Landau frequency vectorL≡eBMc(2.774)and the 3 × 3-generators of the rotation group(Lk )ij ≡ −ikij(2.775)asΩL = i L ·L.(2.776)Inserting this into Eq.
(2.770), we findx̂(t) = x̂(0) +eΩL t − 1 P̂(0),ΩLM(2.777)where the matrix on the right-hand side is again defined by a power series expansiont2t3eΩL t − 1= t + ΩL + Ω2L + . . . .ΩL23!(2.778)P̂(0)ΩL /2=e−ΩL t/2 [x̂(t) − x̂(0)] .Msinh ΩL t/2(2.779)P̂(t) = M N (ΩL t) [x̂(t) − x̂(0)] ,(2.780)We can invert (2.777) to findUsing (2.772), this implieswith the matrixN (ΩL t) ≡ΩL /2eΩL t/2 .sinh ΩL t/2(2.781)By squaring (2.780) we obtainMP̂2 (t)T=[x̂(t) − x̂(0)] K(ΩL t) [x̂(t) − x̂(0)] ,2M2(2.782)K(ΩL t) = N T (ΩL t)N (ΩL t).(2.783)whereUsing the antisymmetry of the matrix ΩL , we can rewrite this asK(ΩL t) = N (−ΩL t)N (ΩL t) =Ω2L /4.sinh2 ΩL t/2The commutator between two operators x̂(t) at different times is, due to Eq. (2.777), ΩL tie−1[x̂i (t), x̂j (0)] = −,MΩLij(2.784)(2.785)1982 Path Integrals — Elementary Properties and Simple SolutionsandTieΩL t − 1 eΩL t − 1x̂i (t), x̂j (0) + [x̂j (t), x̂i (0)] = −+MΩLΩTL ΩL ti sinh ΩL te− e−ΩL ti= −2.= −MΩLMΩLijij!ij(2.786)Respecting this, we can expand (2.782) in powers of operators x̂(t) and x̂(0), thereby time-orderingthe later operators to the left of the earlier ones as follows:M Tx̂ (t)K(ΩL t)x̂(t) − 2x̂T K(ΩL t)x̂(0) + x̂T K(ΩL t)x̂(0)2ih̄ΩL tΩL−.trcoth222H(x̂(t), x̂(0)) =This has to be integrated in t, for which we use the formulasZZΩL tΩLΩ2L /2coth,dt K(ΩL t) = dt=−222sinh ΩL t/2(2.787)(2.788)andZ1sinh ΩL t/2ΩL tsinh ΩL t/2ΩLdt tr= tr logcoth= tr log+ 3 log t,222ΩL /2ΩL t/2(2.789)these results following again from a Taylor expansion of both sides.
The factor 3 in the last termis due to the three-dimensional trace. We can then immediately write down the exponential factorE(x, x0 ; t) in (2.740):ΩL tsinh ΩL t/2iMΩL110 T00(x−x )coth(x−x ) − tr log. (2.790)E(x, x ; t) = 3/2 exph̄ 2222ΩL t/2tThe last term gives rise to a prefactor−1/2sinh ΩL t/2det.ΩL t/2(2.791)As before, the time-independent integration factor C(x, x0 ) in (2.740) is fixed by differentialequations in x and x0 , which involve here the covariant derivatives:ihhie−ih̄∇− A(x) hx t|x0 0i = hx|P̂e−iĤt/h̄ |x0 i = hx|e−iĤt/h̄ eiĤt/h̄ P̂e−iĤt/h̄ |x0 ic= hx t|P̂(t)|x0 0i = L(ΩL t)(x − x0 )hx t|x0 0i,(2.792)ihe00−iĤt/h̄0P̂|x iih̄∇ − A(x) hx t|x 0i = hx|ec= hx t|P̂(0)|x0 0i = L(ΩL t)(x − x0 )hx t|x0 0i.(2.793)Calculating the partial derivative we find−ih̄∇hx t|x0 0i = [−ih̄∇C(x, x0 )]E(x, x0 ; t)+C(x, x0 )[−ih̄∇E(x, x0 ; t)]ΩL tΩL000coth(x−x0 )E(x, x0 ; t).= [−ih̄∇C(x, x )]E(x, x ; t)+C(x, x )M22Subtracting the right-hand side of (2.792) leads toΩLMΩL tM(x − x0 ) − M L(ΩL t)(x − x0 ) = − ΩL (x − x0 ),coth222(2.794)H.
Kleinert, PATH INTEGRALS2.23 Heisenberg Operator Approach to Time Evolution Amplitudeso that C(x, x0 ) satisfies the time-independent differential equationeM−ih̄∇ − A(x) −ΩL (x − x0 ) C(x, x0 ) = 0.c2A similar equation is found from the second equation (2.793):MeΩL (x − x0 ) C(x, x0 ) = 0.ih̄∇0 − A(x) −c2199(2.795)(2.796)These equations are solved byC(x, x0 ) = C exp Z x iMedA( ) +ΩL ( − x0 ) .h̄ x0c2The contour of integration is arbitrary since the vector field in brackets,e 0eΩLe100A( ) − B × ( − x )A ( ) ≡ A( ) +( −x)=cc2c2(2.797)(2.798)has a vanishing curl, ∇ × A0 (x) = 0. We can therefore choose the contour to be a straight lineconnecting x0 and x, in which case d points in the same direction of x − x0 as − x0 so that thecross product vanishes.
Hence we may write for a straight-line connection the ΩL -termZe x0(2.799)C(x, x ) = C exp id A( ) .c x0Finally, the normalization constant C is fixed by the initial condition (2.755) to have the value(2.756).Collecting all terms, the amplitude ishx t|x0 0i =1detsinh ΩL t/2ΩL t/2−1/2Ze xd A( )exp ic x0q32πih̄2 t/MΩL tiMΩL0 T0(x − x )coth× exp(x − x ) .h̄ 222All expressions simplify if we assume the magneticcase the frequency matrix becomes0ωLΩL = −ωL 000so thatandwhose determinant isfield to point in the z-direction, in which00 ,0(2.801)cos ωL t/20ΩL t 0cos ωL t/2cos=20000 ,100 ,10sin ωL t/2sinh ΩL t/2 − sin ωL t/20=ΩL t/200sinh ΩL t/2det=ΩL t/2(2.800)sinh ωL t/2ωL t/22.(2.802)(2.803)(2.804)2002 Path Integrals — Elementary Properties and Simple SolutionsLet us calculate the exponential involving the vector potential in (2.800) explicitly.
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