Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 16
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Kleinert, PATH INTEGRALS331.5 Observablesin which |ha|Ψ(t)i|2 is the probability to measure an arbitrary eigenvalue a. If thisprobability is sharply focused at a specific value of a, the state necessarily coincideswith |ai.Given the set of all eigenstates |ai of Â, we may ask under what circumstancesanother observable, say B̂, can be measured sharply in each of these states. Therequirement implies that the states |ai are also eigenstates of B̂,B̂|ai = ba |ai,(1.217)with some a-dependent eigenvalue ba .
If this is true for all |ai,B̂ Â|ai = ba a|ai = aba |ai = ÂB̂|ai,(1.218)the operators  and B̂ necessarily commute:[Â, B̂] = 0.(1.219)Conversely, it can be shown that a vanishing commutator is also sufficient fortwo observable operators to be simultaneously diagonalizable and thus to allow forsimultaneous sharp measurements.1.5.2Density Matrix and Wigner FunctionAn important object for calculating observable properties of a quantum-mechanicalsystem is the quantum mechanical density operator associated with a pure stateρ̂(t) ≡ |Ψ(t)ihΨ(t)|,(1.220)and the associated density matrix associated with a pure stateρ(x1 , x2 ; t) = hx1 |Ψ(t)ihΨ(t)|x2 i.(1.221)The expectation value of any function f (x, p̂) can be calculated from the tracehΨ(t)|f (x, p̂)|Ψ(t)i = tr [f (x, p̂)ρ̂(t)] =Zd3 xhΨ(t)|xif (x, −ih̄∇)hx|Ψ(t)i.(1.222)If we decompose the states |Ψ(t)i into stationary eigenstates |En i of the HamiltonianPoperator Ĥ [recall (1.175)], |Ψ(t)i = n |En ihEn |Ψ(t)i, then the density matrix hasthe expansionρ̂(t) ≡Xn,m|En iρnm (t)hEm | =Xn,m|En ihEn |Ψ(t)ihΨ(t)|Em ihEm |.(1.223)Wigner showed that the Fourier transform of the density matrix, the Wigner functionW (X, p; t) ≡Zd3 ∆x ip∆x/h̄eρ(X + ∆x/2, X − ∆x/2; t)(2πh̄)3(1.224)341 Fundamentalssatisfies, for a single particle of mass M in a potential V (x), the Wigner-Liouvilleequationp∂t + v · ∇X W (X, p; t) = Wt (X, p; t), v ≡,(1.225)MwhereWt (X, p; t) ≡2h̄Zd3 qW (X, p − q; t)(2πh̄)3Zd3 ∆x V (X − ∆x/2)eiq∆x/h̄ .(1.226)In the limit h̄ → 0, we may expand W (X, p − q; t) in powers of q, and V (X − ∆x/2)in powers of ∆x, which we rewrite in front of the exponential eiq∆x/h̄ as powers of−ih̄∇q .
Then we perform the integral over ∆x to obtain (2πh̄)3 δ (3) (q), and performthe integral over q to obtain the classical Liouville equation for the probabilitydensity of the particle in phase spacep,(1.227)∂t + v · ∇X W (X, p; t) = −F (X)∇p W (X, p; t), v ≡Mwhere F (X) ≡ −∇X V (X) is the force associated with the potential V (X).1.5.3Generalization to Many ParticlesAll this development can be extended to systems of N distinguishable mass pointswith Cartesian coordinates freedom x1 , .
. . , xN . If H(pν , xν , t) is the Hamiltonian,the Schrödinger equation becomesH(p̂ν , x̂ν , t)|Ψ(t)i = ih̄∂t |Ψ(t)i.(1.228)We may introduce a complete local basis |x1 , . . . , xN i with the propertieshx1 , . . . , xN |x01 , . . . , x0N i = δ (3) (x1 − x01 ) · · · δ (3) (xN − x0N ),Zand defined3 x1 · · · d3 xN |x1 , . . . , xN ihx1 , . . . , xN | = 1,hx1 , . . . , xN |p̂ν = −ih̄∂xν hx1 , .
. . , xN |,hx1 , . . . , xN |x̂ν = xν hx1 , . . . , xN |.(1.229)(1.230)The Schrödinger equation for N particles (1.107) follows from (1.228) by multiplyingit from the left with the bra vectors hx1 , . . . , xN |. In the same way, all other formulasgiven above can be generalized to N-body state vectors.1.6Time Evolution OperatorIf the Hamiltonian operator possesses no explicit time dependence, the basisindependent Schrödinger equation (1.163) can be integrated to find the wave function|Ψ(t)i at any time tb from the state at any other time ta|Ψ(tb )i = e−i(tb −ta )Ĥ/h̄ |Ψ(ta )i.(1.231)H.
Kleinert, PATH INTEGRALS351.6 Time Evolution OperatorThe operatorÛ (tb , ta ) = e−i(tb −ta )Ĥ/h̄(1.232)is called the time evolution operator . It satisfies the differential equationih̄∂tb Û(tb , ta ) = Ĥ Û(tb , ta ).(1.233)Its inverse is obtained by interchanging the order of tb and ta :Û −1 (tb , ta ) ≡ ei(tb −ta )Ĥ/h̄ = Û (ta , tb ).(1.234)As an exponential of i times a Hermitian operator, Û is a unitary operator satisfyingÛ † = Û −1 .(1.235)Indeed,Û † (tb , ta ) = ei(tb −ta )Ĥ† /h̄(1.236)= ei(tb −ta )Ĥ/h̄ = Û −1 (tb , ta ).If H(p̂, x̂, t) depends explicitly on time, the integration of the Schrödinger equation(1.163) is somewhat more involved. The solution may be found iteratively: Fortb > ta , the time interval is sliced into a large number N + 1 of small pieces ofthickness with ≡ (tb − ta )/(N + 1), slicing once at each time tn = ta + n forn = 0, .
. . , N + 1. We then use the Schrödinger equation (1.163) to relate the wavefunction in each slice approximately to the previous one:|Ψ(ta + )i ≈|Ψ(ta + 2)i ≈...ih̄Zta +i1−h̄Zta +2i1−h̄Zta +(N +1)1−|Ψ(ta + (N + 1))i ≈taEdt Ĥ(t) Ψ(ta ) ,ta +dt Ĥ(t) |Ψ(ta + )i,ta +N (1.237)!dt Ĥ(t) |Ψ(ta + N)i.The combination of these equations yields|Ψ(tb )i =i Z tb 01−dtĤ(t0N +1 )h̄ tN N +1i× 1−h̄ZtNtN−1×···× 1 −ih̄dt0NZt1taĤ(t0N )!(1.238)dt01 Ĥ(t01 ) |Ψ(ta )i.Thus the time evolution operator is given approximately by the productiÛ (tb , ta ) ≈ 1 −h̄ZtbtNdt0N +1Ĥ(t0N +1 )i×···× 1−h̄Zt1tadt01Ĥ(t01 ).(1.239)361 Fundamentalstbt2tatat1tbFigure 1.3 Illustration of time-ordering procedure in Eq. (1.244).By multiplying out the product and going to the limit N → ∞ we find the seriesÛ(tb , ta ) = 1 −ih̄Z−i+h̄tbtadt1 Ĥ(t1 ) +3 Ztbtadt3Zt3tadt2−ih̄Z2 Zt2tatbtadt2Zt2tadt1 Ĥ(t2 )Ĥ(t1 )(1.240)dt1 Ĥ(t3 )Ĥ(t2 )Ĥ(t1 ) + .
. . ,known as the Neumann-Liouville expansion or Dyson series. An interesting modification of this is the so-called Magnus expansion to be derived in Eq. (2A.25).Note that each integral has the time arguments in the Hamilton operators orderedcausally: Operators with later times stand to left of those with earlier times. It isuseful to introduce a time-ordering operator which, when applied to an arbitraryproduct of operators,Ôn (tn ) · · · Ô1 (t1 ),(1.241)reorders the times successively.
More explicitly we defineT̂ (Ôn (tn ) · · · Ô1 (t1 )) ≡ Ôin (tin ) · · · Ôi1 (ti1 ),(1.242)where tin , . . . , ti1 are the times tn , . . . , t1 relabeled in the causal order, so thattin > tin−1 > . . . > ti1 .(1.243)Any c-number factors in (1.242) can be pulled out in front of the T̂ operator. Withthis formal operator, the Neumann-Liouville expansion can be rewritten in a morecompact way. Take, for instance, the third term in (1.240)Ztbtadt2Zt2tadt1 Ĥ(t2 )Ĥ(t1 ).(1.244)The integration covers the triangle above the diagonal in the square t1 , t2 ∈ [ta , tb ]in the (t1 , t2 ) plane (see Fig.
1.2). By comparing this with the missing integral overthe lower triangleZZtbtadt2tbt2dt1 Ĥ(t2 )Ĥ(t1 )(1.245)H. Kleinert, PATH INTEGRALS371.6 Time Evolution Operatorwe see that the two expressions coincide except for the order of the operators. Thiscan be corrected with the use of a time-ordering operator T̂ . The expressiontbZT̂tatbZdt2t2dt1 Ĥ(t2 )Ĥ(t1 )(1.246)is equal to (1.244) since it may be rewritten astbZZdt2tatbdt1 Ĥ(t1 )Ĥ(t2 )t2(1.247)or, after interchanging the order of integration, astbZtadt1Zt1tadt2 Ĥ(t1 )Ĥ(t2 ).(1.248)Apart from the dummy integration variables t2 ↔ t1 , this double integral coincideswith (1.244).
Since the time arguments are properly ordered, (1.244) can triviallybe multiplied with the time-ordering operator. The conclusion of this discussion isthat (1.244) can alternatively be written asZ tb1 Z tbT̂dt2dt1 Ĥ(t2 )Ĥ(t1 ).2 tata(1.249)On the right-hand side, the integrations now run over the full square in the t1 , t2 plane so that the two integrals can be factorized into1T̂2tbZtadt Ĥ(t)2.(1.250)Similarly, we may rewrite the nth-order term of (1.240) as1T̂n!ZtbtadtnZtbtadtn−1 · · ·Ztbtadt1 Ĥ(tn )Ĥ(tn−1 ) · · · Ĥ(t1 )"Z#ntb1dt Ĥ(t) .= T̂n!ta(1.251)The time evolution operator Û (tb , ta ) has therefore the series expansioniÛ(tb , ta ) = 1 − T̂h̄Ztbtadt Ĥ(t) +1 −i+...+n! h̄nT̂1 −i2! h̄Ztbta2T̂tbZdt Ĥ(t)tandt Ĥ(t)2(1.252)+ ...
.The right-hand side of T̂ contains simply the power series expansion of the exponential so that we can writeiÛ (tb , ta ) = T̂ exp −h̄Ztbtadt Ĥ(t) .(1.253)381 FundamentalsIf Ĥ does not depend on the time, the time-ordering operation is superfluous, theintegral can be done trivially, and we recover the previous result (1.232).Note that a small variation δ Ĥ(t) of Ĥ(t) changes Û(tb , ta ) by(i tbii tb 0dt T̂ exp −dt Ĥ(t) δ Ĥ(t0 ) T̂ exp −δ Û (tb , ta ) = −0h̄ tah̄ th̄Z tbi=−dt0 Û(tb , t0 ) δ Ĥ(t0 ) Û (t0 , ta ).h̄ taZZZt0tadt Ĥ(t))(1.254)A simple application for this relation is given in Appendix 1A.1.7Properties of Time Evolution OperatorBy construction, Û (tb , ta ) has some important properties:a) Fundamental composition lawIf two time translations are performed successively, the corresponding operators Ûare related byÛ (tb , ta ) = Û(tb , t0 )Û (t0 , ta ),t0 ∈ (ta , tb ).(1.255)This composition law makes the operators Û a representation of the abelian groupof time translations.
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