QuantumPhysics.dvi

14 Path Integral Formulation of Quantum Mechanics A quantum system associated with classical mechanics admits a formulation in t ...

Actually, the time-evolution equations in both pictures may be solved with the use of thetime evolution operatorU(t). This opera ...

Translation operators in both bases satisfy e+iaP/ ̄hQe−iaP/ ̄h=Q+a e−iaP/ ̄h|q〉=|q+a〉 e−ibQ/ ̄hPe+ibQ/ ̄h=P+b e+ibQ/ ̄h|p〉=|p+b ...

Putting all together, and usingδ(pb−pa) to carry out the integration overpb, we have 〈qb|U(tb−ta)|qa〉 = ∫ dpa 2 π ̄h e−i(tb−ta)p ...

Figure 16: The evolution operator given as a summation over paths,qa=qandqb=q′ The problem will be, however, that the Hamiltonia ...

= ∫ dpn 2 π ̄h 〈qn|pn〉〈pn|qn− 1 〉exp { −i ε ̄h H(qn,pn) } = ∫ dp n 2 π ̄h exp { i(qn−qn− 1 )pn/ ̄h−i ε ̄h H(qn,pn) } (14.27) The ...

where the combine measure is defined by ∫ Dq ∫ Dp= lim N→∞ N∏− 1 n=1 ∫ R ∫ R dq(tn)dp(tn) 2 π ̄h (14.34) and the paths satisfy t ...

whereLis the standard Lagrangian, formulated in terms ofqand ̇q, and given by L(q,q ̇) = 1 2 mq ̇^2 −V(q) (14.41) The extra dyna ...

Of course, we knw how to do this integral exactly; it is just a Gaussian. But if we were just interested in thea≫1 behavior of t ...

of paths around the classical one. As before, we assume that we have a classical solutionq 0 (t) connecting initial and final po ...

Here,XandZare respectively a real and a complex column matrix of heightN. These variables and their associated measures may be p ...

for the operatorM(t), defined by M(t) =−m d^2 dt^2 −V′′(q 0 (t)) (14.64) and subject to the boundary conditionsy(ta) =y(tb) = 0. ...

15 Applications and Examples of Path Integrals In this section, we shall illustrate the use of path integrals and give examples ...

The product of the eigenvalues may first be defined up to a leveln≤N, ∏N n=1 λn= ∏N n=1 [ mπ^2 n^2 T^2 ( 1 − ω^2 T^2 π^2 n^2 )] ...

Expanding in powers of the exponential, we find, Tr ( U(T) ) = ∑∞ n=0 e−iTEn/ ̄h En= ̄hω ( n+ 1 2 ) (15.16) and recover the ener ...

can never penetrate inside. We assume vanishing electric field throughout, so that Φ = 0. On the outside of the cylinder, the ma ...

15.3 Imaginary time path Integrals If we formally replacet→−iτand considerτreal, then we obtain theimaginary timeorEuclidean for ...

For the free case (V = 0), we haveLE=^12 mq ̇^2 , so that 〈qb|UE(τb−τa)|qa〉= √ m 2 π ̄h(τb−τa) exp { − m(qb−qa)^2 2 ̄h(τb−τa) } ...

Theinternal energyEand theentropySare related to the free energy by F=E−TS (15.37) Both may be expressed in terms of the partiti ...

The general matrix elements of the Boltzmann operator may now be expressed using the Euclidean path integral, 〈q′|e−βH|q〉= ∫ Dqe ...