This may be further generalized to a time-ordered product of operators in .NET framework

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This may be further generalized to a time-ordered product of operators
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If they are all diagonal in the q representation we have
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<qf, tIl (!)1(tl)(!)2(t 2) Iqi' t) =
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f0(q) eiI(J,i) (!)1[q(t 1)] (!)2[q(t2)]-
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(9-15)
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QUANTUM FIELD THEORY
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Using these results we may consider the effect of an infinitesimal change in the dynamics between t; and tf (for instance, a slight variation of the potential or of the boundary conditions). The corresponding variation of the transition amplitude rna y be expressed as
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This has the form of (9-14) and (9-15) with an operator which we denote (5J, depending, if the case may be, on the intermediate times, so that we can write in general (9-16) This expression may be used as an infinitesimal substitute to functional integrals. For instance, if the variation is a result of an increase in the final time from tf to tf + btf for which (5J = -Hbt, formula (9-16) reproduces Schrodinger's equation in the form
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8<qf' t f q;, t;)
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i<qf' tfl H(tf) Iq;, t;) btf
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Let us check Eq. (9-11) for a simple harmonic oscillator
mo:l V(Q)=T Q2
(9-17)
of frequency wand energy levels En = (n + i)w. We apply (9-11) as it stands. On any short interval (t, t + 8t) we approximate an arbitrary path by a linear one in such a way that the corresponding increment in the action is
81(2 1) =
~ f(q2 - ql _ w28( q~ + q2ql + qr]
Set t = tJ - ti. We find
(9-18)
We have to integrate the exponential of a quadratic form ei (m/2)qMq where q stands for the set == q" q 1, . . . , qn-l, qn == qJ and M is such that
n w2 Moo=Mnn=---t t 3n
1:s:k:S:n-1
AIl other matrix elements are zero. The integration variables are ql,"" qn-l. If N is the matrix obtained by deleting from M the rows and columns with indices 0 and n, we have
FUNCTIONAL METHODS
qMq = Moo(q~
+ q~) + 2M o1 (qOql + qn-lqn) + I
n-l qjNj1ql
j,l= 1
Using the classical method to compute gaussian integrals, it follows that
<f[i)= lim ( n~oo
dn- 1q e i(mI2)qNq
einl2
)(n- 1)12
(det N)-1 12
Linear terms in the exponential are taken into account by a translation, with the result that
nm e-inI2)nI2(2n e inI2 )(n-l)12
x exp Defining a to be equal to
2 (detN)-1 1
[Moo(q~ + q;) - M~I(Nljq~ + N;}I,n-lq; + 2Nl'~-lqoqn)J}
the determinant of N is given by det N
2 n-
(I't l_
2 ro t)n- 1 det n 3n
-a -a
The remaining determinant will be denoted In-l(a). For fixed a it satisfies the recursion relation
This is solved in the form
Ip(a)) ( Ip-M) =
After diagonalization of the 2 x 2 matrix we find
1 _ iJ4a2 A+(a) = _ -+ " - c c_ __1
ie i!JJt/n
For large n we thus have
-1(a)~---
n sin rot 2 n- 1 rot
Consequently,
When sin rot < 0 a careful calculation allows one to choose the correct phase, Let us leave this point aside by assuming rot < n. To conclude, we have to evaluate
N -1 11 ,
t I n- 2(a) t (rot ) -3 ---~- l--cotrot +O(n ) 2n(l-ro 2t 2/3n 2)In_l(a) n n
QUANTUM FIELD THEORY
The final expression
<I Ii> = ( mwe- ' )1 2exp {imW [ (q} + qr) cot wt . ~~M 2
i /2
..,....!-':..
2q q
(9-19)
coincides, of course, with the result obtained through more traditional means. It is interesting to note that in this special case the coefficient of the exponential is equal to the value of the action for the actual classical trajectory over the finite time interval. This is a particular feature of quadratic hamiltonians. As a second exercise the reader may apply the same method to a time-dependent hamiltonian describing the one-dimensional motion of a particle submitted to an external force:
H= - - QF(t)
(9-20)
The corresponding amplitude is
<II i>F =
m (qJ - qy -2
m e- i' /2 J1 /2 eiI(f,i) [ 2n(tJ - til
(9-21)
where again I(f, i) stands for the action evaluated along the classical trajectory
I(f, i)
+ itf dt F(t) ( t --t + qi qJ - '
~ ~-~
tJ - t) --
rtf ftf dt' dt" F(t') [(t' - ti)(t" - til _ Inf(t' - t" t" - ti)JF(t") 2m Jt, t, t J - ti
(9-22)
The expression G(t', t") = t't"fT - Inf(t', t") is the symmetric Green function of the classical problem ij = F(t)fm already encountered in Chap. 1, satisfying the boundary conditions G(O, t") = G(T, t") = 0, The analog ofEq. (9-16) yields (9-23) This result enables us to give an algebraic definition of the transition amplitude for an arbitrary hamiltonian of the form (9-2). In this case we may write
f ~(q) exp
{i rtf dt [m Jt, 2
V(q)J}
exp {- i rtf dt
v[-. J}</li>FI F~ 8 z8F(t)
(9-24)
This formal expression becomes operational if we expand the exponential operator into a power series and generate the perturbation theory. All the previous considerations can be extended to finitely many degrees of freedom without difficulty. Let us now mention an apparently more general derivation of the evolution kernel (9-12). The idea is to use both the Iq >and Ip> bases of the Hilbert space. Let us indeed assume that the mixed matrix element of the hamiltonian may be given the form
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