Neutron Scattering

The route from expression (5 .27) to the Van Hove correlation function starts with an
integral representation of the delta function :
S (liw + EA - Ea,) =1 dt exp -i + EA - EA, t I (5 .28)
27rh J Cw )
which results from the fact that the delta function is the Fourier transform of a constant
one .
With this expression the matrix element in equation (5 .27) can be written as a
Fourier transform in time:
E(A' Ibi exp(iQ - ri ) A)
2
S(hw+EA -EA ,)
dt exp (-iwt) exp -i t )
exp -i A, t )
27rh
Ebi(A ' Iexp(iQ - ri)IA)Ebi (AI exp( -iQ - rj )IA' )
i
j
1 °° dt exp (-iwt) E biM (AI exp (-iQ - r j ) I A')
27rh oo i,j
(A'I exp(iEA,t/h) exp(iQ - r i ) exp(-iEAt/h)la) (5 .29)
If H is the Hamiltonian of the scattering system, the fact that IA) are energy eigenstates
is expressed by
Iterating this equation n times yields :
HIA) = Eal AÎ . (5 .30)
HnIA) = EA
n
lA) . (5 .31)
By expanding the exponential into a power series one finally obtains from this relation
exp(iHt/tt) IA) = exp(iEat/1i) IA) . (5 .32)
With this result and the analogous one for A' it
in (5 .29) by the Hamiltonian H :
is possible to replace the eigenvalues EA
. . . (A' I exp(iHt/li) exp(iQ - r i ) exp(-iHt/li) IA) . (5 .33)
In the picture of time dependent Heisenberg operators the application of the operator
exp(iHt/h) and its conjugate just mean a propagation by time t :
exp (iQ - ri (t)) = exp(iHt/1i) exp(iQ - ri (0)) exp(-iHt/1i) (5 .34)
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