tjJ(t, x) = i in .NET framework

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tjJ(t, x) = i
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d 3y [SF(t - tl, x - y)y tjJ(+\tt, y)
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- SF(t - tz, x - y)y tjJ(-\tz, y)]
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(2-116)
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2-5-2 Propagation in an Arbitrary External Electromagnetic Field
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In practice, we deal with the propagation in the presence of obstacles: diffusion processes, external fields, interactions with other particles. Let us treat the propagation in an external electromagnetic field (2-117) With very few exceptions, we are unable to find a compact expression for SA. Fortunately, it frequently occurs that the term eJ is small enough to be treated as a perturbation, and SA may be expressed as an (asymptotic) expansion in eJ. To derive it, we multiply both sides of (2-117) by SF(X3, xz) and integrate over Xz:
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From (2-111), it follows that
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SF(X3,XZ)( -i'jz - m) = b4 (X3 - xz)
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Therefore, the integral equation which determines SA reads
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SA( X3, Xl) = SF(X3, Xl)
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d4xz J(XZ)SA(XZ,
(2-118)
QUANTUM FIELD THEORY
SF(xr-Xi)
+ ..........~)(--. + -"""*)(-"""*)(-...... + ...
Figure 2-4 Diagrammatic representation of the perturbative expansion (2-119). The solid line between Xk and Xl stands for the propagator SF(Xk - Xl), the cross for e41(x).
Equation (2-118) is adapted to a perturbative expansion obtained by iteration:
SAXj, Xi) = SF(Xj, Xi)
d4xl SF(Xj, Xl)ej(Xl)SF(Xr, Xi)
d4xl d4X2 SF(Xj, Xl)ej(Xl)SF(Xl,
X2)ej(X2)S~X2' Xi) + ...
(2-119)
This expansion is depicted diagrammatically on Fig. 2-4.
Introducing the Fourier transforms of SA and A(x):
(we use the same notation in configuration and momentum space for the sake of simplicity) a perturbative expansion of SA(Pf, Pi) may also be written. Since SF(X" Xi) = SF(xf - Xi) is translation invariant,
SF(Pf, Pi)
= (2n)4,54(pf -
Pi)SF(Pi)
where SF(P) has been given in (2-114) and
SA(Pf,Pi) = SF(Pf)(2n)4,54(pf - Pi)
d 4pi SF(Pf)e41(Pi)SF(Pi)(2n)4,54(Pf - Pi - Pi)
d 4pi d 4p2 SF(Pf)e41(Pi)SF(P2
+ Pi)e41(P2)SF(Pi)(2n)4,54(pf -
Pi - P2 - Pi)
+ ...
2-5-3 Application to the Coulomb Scattering
Coulomb scattering will serve as a testing ground of the propagator method. The process under study is the scattering of a charged electron of mass m by a center with charge - Ze and infinite mass. The latter creates a potential Ao = - Ze/4nr, A = 0, where r stands for the vector joining the center to the charge. In classical nonrelativistic mechanics, the trajectories are hyperbolas. The scattering angle () is related to the impact parameter b (see Fig. 2-5 for notations) by
THE DIRAC EQUATION
1. Geometrical relations
tan2
cos-
2. Energy conservation
e=-=-=----2m 2m 2m c- a
(PA is the momentum at point A.) 3. Angular momentum conservation
1= Pib = pfb = PA(C - a)
After elimination of Pi, PA, c, and a, we get
b= 2e tan el2
(2-120)
We consider a uniform flux of electrons, of density p and incident velocity v = p;/m = Pf 1m. The number of scattered particles in the solid angle dO. = 2n d cos e per unit time is equal to the number of incident particles on the ring of area 2nb db, namely,
dN Tt=pv2nbdb=pv
(Za)2 sindO.e/2 2e
Therefore, the differential cross section, defined as the ratio of dN Idt dO. to the incident flux, is
d(J dO.
(2-121)
Figure 2-5 Coulomb scattering off a charge located at F.
QUANTUM FIELD THEORY
Here q = PI - Pi is the momentum transfer and Jq'J = 2Pi sin e12. This is the classical Rutherford formula. We turn to the relativistic quantum case. We shall use the expression derived in (2-116), after substituting SA for SF and taking the following boundary conditions: for t1 = -00, I/I(+)(t1o x) is an incident plane wave of positive energy electrons, whereas for t2 = +00, I/I(-)(t2, x) vanishes. Since we do not know SA, we content ourselves with the first two orders of the perturbative expansion (2-119). Let I/Iinc(t, x) be the solution of the free Dirac equation which reduces to the incident wave when t -+ t1 = -00. According to (2-119), the perturbative wave function reads where
I/Idiff(X) =
(2-122)
tlY~ro i
3 d y
4 d z SF(X - z)ej(z)SF(Z - y)yol/linc(t1, y)
(2-123)
y=(t1,Y)
Since for ZO > t 1,
we have
I/Idiff(X)
= d4 z SF(X - z)ej(z)I/Iinc(Z)
As XO tends to + 00, 1/1diff behaves as a pure positive energy solution of the free Dirac equation. Indeed, using the expression (2-112), we see that only the first term contributes in this limit, and we get
,I, , ( )
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