 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
(  0  2ieA . a + e 2 A 2 in .NET framework
(  0  2ieA . a + e 2 A 2 PDF417 Decoder In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. Make PDF417 In VS .NET Using Barcode printer for Visual Studio .NET Control to generate, create PDF 417 image in Visual Studio .NET applications. Let us exhibit solutions of the form
Recognizing PDF417 2d Barcode In .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET framework applications. Encoding Bar Code In VS .NET Using Barcode creation for .NET Control to generate, create bar code image in .NET framework applications. m 2  ie 4')t/! Scan Barcode In Visual Studio .NET Using Barcode reader for .NET Control to read, scan read, scan image in VS .NET applications. PDF417 Generation In C# Using Barcode creator for VS .NET Control to generate, create PDF417 image in .NET applications. (277) Paint PDF 417 In .NET Framework Using Barcode encoder for ASP.NET Control to generate, create PDF417 2d barcode image in ASP.NET applications. PDF417 Maker In Visual Basic .NET Using Barcode creation for Visual Studio .NET Control to generate, create PDF417 image in .NET framework applications. (278) with <p a Dirac spinor and P a fourvector which is not orthogonal to n. By adding to P some quantity of the form An, we may realize the condition Draw GS1 DataBar14 In .NET Using Barcode generation for VS .NET Control to generate, create DataBar image in Visual Studio .NET applications. UCC128 Creator In VS .NET Using Barcode encoder for VS .NET Control to generate, create EAN / UCC  13 image in VS .NET applications. p2 = m2
Generating Bar Code In .NET Using Barcode printer for .NET Control to generate, create bar code image in VS .NET applications. European Article Number 8 Encoder In .NET Using Barcode drawer for VS .NET Control to generate, create EAN 8 image in .NET framework applications. The interpretation of this fourvector is the following. In the frame where eO = 0, and thus A O = 0, E and A along Ox, B along Oy, and n along Oz, the operators PI = ia b P2 = ia 2, and i(a o + a3 ) = Po + P3 commute with the Dirac hamiltonian. The substitution of (278) into Eq. (277) leads to the condition Decode EAN13 In Visual Basic .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Generate UCC  12 In Java Using Barcode printer for Java Control to generate, create GTIN  128 image in Java applications. 2in p<p'(~) Print Barcode In None Using Barcode encoder for Microsoft Word Control to generate, create bar code image in Word applications. Decoding UPCA Supplement 5 In Visual Basic .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. which is readily integrated as
ANSI/AIM Code 39 Maker In Java Using Barcode generation for Java Control to generate, create Code 39 image in Java applications. Painting Code 128B In None Using Barcode generator for Software Control to generate, create Code 128 Code Set B image in Software applications. + (e 2A2  Making Universal Product Code Version A In ObjectiveC Using Barcode printer for iPhone Control to generate, create UPCA Supplement 5 image in iPhone applications. UPC  13 Recognizer In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. 2eA P  ieU')<p(~) = <pW= (m)1/2 exp {e 4 _ if< d~'[eA(~') P _ 2n P 0 P
e2A2(~')J}u
2n . P
(279) where u is a constant bispinor. Since ( 4)2 =  n2A2 = 0, we may write
t/!p(x) e )(m)1/2 ue iI ( 1 + 2n P 4 Po
where I stands for the action of a classical particle in a plane wave (damped at infinity), with P = mu(oo) = p(oo) [compare Eq. (184)]: 1= p x  fn.x d~[~ A(~) p  ~ A2(~)J o n p 2n P
For t/! to satisfy the original Dirac equation and not only the squared equation (273), u has to obey some auxiliary condition. After some algebra, we find that (i~  e/1 m)t/!p(x) = (m)1/2 ( 1 + 2p. n U
e)(p  m)u e'l
Therefore
(p  m)u =
THE DIRAC EQUA nON
and u = u(p) is a solution of the free Dirac equation. The reader will verify that the solution (279) has the correct normalization and that the associated current reads
r(x) = iJ!p(x)y"I/Ip(x) = ~ [p"  eA" + n" (e ~  eA2)] pO n' p 2n' p
If A(~) is a quasiperiodic function of ~ (slowly damped at infinity), the average value ofr is
showing the same phenomenon as its classical counterpart. These expressions were originally derived by Volkow in 1935. 224 FoldyWouthuysen Transformation
In the last subsections, the physical meaning of the Dirac equation has been investigated using a nonrelativistic approximation. It is worth while showing that this may be pursued in a systematic way. This is the purpose of the FoldyWouthuysen transformation. To be more explicit, we want to find a unitary transformation eiSI/I ' which decouples the small and large components and where S may be time dependent. Let us call odd the operators such as IX which couple large and small components, even those which do not (for example, I, /3, ... ). Since 1/1 ' satisfies the equation . (280) our problem is to find S so as to get rid of the odd operators in H' at a given order in 11m. In practice, we shall do it up to terms of order (kinetic energy1m or (kinetic energy x potential energy/m 2 ). This will lead to a further insight in relativistic corrections entailed by the Dirac equation. In the free case, we can construct S exactly. It is time independent and can be chosen as S= z/38=  z  8 IX'P
Since (y' p/ipJ)2 = I, we may compute e iS and H' in a closed form: e's = e(Y'p/!p!l8
cos 8 + /3  IX'P
SIll
QUANTUM FIELD THEORY
(COS 8 + f3 O(I~I sin 8}0(. P + f3m) (cos 8  f3 O(I~I sin 8) 0(. p(cos 28  1:1 sin 28) + f3(m cos 28 + Ipi sin 28) The choice of 8 such that sin 28
cos 28 =]f
eliminates the odd operator 0( . p and leaves
(p2 + m E
f3(m 2
+ p2)1/2 as we would expect. In other words, we have decomposed H into a direct sum 2 of two nonlocal hamiltonians + p2. It is clear that these square roots cannot be represented in configuration space by a finite set of differential operators. In the interacting case, we therefore expect S to be of order ml, and we expand H' to the desired order: + i[S, H]  ~ [S, [S, H]]  ~ [S, [S, [S, H]]] + 24 [S, [S, [S, [S, H]]]]  S  2 [S, S] + "6 [S, [S, S]] + ...
Here use has been made of the general identity
eA Be A
L.., n=O
,[A,[A,[ [A,B]] ]] (281) where, in the generic term of the sum, A appears n times. This identity is easily derived by computing the successive derivatives of esA BesA at s = 0, and using them in the (formal) Taylor expansion at s = 1. We start from H = f3m + (!) + $ where (!) denotes the odd operator (!) = 0( . (p  eA) and $ the even one eAo. The solution of the free case suggests that we take S =  if3(!)/2 to first order. Then according to the above expression for H', we calculate + (!)' + $'

