 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
THE DIRAC EQUA nON in .NET framework
THE DIRAC EQUA nON Recognizing PDF 417 In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Encoding PDF417 In .NET Framework Using Barcode creator for VS .NET Control to generate, create PDF417 2d barcode image in .NET framework applications. "'n~ 1,]~ 1/2.m~ 1/2 . _ (2mZa)3 /2 ( 1 + l' )1/2 , _ 1 mZ., (4n)1/2 2r{l + 21') (2mZar) e
Recognize PDF 417 In Visual Studio .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET framework applications. Create Bar Code In Visual Studio .NET Using Barcode generation for Visual Studio .NET Control to generate, create barcode image in .NET applications. (294) Barcode Decoder In Visual Studio .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Generating PDF417 2d Barcode In C#.NET Using Barcode generation for VS .NET Control to generate, create PDF417 2d barcode image in .NET applications. "'n~l.j~ 1/2.m~ 1/2 = PDF417 Drawer In Visual Studio .NET Using Barcode creator for ASP.NET Control to generate, create PDF417 2d barcode image in ASP.NET applications. PDF 417 Printer In Visual Basic .NET Using Barcode maker for Visual Studio .NET Control to generate, create PDF 417 image in .NET applications. (2mZa)3 /2 ( 1 + l' )1/2 (4n)1/2 2r{l + 21') (2mZar),1 mZ i(1  1') Za sin
Linear 1D Barcode Encoder In Visual Studio .NET Using Barcode printer for .NET Control to generate, create Linear Barcode image in Visual Studio .NET applications. 2D Barcode Maker In VS .NET Using Barcode generation for Visual Studio .NET Control to generate, create Matrix Barcode image in .NET framework applications. i{l1') \'''' cos Za
Code 128 Code Set A Drawer In .NET Framework Using Barcode creator for .NET Control to generate, create Code 128 image in .NET framework applications. UPC  E1 Generation In VS .NET Using Barcode creation for .NET framework Control to generate, create UPCE image in Visual Studio .NET applications. (J eiq>
GS1  13 Generator In VS .NET Using Barcode creation for Reporting Service Control to generate, create EAN13 Supplement 5 image in Reporting Service applications. Code39 Recognizer In VB.NET Using Barcode reader for .NET Control to read, scan read, scan image in VS .NET applications. Note that l' = )1  Z2 a2 ~ 1  (Z2 a2/2). In the nonrelativistic limit l' > 1, we recover Schrodinger wave functions multiplied by Pauli spinors. On the other hand, these wave functions are singular at the origin, but this effect is noticeable only for UPCA Supplement 2 Creator In Java Using Barcode drawer for Android Control to generate, create Universal Product Code version A image in Android applications. Draw Barcode In None Using Barcode drawer for Online Control to generate, create barcode image in Online applications. 2mZar ;S e 2/Z'.' "'" 1016300/Z' Code 128 Code Set A Creator In None Using Barcode printer for Online Control to generate, create Code 128 image in Online applications. Make Code128 In Java Using Barcode maker for BIRT Control to generate, create Code 128 Code Set B image in BIRT reports applications. i.e., in a rather tiny region! Barcode Creator In Java Using Barcode generator for Eclipse BIRT Control to generate, create barcode image in Eclipse BIRT applications. Generate Code 128B In None Using Barcode creation for Word Control to generate, create Code 128C image in Word applications. To compare these results with the experimental levels, several other effects have to be taken into account. Hyperfine structure As a first approximation, we have neglected the magnetic field induced by the magnetic moment of the nucleus (we consider henceforth the hydrogen atom Z = 1). The coupling of the proton spin with the total electron angular momentum splits the levels into doublets. To get an estimate of this effect, let us use a nonrelativistic approximation for s states. The new interaction is e H hf =   ( 1 B 2m e where B = curl A 1 1 A=   p . x V4n P r
(1e denotes the electron spin, and p.p the proton magnetic moment. We recall that for a current distribution j(x), the magnetic moment is p. = r x j(x) d 3 x and L1A =  j; therefore for a moment localized at the origin j =  p. x V15 3 (x). Hence e l l H hf =    ( 1 e [V X (p. X V)]4n 2m p r
e 8nm
(1 e
1 [p. L1  (p. . V) V] P P
QUANTUM FIELD THEORY
In s states, we only need the angular average of Hhf, 'Ili'llj can be replaced by jl1bij, and
=  3m U e' /lp b (r) (295) Introducing the proton gyro magnetic ratio gp, /lp = (gpe/2mp)(u p we obtain /2), _ e2 <Hhf > 12 = gpue'up!l/J(OW memp For the ground state, n = 1 [compare (284)] : !l/J(OW
(me a)3 and U e uptakes the values 1 in the triplet state and  3 Finally, we get
I1Ehf
n= l,j= 1/2 the singlet one.
4 (tripletsinglet) = 3 mea4 me gp = 5.89 mp
10 6 eV = 1.42
109 Hz
If we compare this splitting to the fine splitting, we see that the dominant effect is a reduction by a factor me/mp ~ IIp/Ile. The previous estimate may, of course, be extended to higher waves (/ 2: 1). Radiative corrections There are several kinds of such corrections. First, the excited states are unstablethey acquire a widthand the atom may undergo a spontaneous transition to a lower state. In a nonrelativistic dipole approximation, the probability per unit time of radiative transition between two states .l. and Il per unit time is WjL_,l
3 ~:+; 1<IlIIDIIA>1
4 (E
E )3 where <1l11 D 11.l.> is the reduced matrix element of the dipole operator D = er, as defined by the WignerEckart theorem: In particular, for a transition 2P > IS, we find
WIS_2P
= 6.2 x 10 8 = 4.1 x 107 eV
Second, the charged particles interact with the fluctuations of the quantized electromagnetic field. The latter vanishes only on an average. As a result the levels are slightly shifted. Although a complete and systematic treatment of these effects requires the methods of quantum field theory to be described later, we may give, following Welton, a qualitative description of the main effect: the Lamb shift. It is based upon the same kind of argument as the discussion of THE DIRAC EQUATION
the Darwin term in the last section, but here the fluctuation of the position is due to the electromagnetic field, rather than to a relativistic zitterbewegung. Hence the new contribution to the hamiltonian has the form LlH Lamb =
i c5r)2 > Ll V
where V =  (ZIX/r) and therefore, according to Poisson's law, Ll V = 4nZIXc5 3(r). In this simple picture, treating this perturbation to first order, only s waves are affected and the nth level is shifted by the amount LlELamb(n) 2nZIX 3 <(c5r) 2> 1l/In(O) 12 = (2mZIX)3 !!... c5r >
12 n3 (For higher waves, the shift is reduced by the vanishing of the wave function at the origin, and is considerably smaller.) The estimate of c5r)2 > relies on a classical description of the motion of the electron in the fluctuating field (the nucleus, which is heavier, does not move). The electron oscillates according to mc5'i
The Fourier component Ero with frequency w of the electric field contributes
mc5 r ro
e Ero w
and assuming that there is no correlation between various modes
c5r > =
~ foo 2
dw w4
<E~>
Anticipating the quantum treatment of the electromagnetic field, we suppose that the field is an incoherent superposition of plane waves and remember that the vacuum energy of such a field is the sum of zero point energies

