_~ _ ~ ~ + L2 - Z2 a 2 _ 2ZaE _ (E2 _ m2)Jep = 0 in .NET framework

Encoder PDF-417 2d barcode in .NET framework _~ _ ~ ~ + L2 - Z2 a 2 _ 2ZaE _ (E2 _ m2)Jep = 0

_~ _ ~ ~ + L2 - Z2 a 2 _ 2ZaE _ (E2 _ m2)Jep = 0
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or2 r Or r2 r
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(2-85)
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This equation is formally identical to the Schrodinger equation, after the substitutions
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L2 -+ L2 - Z2 a 2 a-+aE
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The orbital quantum number is shifted by bt, I -+ A = I - bl , where b = 1+ i - [(l + i)2 - Z 2a 2]l/2
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and the principal quantum number n is similarly displaced by the same amount, since n' = n - (l + 1) must be an integer. Therefore, the energy levels are given by E;I - m 2 mZ 2a 2 E;I 1 - --2- m 2 (n - blf 2m or
(2-86)
QUANTUM FIELD THEORY
The second term is the nonrelativistic binding energy and the third one breaks the 0(4) degeneracy. We are not going to discuss the pathologies of this case, namely, the singular behavior of the wave function at the origin due to the attractive term - (Z2 a2/r2) or the catastrophe which occurs when Z > 137/2 (b l and thus Enl become complex I). Equation (2-86) is in poor agreement with the experimental situation, meaning that the effects of spin cannot be neglected.
2-3-2 Dirac Theory
We turn now to the predictions of the Dirac equation. Before constructing the wave functions, we first proceed to a simple derivation of the spectrum, as in the Klein-Gordon case. For this purpose, we square the equation, as in Eq. (2-73), and insert the nonvanishing component of the potential Ao = -(Ze/4nr), while F Oi = - FiO = - oiAo = Ei. It is convenient to work in the chiral representation
(2-20) for the y matrices where a Oi is diagonal: a Oi
term in Eq. (2-73) reads
- aJlV F JlV = - iea " E = + 2
= i(
~i). Then the spin
a" f -+ iZa - r2
\' where f is the. unit vector r/r. The analog of Eq. (2-8/) is an equation for twocomponent splllors :
02 2 0) [ - (-or2 + -r -or + (L 2 -
- . ,1 2ZaE 2 Z 2a 2 + IZM" r)- - - - - (E 2 - m) r2 r
'I' =
The total angular momentum J = L + S = L + a/2 commutes with the hamiltonian and with L2. In the subspace where P = j(j + 1), J z = m (j = i, ~, ... ; - j :::;: m :::;: j), and L2 = l(l + 1), the integer I takes two values: 1= j + i == 1+ and I = j - i == 1-. Since a" r has no diagonal matrix element, <l la" rll = 0, is hermitian and has a square equal to one, the operator [L2 - Z2 a2 =+= iZM "r] assumes in this subspace the (ollowing form:
' L 2 - Z2 a 2 - Zaa"r= ((j +1 Let 2(2
+ i)(j + i) -. Z2 a2
=+=iZa ';(
+ 1) be its eigenvalues 2 = [(j + i - Z 2a 2]1/2
which may be written
with
1 b- = ] . + - J
J(.] + -
2 = (j
i) -
bj Z2 2 - - + O(Z 4a 4) 2j + 1 a
- Z 2a 2 ~
Again, n is also shifted by bj so as to keep n' = (n - bj) - 2 - 1 integer. The
THE DIRAC EQUATION
condition n' :2 0 restricts j to th~ range j :::;; n - i for A = j + i - (}j and j :::;; n j - i - (}j. Therefore, there is a twofold degeneracy except for the state j = n - i. The final result is
i for A =
)1 +
m [Z2!X2/(n _ (}j)2]
mZ 2!X2
= m-
mZ 4!X4 n\2j + 1)
3 mZ 4!X4
+ 8"
- r + O(!X
(2-87) with n = 1,2, ... and j = i, i, ... , n - i. A catastrophe occurs now for Z = 137; (}1/2 becomes imaginary beyond this value. The degenerate states may be distinguished by their orbital angular momentum l, which takes the values j i (except for j = n - i where 1= n - 1); this is in turn related to the transformation property of the state under parity. The energies of the low-lying states have been represented on Fig. 2-2, where we have used the customary, nonrelativistic spectroscopic notation nlj. The new phenomenon is the occurrence of the fine structure, i.e., the difference in energy between levels of different j, for the same value of n. Typically, for Z= 1,
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