E(2P3/2) - E(2P 1 / 2) ~ m!X n 4 in .NET framework

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E(2P3/2) - E(2P 1 / 2) ~ m!X n 4
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This fine splitting may be seen as a consequence of the spin-orbit coupling in (2-82):
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This term vanishes for an s wave, whereas for a p wave, L (1 = (J 2 - L2 - (12/4) takes the values 1 and -2 for j = 3/2 and j = 1/2 respectively. On the other hand, the expectation value of l/r 3 is, on dimensional grounds, of the form <nti l/r 3Jnl> = knl(mZa)3, where knl is a pure number. Any textbook on quantum mechanics tells us that knl = 8/(21 + 1)n 3[(21 + 1)2 - 1] (=.,p. for n = 2 and 1= 1):
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in agreement with the previous estimate. (This holds also.for higher values of nand /.)
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Let us now construct the spinors which are the energy eigenstates of this problem. Returning to the Dirac representation (2-10), we write the bispinors
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(:) and look for two-component spinors which are eigenstates of J2, J z,
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and L2 with eigenvalues j(j + 1), m, and l(l + 1) respectively. Let cp},;,) be the eigenstate for j = 1 i. As eigenstates of L2, Cp},;,) must have the form
CPj,m ( ) _
(all yll YI',) b
Il' I
(no summation over Jl and Jl')
QUANTUM FIELD THEORY
where we use the standard notation 11" for spherical harmonics. They must also be eigenstates of J z = Lz + Sz with eigenvalue m and eigenstates of L (1 = J2 - V - i with eigenvalue I for cp~-:;J and -(I + 1) for cp~1. The first condition gives f.1. = m - i, and f.1.' = m + i, the second one, together with the normalization
3S1/2
--c:::::=
r-;;z
2S 1/2
--c:::::=
Lamb shift
2Pl/2
--c:::::=
1057 MHz
Fine structure 10.9 GHz
Triplet
ISI/ 2
--C=
Singlet
Hyperfine structure 1420 MHz
Figure 2-2 Low-lying energy levels of hydrogen.
THE DIRAC EQUATION
condition
JaJ2 + JbJ2, determines a and b. We finally get the spinor harmonics
(21 +
1)-1/2
(+) =
({J},m ({J(-)
((I + m + W/2 (/- m +
i)1/2
j=l+i
(2-88)
+ It 112 (
(1- m (I + m
+ W/2
+ i)1/2
yrYr+
1/2 1/2
j = I-i, I> 0
(2-89)
The phase has been chosen in such a way that ({J),t'> = (1 . f ({JJ:;',>
(2-90)
Since (1' f is a pseudo scalar operator, ({J);:;) and ({J),;;) have opposite parities (also, their angular momenta 1 differ by one unit). It is convenient to introduce a common notation. Let ({J;,m denote ({J);:;) if j = 1 + i and ({J);;;) if j = 1 - i. We verify by inspection that ({J;,m has parity (_1)1. Since the Dirac equation in the Coulomb potential
EI/I =
1 i IX V + 13m -
-r 1/1 == H 1/1
(2-91)
is invariant under a space reflection, odd and even eigenstates may be constructed, namely,
It is clear that the spinors
'/'jm
./,1
- - ({Jjm iGlj(r) 1 r
Flj(r) ') 1 [ - - - ((1' r ({Jjm
(2-92)
have parity (-1/ In (2-92), the factors i and l/r have been introduced for later convenience. Noting that H in (2-91) reads
m-Za r
(1' P
(1'P
-m _
we do the intermediate calculations
(1' pf(r)({J)m =
f((1' f
p)f(r)({J)'"
f (r' p + i(1' L)f(r)({J)". for j = 1 i
= - i
(1/ {r
d~~) + [1 =+= (j + 1/2)] f(r)} ({J}m
QUANTUM FIELD THEORY
and, similarly,
(u' p)(u' r)f(r)cp},.,
~ [r :r + 1
Our lengthy separation of variables results in the pair of radial equations
To solve this pair of equations, we introduce the notation 2 = Jm 2 - E2 (since E2 < m 2), the new variable p = 22r, and, for I and j given, the new functions F 1 (p) and F 2(P):
E)1/2 G(r) = ( 1 + -;;:; e- p/ 2(F 1 + F2)(P) E)1/2 F(r) = ( 1 - m e- p / 2(F 1 - F2)(P)
]f(r)CP}m
E- m+ -
dFlj (r) - (. + - FIAr) Gdr) = - + ] -J dr 2 r
GIAr) + (E + m + -Za) Fdr) = dGlj (r) _ (.] + -21) -rr dr
(2-93)
By eliminating F2 in favor of F 1, we get a second-order differential equation, the solution of which is
y - ZaE/2 ( ZaE ) F 1 = -2+Zam/2 PYF y + I - --,2y+l;p 2 ZaE F2 = p' F ( y - -2-' 2y
+ 1; P )
with
== [(j +jY -
Z2 a2] 1/2 =j
+ -5;- OJ
Here F(a, b; p) denotes the degenerate hypergeometric function solution of
d2 d ] P-2 +(b-p)--a F(a,b;p)=O [ dp dp
For large p this function behaves as [r(b)/r(a)]pa-he P Demanding that F/j and G/j be normalizable, that is, J~ dr(Flj + G"t) = 1, implies that [r(y - ZaE/2)] -1 must vanish. This is the desired quantization condition
-2- -y= n,
(nonnegative integer)
== n - (j +-5;)
which leads to (2-87):
Collecting all the factors, it is then possible to write the expression of the normalized solutions Flj and G/j, and therefore of the t/lJ",. We only quote here the form of the ground-state wave functions
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