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THE DIRAC EQUA nON
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-i{l-1') \--'----''-'- cos Za
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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
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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 -j-l1bij, and
= - 3m U e' /lp b (r)
(2-95)
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 (2-84)] :
!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 (triplet-singlet) = -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 unstable-they acquire a width-and 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 Wigner-Eckart theorem:
In particular, for a transition 2P -> IS, we find
WIS_2P
= 6.2
x 10 8
= 4.1
x 10-7 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
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