0198506961.pdf

2.3 Fine structure 41

Lamb shift

Spin−orbit

Fig. 2.7The fine structure of the

n =2andn = 3 shells of hydro-

gen and the allowed transitions between

the levels. According to the Dirac

equation, the 2 S 1 / 2 and 2 P 1 / 2 levels

should be degenerate, but they are not.

The measured positions show that the

2s^2 S 1 / 2 level is shifted upwards relative

to the positionEDirac(n=2,j=1/2)

and is therefore not degenerate with

the 2p^2 P 1 / 2 level. Such a shift occurs

for all the s-electrons (but the size of

the energy shift decreases with increas-

ingn). The explanation of this shift

takes us beyond relativistic quantum

mechanics into the realm of quantum

electrodynamics (QED)—the quantum

field theory that describes electromag-

netic interactions.

(a) (b) (c)

Fig. 2.8The conservation of total an-

gular momentum in electric dipole tran-

sitions that gives the selection rule in

eqn 2.59 can be represented as vector

addition. The photon has one unit of

angular momentum, and so to go from

levelj 1 toj 2 the vectors must form a

triangle, as shown for the case of (a)

j 1 =1/2toj 2 =1/2, (b)j 1 =1/2to

j 2 =3/2and(c)j 1 =3/2toj 2 =3/2.

surement of a large frequency. Nowadays this can be achieved by laser
spectroscopy (Chapter 8) but the near degeneracy of the twoj=1/ 2
levels withn = 2 was crucial in Lamb’s experiment.^58 Another im-^58 Higher shells have smaller shifts be-
portant feature in that experiment was the metastability of the 2 S 1 / 2 tween thej=1/2 levels.
level, whose lifetime was given in Section 2.2.3. That level decays∼ 108
times more slowly than that of 2 P 1 / 2 .Inanatomicbeamofhydro-
gen (at room temperature) the atoms have typical velocities of about
3000 m s−^1 and atoms excited into the 2p configuration travel an aver-
age distance of only 5× 10 −^6 m before decaying with the emission of
Lyman-αradiation. In contrast, metastable atoms travel the full length
of the apparatus (1 m) and are de-excited when they collide with a
detector (or the wall of the vacuum chamber). Hydrogen, and hydro-
genic systems, are still used for experimental tests of fundamental theory
because their simplicity allows very precise predictions.

2.3.5 Transitions between fine-structure levels

Transitions in hydrogen between the fine-structure levels with principal
quantum numbersn= 2 and 3 give the components of the Balmer-α
line shown in Fig. 2.7; in order of increasing energy, the seven allowed