0198506961.pdf

38 The hydrogen atom

where the spin–orbit constantβis (from eqns 2.51 and 2.23)

β=

^2

2 m^2 ec^2

e^2

4 π 0

1

(na 0 )^3 l

(

l+^12

)

(l+1)

. (2.55)

A single electron hass=^12 so, for eachl, its total angular momentum

quantum numberjhas two possible values:

j=l+

1

2

or l−

1

2

.

From eqn 2.54 we find that the energy interval between these levels,

∆Es−o=Ej=l+^12 −Ej=l−^12 ,is

∆Es−o=β

(

l+^12

)

=

α^2 hcR∞

n^3 l(l+1)

. (2.56)

(^50) As shown in Section 1.9,meαca 0 = Or, expressed in terms of the gross energyE(n) in eqn 1.10, 50

andhcR∞=(e^2 / 4 π 0 )/(2a 0 ).
∆Es−o=
α^2
nl(l+1)
E(n). (2.57)
This agrees with the qualitative discussion in Section 1.4, where we
showed that relativistic effects cause energy changes of orderα^2 times
the gross structure. The more complete expression above shows that the
energy intervals between levels decrease asnandlincrease. The largest
interval in hydrogen occurs forn=2andl= 1; for this configuration
the spin–orbit interaction leads to levels withj =1/2andj =3/2.
The full designation of these levels is 2p^2 P 1 / 2 and 2p^2 P 3 / 2 , in the no-
tation that will be introduced for theLS-coupling scheme. But some of
the quantum numbers (defined in Chapter 5) are superfluous for atoms
with a single valence electron and a convenient short form is to denote
these two levels by 2 P 1 / 2 and 2 P 3 / 2 ; these correspond tonPj,where
P represents the (total) orbital angular momentum for this case. (The
capital letters are consistent with later usage.) Similarly, we may write
2S 1 / 2 for the 2s^2 S 1 / 2 level; 3 D 3 / 2 and 3 D 5 / 2 for thej=3/2and5/ 2
(^51) Another short form found in the lit- levels, respectively, that arise from the 3d configuration. (^51) But the full
erature is 2^2 P 1 / 2 and 2^2 P 3 / 2. notation must be used whenever ambiguity might arise.

2.3.3 The fine structure of hydrogen

As an example of fine structure, we look in detail at the levels that arise

from then=2andn= 3 shells of hydrogen. Equation 2.54 predicts

that, for the 2p configuration, the fine-structure levels have energies of

Es−o

(

2P 1 / 2

)

=−β2p,

Es−o

(

2P 3 / 2

)

=^12 β2p,

as shown in Fig. 2.5(a). For the 3d configuration

Es−o

(

3D 3 / 2

)

=−

3

2

β3d,

Es−o

(

3D 5 / 2

)

=β3d,