42 The hydrogen atom
transitions between the levels with differentjare as follows:2P 3 / 2 −3S 1 / 2 ,
2P 3 / 2 −3D 3 / 2 ,
2P 3 / 2 −3D 5 / 2 ,
2S 1 / 2 −3P 1 / 2 ,
2P 1 / 2 −3S 1 / 2 ,
2S 1 / 2 −3P 3 / 2 ,
2P 1 / 2 −3D 3 / 2.These obey the selection rule ∆l=±1 but an additional rule prevents a
transition between 2 P 1 / 2 and 3 D 5 / 2 , namely that the change of the total
angular momentum quantum number in an electric dipole transition
obeys
∆j=0,± 1. (2.59)
This selection rule may be explained by angular momentum conserva-
tion (as mentioned in Section 2.2.2). This rule can be expressed in terms
of vector addition, as shown in Fig. 2.8; the conservation condition is
equivalent to being able to form a triangle from the three vectors rep-
resentingjof the initial state, the final state, and a unit vector for the
(one unit of) angular momentum carried by the photon. Hence, this se-
lection rule is sometimes referred to as the triangle rule. The projection
ofjalong thez-axis can change by ∆mj=0,±1. (Appendix C gives a
summary of all selection rules.)Further reading
Much of the material covered in this chapter can be found in the intro-
ductory quantum mechanics and atomic physics texts listed in the Ref-
erences. For particular topics the following are useful: Segr`e (1980) gives
an overview of the historical development, and Series (1988) reviews the
work on hydrogen, including the important Lamb shift experiment.Exercises
(2.1)Angular-momentum eigenfunctions
(a) Verify that all the eigenfunctions withl=1
are orthogonal toY 0 , 0.
(b) Verify that all the eigenfunctions withl=1
are orthogonal to those withl=2.(2.2)Angular-momentum eigenfunctions
(a) Find the eigenfunction with orbital angular
momentum quantum numberland magnetic
quantum numberm=l−1.
(b) Verify thatYl,l− 1 is orthogonal toYl− 1 ,l− 1.