1000 Solved Problems in Modern Physics

118 2 Quantum Mechanics – I

The splitting of levels as in sodium is shown in Fig. 2.5. Transitions take place

with the selection rule

ΔM= 0 ,± 1.

2.50 Under the assumption of Russel–Saunders coupling, the ratios of the intervals
in a multiplet can be easily calculated as follows. The magnetic field produced
byLis proportional to [L(L+1)]^1 /^2 , and the component ofSin the direction
of this field is [S(S+1)]^1 /^2 cos(L,S). The energy in the magnetic field is
W=W 0 −BμB (1)
whereμBis the component of the magnetic moment in the field direction and
W 0 is the energy in the field-free case. From (1) the interaction energy is
μBB=A[L(L+1)]^1 /^2 [S(S+1)]^1 /^2 cos(L,S)(2)
whereAis a constant. From Fig. 2.6 It follows that

cos(L,S)=

J(J+1)−L(L+1)−S(S+1)

2

√

L(L+1)

√

S(S+1)

Consequently the interaction energy isA[J(J+1)−L(L+1)−S(S+1)]/ 2

AsLandSare constant for a given multiplet term, the intervals between

successive multiplet components are in the ratio of the differences of the

correspondingJ(J+1) values. Now the difference between two successive

J(J+1) values is

Fig. 2.6Russel-Saunders
coupling

(J+1)(J+2)−J(J+1) or 2(J+1)

and therefore proportional toJ+1. This is known as Lande’s interval rule.

For the calcium triplet

(J+2)/(J+1)= 60 × 10 −^4 / 30 × 10 −^4 = 2

whenceJ=0. The three levels of increasing energy haveJ= 0 ,1 and 2.

NowJ = 0 ,1 and 2 are produced from the combination ofLandS. With

the spectroscopic notation^2 S+^1 LJthe terms for the three levels are^3 P 0 ,^3 P 1

and^3 P 2.

2.51ΔE=μBB

hΔv=hcΔλ/λ^2 =μBB

B=

hcΔλ

μBλ^2

=

(6. 63 × 10 −^34 )(3× 108 )(1. 7 × 10 −^12 )

(9. 17 × 10 −^24 )(350× 10 −^9 )^2

= 0 .3T