Exercises for Chapter 3 59

otic particles bound to a helium nucleus (and no

electrons). [Hint.It is not necessary to specify the

values of total spin associated with the levels.]

(3.5)The integrals in helium
(a) Show that the integral in eqn 3.24 gives the
value stated in eqn 3.7.
(b) Estimate the ground-state energy of helium us-
ing the variational principle. (The details of
this technique are not given in this book; see
the section on further reading.)

(3.6)Calculation of integrals for the1s2pconfiguration
(a) Draw a scale diagram of RZ1s=2(r),RZ2s=1(r)
andRZ2p=1(r). (See Table 2.2.)
(b) Calculate the direct integral in eqn 3.31 and
show that it gives

J1s2p=−

e^2 / 4 π 0

2 a 0

13

2 × 55

.

Give the numerical value in eV (cf. that given

in the text).

(3.7)Expansion of 1 /r 12
Forr 1 <r 2 the binomial expansion of
1
r 12

(

r 12 +r^22 − 2 r 1 r 2 cosθ 12

)− 1 / 2

is

1

r 12

=^1

r 2

{

1 − 2 r^1

r 2

cosθ 12 +

(

r 1

r 2

) 2 }−^1 /^2



1

r 2

{

1+

r 1

r 2

cosθ 12 +...

}

. (3.36)

(Whenr 1 >r 2 we must interchanger 1 andr 2 to ob-

tain convergence.) The cosine of the angle between

r 1 andr 2 is

cosθ 12 =̂r 1 ·̂r 2

=cosθ 1 cosθ 2 +sinθ 1 sinθ 2 cos (φ 1 −φ 2 ).

(a) Show that the first two terms in the binomial

expansion agree with the terms withk=0and

1 in eqn 3.30.

(b) The repulsion between a 1s- and annl-electron

is independent ofm. Explain why, physically

or mathematically.

(c) Show that eqn 3.32 leads to eqn 3.34 forl=1.

(d) For a 1snlconfiguration, the quantityK(r 1 ,r 2 )

in eqn 3.34 is proportional tor 1 l/rl 2 +1 when

r 1 <r 2. Explain this in terms of mathemat-

ical properties of theYl,mfunctions.

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