1000 Solved Problems in Modern Physics

234 3 Quantum Mechanics – II

Nowψican be expanded as a sum of partial waves

ψi=eikrcosθ=

∑∞

l= 0

Aljl(kr)pl(cosθ)(8)

wherejl(kr) are the spherical Bessel functions andpl(cosθ) are the Legen-

dre polynomials of degreel.Forr→∞,jl(kr)≈kr^1 sin

(

kr−π 2 l

)

.TheAl

are some constants which can be evaluated as follows.

Multiply both sides of (8) byPl(cosθ)sinθdθand integrate. Put cosθ=t

Aljl(kr)2/(2l+1)=

∫+ 1

− 1

eikrtpl(t)(d)t

where we have used the orthonormal property of Legendre polynomials.

Integrating the RHS by parts

(1/ikr)

[

eikrtpl(t)

]+ 1

− 1 −(1/ikr)

∫

eikrtpl′(t)dt

where prime (′) means differentiation with respect to t. The second term is of

the order of 1/r^2 which can be neglected. Therefore

[

2

2 l+ 1

]

Aljl(kr)≈

(

1

ikr

)

[

eikr−e−ikr(−1)l

]

(9)

where we have usedpl(1)=1 andpl(−1)=(−1)l

Also, using the identity

eiπl/^2 =il (10)

(9) becomes

[

2

2 l+ 1

]

Aljl(kr)≈

[

2 il

kr

][

ei(kr−

πl

2 )−e−i(kr−

πl

2 )

]

2 i

=

2 ilsin

(

kr−π 2 l

)

kr

Thus

Aljl(kr)=

(2l+1)ilsin

(

kr−π 2 l

)

kr

(11)

Similarly, we can expand the total wave function into components

ψ(r,θ)=

∑∞

l= 0 BlRl(r)pl(cosθ)

=

∑

r→∞

(

Bk

kr

)

sin

(

kr−

πl

2

+δl

)

pl(cosθ)

where Blare arbitrary coefficients andδlis the phase-shift of thelth wave.

From (6)

f(θ)=re−ikr

[∑

Bl

(

1

kr

sin

(

kr−

πl

2

+δl

)

pl(cosθ)

)

−

Σil(2l+1)

kr

sin

(

kr−

πl

2

)

pl(cosθ)

]