1000 Solved Problems in Modern Physics

(Tina Meador) #1

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)

]
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