1000 Solved Problems in Modern Physics

3.3 Solutions 201

The boundary condition thatu/rbe finite atr=0 demands thatb=0.

Thus,ψis proportional torl. The probability that a particle be in a spherical

shell of radiirandr+drfor smallr, is proportional tor^2 l+^2 dr. The largerl

is, the smaller is the probability that the particle be in the vicinity of the origin.

For the case of collision problems, there is a classical analogy: the larger the

orbital angular momentum the larger the impact parameter.

Thusu(r)∼rl+^1 (r→0)

For→∞, we obtain, as an approximation to differential equation (3), as

d^2 u

dr^2

−

2 μγ^2 r^2 u

^2

= 0

If we try a solution of the form,

u(r)=u 0 e−Br

(^2) / 2
the asymptotically valid solution is satisfied provided we change

B=

γ(2μ)

1

2



=

μω



Inorder to solve (3) for allr, we may first separate the asymptotic behaviour

by writing

u(r)=rl+^1 eBr

(^2) / 2
V(r)(5)
Insert (5) in (3), and dividing byrl+^1 e−Bν
(^2) / 2
We get
d^2 ν
dr^2

+

2dv

dr

[(

l+ 1

r

)

−Br

]

−Bv

[

2 l+ 3 −

2

ω

(V 0 +E)

]

DefineC=l+^32

4 A= 2 l+ 3 −

2

ω

(V 0 +E)(6)

d^2 v

dr^2

+

dv

dr

[(

2 C− 1

r

)

− 2 Br

]

− 4 ABv=0(7)

Set

Fig. 3.22The parabolic
potential of the three
dimensional harmonic
oscillator