QuantumPhysics.dvi

symmetry of the problem then allows one to decouple the system into partial waves, producing
conceptual and practical simplifications.

We consider a Hamiltonian of the form, expressed in coordinate representation,

H=−

̄h^2

2 m

∆ +V(r) (12.65)

We assume thatV(r)→0 asr→∞. In order to avoid trailing factors of ̄handm, we express the
Schr ̈odinger equation in terms of the following variables instead,

U(r) =

2 m

̄h^2

V(r) E=

̄h^2 k^2

2 m

(12.66)

wherekis a real wave number variable. Spherical symmetry now implies thatL^2 andLzcommute
withH, so that a basis of eigenfunctionsψℓm(r) ofHmay be chosen in a basis whereL^2 andLz
are diagonal,

ψℓm(r) =Rℓ(r)Yℓm(θ,φ) (12.67)

The Schr ̈odinger equation then becomes,

R′′ℓ(r) +

2

r

R′ℓ(r)−

ℓ(ℓ+ 1)

r^2

Rℓ(r) +k^2 Rℓ(r)−U(r)Rℓ(r) = 0 (12.68)

It is this equation that we seek to solve perturbatively in powers ofU.

12.10.1Bessel Functions

ForU = 0, (12.68) is the differential equation that defined (spherical) Bessel functions. The two
linearly independent solutions of this second order differential equation are denoted by^12

R(1)ℓ (r) =jℓ(kr) regular asr→ 0

R(2)ℓ (r) =hℓ(kr) singular asr→ 0 (12.69)

These functions admit convenient integral representations, and we have

jℓ(ρ) =

ρℓ

2 ℓ+1ℓ!

∫ 1

− 1

dz eiρz(1−z^2 )ℓ

hℓ(ρ) = −

ρℓ

2 ℓℓ!

∫1+i∞

1

dz eiρz(1−z^2 )ℓ (12.70)

(^12) Note that ifjℓis a solution, then so isj− 1 −ℓ. Forℓnot an integer, these solutions are linearly independent,
but for integerℓthese two solutions are proportional to one another, however, and cannot be used as a basis
for both linearly independent solutions.