QuantumPhysics.dvi

The operatorL^2 may be expressed solely in terms of the coordinatesθandφ, but we shall not

need its explicit form here. Instead, we specialize to a definite eigenstate ofL^2 , labeled by

its eigenvalue ̄h^2 ℓ(ℓ+ 1), withℓ= 0, 1 , 2 ,···, corresponding tos,p,d-waves etc respectively.

The remaining equation for the radial wave functionφℓ,E(r) takes the form,

−

̄h^2

2 m

(

φ′′ℓ,E+

2

r

φ′ℓ,E−

ℓ(ℓ+ 1)

r^2

φℓ,E

)

+

q 1 q 2

r

φℓ,E=Eφℓ,E (5.102)

It is always a good idea to extract all dimensionful quantities, and leave a differential equation

in terms of dimensionless variables. To do so, we rescalerby

r=λx λ^2 =±

̄h^2

2 mE

(5.103)

where we have the + sign for scattering states withE >0, and−for bound states with

E <0. The equation is then governed by a single dimensionless parameterfor fixedℓ,

ε=

2 mq 1 q 2 λ

̄h^2

(5.104)

and takes the final form in terms of the reduced radial wave functionfℓ,ε(x) =φℓ,E(r)/r,

given as follows,

x^2 fℓ,ε′′ −ℓ(ℓ+ 1)fℓ,ε−εxfℓ,ε±x^2 fℓ,ε= 0 (5.105)

5.8.1 Bound state spectrum

For the bound state problem, the sign is−, and solutions that are normalizable at∞must

behave ase−xand at zero likexℓ+1. Extracting this behavior then leaves,

fℓ,ε(x) =gℓ,ε(x)xℓ+1e−x (5.106)

wheregℓ,ε(x) now satisfies the equation,

xgℓ,ε′′ + (2ℓ+ 2− 2 x)g′ℓ,ε−(2 + 2ℓ−ε)gℓ,ε= 0 (5.107)

The solutions leading to normalizable wave functions are polynomial in x. To admit a

polynomial solution of degreeνrequires the relation

ε= 2(ℓ+ 1 +ν)≡ 2 n (5.108)

We find the well-known result that the spectrum of bound states ofthe Coulomb problem is

in fact independent ofℓ, and depends only on the principal quantum numbern, as defined

above. Substituting these values into the above formulas to obtainthe energy, we find,

En=−

1

2

mc^2

1

n^2

q^21 q 22

̄h^2 c^2

(5.109)