QuantumPhysics.dvi

For an electron we haveq 1 =−e, and for a nucleus of withZprotons we haveq 2 =Ze. In

addition, in cgs units (which is what we have been using here), the combination

α=

e^2

̄hc

(5.110)

is dimensionless and is referred to as thefine structure constant, with value approximately

given byα= 1/ 137 .04. Thus, the bound state energy levels of this system are given by,

En=−

1

2

mc^2 α^2 Z^2 (5.111)

which is equivalent to the standard formula.

The solutionsgℓ, 2 n(x) forninteger areassociated Laguerre polynomials, denotedLpq−p(x),

and given in terms of Laguerre polynomialsLq(x) by,

gℓ, 2 n(x) = L^2 nℓ−+1ℓ− 1 (x)

Lpq−p(x) = (−1)p

dp

dxp

Lq(x)

Lq(x) = ex

dq

dxq

(

xqe−x

)

(5.112)

The spherical harmonics, which are the final ingredient of the full wave functions of the

Coulomb problem, will be derived explicitly in the chapter on angular momentum.

5.8.2 Scattering spectrum

The equation is now,

x^2 fℓ,ε′′ −ℓ(ℓ+ 1)fℓ,ε−εxfℓ,ε+x^2 fℓ,ε= 0 (5.113)

The asymptotic behavior of the solutions asx→ ∞is given by an oscillatory exponential

e±ix, as is expected for a scattering problem. Since the differential equation is real, its two

independent solutions may be taken to be complex conjugates of one another, so we shall

choose the asymptotic behavior to bee−ix. Extracting also the familiar factor ofxℓ+1for

regularity atx= 0, we have

fℓ,ε(x) =gℓ,ε(x)xℓ+1eix (5.114)

so that the equation becomes,

xg′′ℓ,ε+ (2ℓ+ 2− 2 ix)gℓ,ε′ −(2i(ℓ+ 1) +ε)gℓ,ε= 0 (5.115)