The Rotational
Eigenfunction
The squared kinetic moment operator L² = Lx² + Ly² + Lz² can be written in the following way in polar coordinates:
L² = -
|
And here are components Lx, Ly, Lz in polar coordinates:
Lx = − Ly = Lz = |
And now we are going to find eigenfunctions of:
L² Yl,m
= l (l + 1) Lz Yl,m
= m |
The eigenfunction of kinetic moment z-component:
Lz Yl,m(J,j)
= h/i∂Yl,m(J,j)/¶j
= m h Yl,m(J,j)
we can devide variables:
Yl,m(J,j) = | P(J) | · | φ(j) |
function of J | function of j |
The solution φ(j) =
C eimj can be readily checked by
substituting h/i¶f/¶j = mh φ
to the ODE. The normalization gives us the C value:
C : normalization constant | { | oò2πC e-imj C eimj = 1 |
oò2π C² dj = 1 → C = 1/(2π)½ |
Since j changes from 0 to 2π we will have because of the unambiguity:
Yl,m(J,j) = Yl,m(J,j+2π) Þ f(j) = φ(j +2π):
C eimj = C eim(j+2p)
This is true only for integer m values:
m = 0, ±1, ±2, ... ±l |
f(j) = 1/(2π)½ eimj |
The component Lz can not be higher than kinetic moment
L that's why the maximum possible value is m = ±l.
The kinetic moment can have 2l+1
possible projections relative to corresponding axis. Since the rotational energy
doesn't have term with quantum number l but m the energy levels
are
(2l+1)-fold degenerated.
The function doesn't have J - dependence also. For m = 0 they are called the Legendre Polynomials; for m ¹ 0 - the conjugated Legendre Polynomial (Plm (cosJ)). The total function Yl,m(J,j) is the Ball area function.
Applying our operators L+ ,L− to the corresponding functions of J one can easily calculate. Here we will write down these two operators in polar coordinates:
L−
= − L+
= |
From the condition L−Yl,m = 0 we obtain for mmin = −l of the ball area function Yl,-l :
L−Yl,-l
= −h e−ij[∂/¶J-
i cotJ¶/¶j]Pl,-l e-ilj/(2π)½
= −h/(2π)½ e−ij
e−ilj[∂/¶J-
l cotJ]Pl,-l
= 0
∂P(J)/¶J = l · cotJ· P(J)
Solution: Pl,-l(J) = C · (sinJ)l
C is the normalization constant again.
If now we apply L+ consequently to Yl,m ,
L+Yl,m = ei(m+1)j [∂/¶J- m cotJ] Pl,m
ei(m+1)j ºfm+1
then it's possible to construct the kinetic moment eigenfunctions Pl,−l+1,
Pl,−l+2, ... Pl,l
consequently. We can obtain the normalization constants by integration of
corresponding volumes dΩ:
oò2π | oòp | |Yl,m|² | sinJ dJ dj | = 1 |
j | J | |¾¾¯¾¾| | ||
dΩ |
The results are summarized in chapter the rotation
and kinetic moment² and the corresponding normalized ball area functions are
represented for l = 0, 1, 2, 3, 4 there.