The Rotational Eigenfunction

The squared kinetic moment operator L² = Lx² + Ly² + Lz² can be written in the following way in polar coordinates:

L² = - h²/sin J∂/¶J(sin J¶/¶J) + 1/sin²JLz²

And here are components Lx, Ly, Lz in polar coordinates:

Lx = −h/i{sin j¶/¶J + cot J cos j¶/¶j} Ly = h/i{cos j¶/¶J-cot J sin j¶/¶j} Lz = h/i{∂/¶j}

And now we are going to find eigenfunctions of:

L² Yl,m = l (l + 1)h² Yl,m Lz Yl,m = m h Yl,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 eim j

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 (cos J)). 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− = −h e−ij[∂/¶J- i cot J¶/¶j] L+ = h eij[∂/¶J + i cot J ∂/¶j]

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 cot J¶/¶j]Pl,-l e-ilj/(2π)½ = −h/(2π)½ e−ij e−il j[∂/¶J- l cot J]Pl,-l = 0

∂P(J)/¶J = l · cot J· P(J)

Solution: Pl,-l(J) = C · (sin J)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 cot J] 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 | sin J 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.