Die Rotation

The Rotational Eigenfunction

The squared kinetic moment operator L² = L_x² + L_y² + L_z² can be written in the following way in polar coordinates:

L² = - ^h²/_sinJ^∂/_¶J(sinJ^¶/_¶J) + ¹/_sin²JL_z²

And here are components L_x, L_y, L_z in polar coordinates:

L_x = −^h/_i{sinj^¶/_¶J + cotJ cosj^¶/_¶j}

L_y = ^h/_i{cosj^¶/_¶J-cotJsinj^¶/_¶j}

L_z = ^h/_i{^∂/_¶j}

And now we are going to find eigenfunctions of:

L² Y_l,m = l (l + 1)h² Y_l,m

L_z Y_l,m = m h Y_l,m

The eigenfunction of kinetic moment z-component:

L_z Y_l,m(J,j) = ^h/_i^∂Yl,m^(J,j)/_¶j = m h Y_l,m(J,j)

we can divide variables:

Y_l,m(J,j) =	P(J)	·	φ(j)
function of J			function of j

The solution φ(j) = C e^imj 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	{	_oò^2πC e^-imj C e^imj = 1
		_oò^2π C² dj = 1 → C = ¹/_(2π)½

Since j changes from 0 to 2π we will have because of the unambiguity:

Y_l,m(J,j) = Y_l,m(J,j+2π) Þ f(j) = φ(j +2π):

C e^imj = C e^im(j+2p)

This is true only for integer m values:

m = 0, ±1, ±2, ... ±l

f(j) = ¹/_(2π)½ e^imj

The component L_z can not be higher than kinetic moment L. This is 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 a term with quantum number l but instead m the energy levels are (2l+1)-fold degenerate.

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 (P_l^m (cosJ)). The total function Y_l,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 these two operators in polar coordinates:

L₋ = −h e^−ij[^∂/_¶J- i cotJ^¶/_¶j]

L₊ = h e^ij[^∂/_¶J + i cotJ ^∂/_¶j]

From the condition L₋Y_l,m = 0 we obtain for m_min = −l of the ball area function Y_l,-l :

L₋Y_l,-l = −h e^−ij[^∂/_¶J- i cotJ^¶/_¶j]P_l,-le^-ilj_/(2π)½ = −h/_(2π)½ e^−ij e^−ilj[^∂/_¶J- l cotJ]P_l,-l = 0

^∂P(J)/_¶J = l · cotJ· P(J)

Solution: P_l,-l(J) = C · (sinJ)^l

C is the normalization constant again.

If now we apply L₊ consequently to Y_l,m ,

L₊Y_l,m = e^i(m+1)j [^∂/_¶J- m cotJ] P_l,m

e^i(m+1)j ºf_m+1

then it's possible to construct the kinetic moment eigenfunctions P_l,−l+1, P_l,−l+2, ... P_l,l. We can obtain the normalization constants by integration of corresponding volumes dΩ:

_oò^2π	_oò^p	\|Y_l,m\|²	sinJ dJ dj	= 1
j	J		^\|¾¾_¯¾¾^\|
			dΩ

The results are summarized in the chapter the rotation and kinetic moment² and the corresponding normalized ball area functions are represented for l = 0, 1, 2, 3, 4 there.

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.