Eigenfunction for rotational wavefunction

[Knot Head]

Homework Statement

Show that the rotational wavefunction 3cos2? -1 is an eigenfunction of the
Hamiltonian for a three dimensional rigid rotor. Determine the corresponding eigenvalue.

Homework Equations

the eigenstates are |l,m>
the quantum number of the total angular momentum is l
the quantum number along the z-axis is m
L^2|l,m>=l(l+1)|l,m>, where l=0,1/2,1,3/2...
L_z|l,m>=m|l,m>,where m=-l,-(l,-1),...,l-1,l
-l$$\leq$$m$$\leq$$l

The Attempt at a Solution

H should=L^2/(2I) but not in 3D, so I need to put the position and momentum operators into components: L$$_{x}$$=YP$$_{z}$$-ZP$$_{y}$$
L$$_{y}$$=ZP$$_{x}$$-XP$$_{z}$$
L$$_{z}$$=XP$$_{y}$$-YP$$_{x}$$
and {L$$^{2}$$,L$$_{x,y,z}$$=0
if we call the eigenstates |$$\alpha$$,$$\beta$$>
the eigenvalue of L$$^{2}$$is L$$^{2}$$|$$\alpha,\beta$$>=$$\hbar^{2}$$$$\alpha$$,$$\beta$$>=$$\hbar^{2}$$$$\alpha$$

 

[NateD]

(\alpha+1)the eigenvalue of L_z is L_{z}|\alpha,\beta>=\hbar\beta|\alpha,\beta>if we call the wavefunction 3cos2?-1, then it is an eigenfunction of the Hamiltonian for a three dimensional rigid rotor with eigenvalue \hbar^{2}\alpha(\alpha+1)