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tasos
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Homework Statement
We have the hamiltonian [tex]H = al^2 +b(l_x +l_y +l_z)[/tex]
where a,b are constants.
and we must find the allowed energies and eigenfunctions of the system.
Homework Equations
The Attempt at a Solution
[/B]
I tried to complete the square on the given hamiltonian and the result is:
[tex]H = a\mathcal{L} ^2 +\frac {3}{4} \frac {b^2}{a}[/tex]
Where [tex]\mathcal{L} ^2[/tex] here is the new operator "of angular momentum" with components :
[tex]\mathcal{L} ^2=(\mathcal{L} _x +\mathcal{L} _y +\mathcal{L} _z)[/tex]
[tex]\mathcal{L} _x=(l_x + \frac {b}{2a}), \mathcal{L} _x=(l_x + \frac {b}{2a}), \mathcal{L} _x=(l_x + \frac {b}{2a})[/tex]
I calculated all the commutators of [tex] (\mathcal{L}^2_x),(\mathcal{L}_x),(\mathcal{L}_y),(\mathcal{L}_z),(\mathcal{L}_+),(\mathcal{L}_-) [/tex]
and i found the same results from angular momentum theory.
So i assumed that the eigenvalues here are [tex] ħl(l+1)+ \frac {3}{4} \frac {b^2}{a} [/tex]
from the eigenvalues equation [tex] Hf = λf[/tex]
and since we have the same theory for "Big L" of angular momentum. We have the same eigenvalues
for [tex] (\mathcal{L}^2 , \mathcal{L}_z)≡ (ħl(l+1), ħm [/tex]
And about the eigenfunctions we have the spherical harmonics [tex]Y_l^m[/tex]
Is this corrrect or i lost on the way?
Thnx in adv.
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