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Advanced Physics Homework Help
Given HamiltonianFind eigenvalues and eigenfunctions
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[QUOTE="tasos, post: 4913820, member: 524875"] [h2]Homework Statement [/h2] 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 [B][U]allowed energies[/U][/B] and [B][U]eigenfunctions[/U][/B] of the system. [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] [/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. [/QUOTE]
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Given HamiltonianFind eigenvalues and eigenfunctions
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