- #1

tasos

- 9

- 0

## 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

**and**

__allowed energies__**of the system.**

__eigenfunctions__## 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.

Last edited: