Given HamiltonianFind eigenvalues and eigenfunctions

[tasos]

Homework Statement

We have the hamiltonian H = al^2 +b(l_x +l_y +l_z)
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:
H = a\mathcal{L} ^2 +\frac {3}{4} \frac {b^2}{a}

Where \mathcal{L} ^2 here is the new operator "of angular momentum" with components :
\mathcal{L} ^2=(\mathcal{L} _x +\mathcal{L} _y +\mathcal{L} _z)
\mathcal{L} _x=(l_x + \frac {b}{2a}), \mathcal{L} _x=(l_x + \frac {b}{2a}), \mathcal{L} _x=(l_x + \frac {b}{2a})

I calculated all the commutators of (\mathcal{L}^2_x),(\mathcal{L}_x),(\mathcal{L}_y),(\mathcal{L}_z),(\mathcal{L}_+),(\mathcal{L}_-)
and i found the same results from angular momentum theory.

So i assumed that the eigenvalues here are ħl(l+1)+ \frac {3}{4} \frac {b^2}{a}
from the eigenvalues equation Hf = λf

and since we have the same theory for "Big L" of angular momentum. We have the same eigenvalues
for (\mathcal{L}^2 , \mathcal{L}_z)≡ (ħl(l+1), ħm
And about the eigenfunctions we have the spherical harmonics Y_l^m
Is this corrrect or i lost on the way?

Thnx in adv.

 

[vela]

You might want to test out your idea for the $l=1$ case (referring to $\hat{L}^2$ in the original Hamiltonian). It should be much work to write down the 3x3 matrix representing $\hat{H}$ explicitly and diagonalize it. Then you can see if you get the results you expect.

 

[tasos]

new calculations showed that [\mathcal{L}^2,\mathcal{L}_+]\neq0 does not commute, so i don't see how this process can help.

Vela i don't understand.

Whats the method you are saying i must follow.

 

[vela]

I didn't suggest a method to solve the problem. I said you should try a relatively concrete example to see if your results panned out since you were wondering if your logic was valid. If you were to look at the case $l=1$, you can use the 3x3 matrices that represent $\hat{L}^2, \hat{L}_x, \hat{L}_y, \hat{L}_z$ in the usual basis and calculate the matrix representing $\hat{H}$ explicitly. Then you could simply find the eigenvalues of that matrix and compare them to what you think they should have been based on your other calculation. If they didn't match, you know you made a mistake somewhere.

I haven't thought about the problem, much less worked it out, so take this with a grain of salt. I noticed we can express the Hamiltonian as
$$\hat{H} = a\hat{L}^2 + b\vec{n}\cdot\vec{L}$$ where $\vec{n} = (1, 1, 1).$ The Hamiltonian appears to pick out $\vec{n}$ as a preferred direction, so perhaps reorienting the coordinate system to point along that direction would be fruitful.

 

[tasos]

if we use the Hamitlonian operator on the YLM basis we get
\hat{H}|Y_l^m> = \dfrac{b}{a}(1-i)l_+ + \dfrac{b}{2}(1+i)l_- + al_z(l_z + \dfrac{b}{a} )|Y_l^m> =
c_-\dfrac{b}{a}(1-i) Y_l^(m+1) +c_+\dfrac{b}{a}(1+i) Y_l^(m-1) + (a\hbar m^2 + b\hbar m)Y_l^m

So i sould demand that the Y_l^(m-1), Y_l^(m+1) must be zero so that the hamiltonian operator gives us exact energies and the exact eigenvalues of angular momentum.

 

[tasos]

Also I've seen a lot of exercises with angular momentum that we use\<a href="http://latex.wikia.com/wiki/Hat_(LaTeX_symbol)" target="_blank" class="link link--external" rel="nofollow ugc noopener">hat{n</a>}l
But in general case with no given value of l i can't see how i can work this out.