- #1
Markus Kahn
- 112
- 14
Homework Statement
Consider a 2-particle system where the two particles have angular momentum operators ##\vec{L}_1## and ##\vec{L}_2## respectively. The Hamiltonian is given by
$$H = \mu\vec{B}\cdot (\vec{L}_1+\vec{L}_2)+\gamma \vec{L}_1\cdot \vec{L}_2.$$
Determine explicitly the eigenvalues and eigenvectors of ##H## when the spins are ##l_1=l_2=1##.
Homework Equations
All relevant Clebsch-Gordan coefficients are given in the form ##\langle 1,1, m_1,m_2 \vert 1,1, j,m \rangle \equiv \langle m_1,m_2 \vert j,m\rangle##.
The Attempt at a Solution
We know that any state vector in the Hilbert-space is of the form ##\vert 1,1, m_1,m_2 \rangle = \vert1,m_1\rangle \otimes \vert 1,m_2\rangle##. Since the Hilbert space is a tensor product we can rewrite it as a direct sum of irreducible representations according to the Clebsch-Gordan-series and find
$$\vert j,m \rangle \equiv\vert 1,1, j,m \rangle = \sum_{\substack{m_1,m_2\\ m=m_1+m_2}} \langle 1,1,m_1,m_2\vert 1,1, j,m\rangle \vert 1,1, m_1,m_2\rangle\equiv \sum_{\substack{m_1,m_2\\ m=m_1+m_2}} \langle m_1,m_2\vert j,m\rangle \vert m_1,m_2\rangle .$$
Now I tried to figure out how ##H## acts on ##\vert m_1, m_2\rangle##. My first problem is that I don't know how ##\vec{L}_1, \vec{L}_2## act on the state vector.. To solve this I would need to assume that ##\vec{B}=B_z \vec{e}_z## and therefore get
$$\vec{B}(\vec{L}_1+\vec{L}_2)\vert m_1,m_2\rangle = B_z (L_1^z+L_2^z)\vert m_1,m_2\rangle = B_z(m_1+m_2)\vert m_1,m_2\rangle .$$
Is this a reasonable assumption or can one solve this without any assumptions for the magnetic field?
Now we still have the second part of the Hamiltonian left. My idea here was: ##L^2 \equiv (\vec{L}_1+\vec{L}_2)^2 = L_1^2+L_2^2+2L_1L_2 ## and therefore ##L_1L_2 = \frac{1}{2} (L^2-L_1^2-L_2^2)##. Now I only need to figure out how ##L^2## acts on ##\vert m_1,m_2\rangle##. Sadly, I can't really figure that part out...
And my last problem is: even if I would figure out how ##L^2## acts on the state, I have a hard time making the last steps to the original goal of finding all eigenvalues and -vectors explicitly.EDIT: I think I'm able to answer my first question:
Is this a reasonable assumption or can one solve this without any assumptions for the magnetic field?
Since we can chose the coordinate system in which we work, we can always chose on in which the magnetic field points in the ##z##-direction. The only assumption would then be that the magnetic field is constant...
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