- #1

Davidllerenav

- 424

- 14

- Homework Statement
- The spin-dependant Hamiltonian of an electron-positron system in the presence of a uniform magnetic field in the z-direction ##(\vec{B}=B\vec{k})## can be written as

##\hat{H}=\lambda \hat{\vec{S}}_1\cdot\hat{\vec{S}}_2+\left(\frac{eB}{mc}\right)(\hat{S}_{1_z}-\hat{S}_{2_z})##

##\lambda## is a constant and ##\hat{\vec{S}}_1## and ##\hat{\vec{S}}_2## are the Spin operators of the electro and positron respectively. Find the energy eigenvalues and eigenvectors by diagonalizing the Hamiltonian

- Relevant Equations
- ##\hat{\vec{S}}=\hat{\vec{S}}_1+\hat{\vec{S}}_2##

##s_1=1/2##

##s_2=1/2##

##m_1=\pm 1/2##

##m_2=\pm 1/2##

##m=m_1+m_2##

##s=0,1##

What I did was first noting that ##\hat{\vec{S}}_1\cdot\hat{\vec{S}}_2=\frac{1}{2}(\hat{\vec{S}}^2-\hat{\vec{S}}_1^2-\hat{\vec{S}}_2^2)##, but these operators don't commute with ##\hat{S}_{1_z}## and ##\hat{S}_{2_z}##, this non the decoupled basis ##\ket{s_1,s_2;m_1,m_2}## nor the coupled one ##\ket{s,m}## are eigenfunctions of this Hamiltonian. So I need to find the eigenvectors.

I tried to find the matrix of the Hamiltonian, but I'm confused about how should I write the components of the matrix, since I've only done it when I have a given ##s##, but here I have ##s=0,1##. All I know is that the matrix must be a ##4\times 4## one.

I tried to find the matrix of the Hamiltonian, but I'm confused about how should I write the components of the matrix, since I've only done it when I have a given ##s##, but here I have ##s=0,1##. All I know is that the matrix must be a ##4\times 4## one.