Diagonalizing of Hamiltonian of electron and positron system

[Davidllerenav]

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.

 

[TSny]

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.

You can choose any convenient set of basis states for the system. For example, you could choose the four direct-product states

$\ket{1} = \ket{\uparrow}_e \ket{\uparrow}_p \equiv \ket{\uparrow \uparrow}$
$\ket{2} = \ket{\downarrow}_e \ket{\uparrow}_p \equiv \ket{\downarrow \uparrow}$
$\ket{3} = \ket{\uparrow}_e \ket{\downarrow}_p \equiv \ket{\uparrow \downarrow}$
$\ket{4} = \ket{\downarrow}_e \ket{\downarrow}_p \equiv \ket{\downarrow \downarrow}$

The arrows denote spin up or down along the z-axis.

Can you find the matrix elements of $H$ in this basis, such as $H_{32} = \bra{3} H \ket{2}$?

Note, for example, that $$\hat S_{1_y} \hat S_{2_y} \ket{2} = \hat S_{e_y} \hat S_{p_y}\ket{2} = \hat S_{e_y}\hat S_{p_y}\ket{\downarrow \uparrow} \equiv \left( \hat S_{e_y} \ket{\downarrow}_e \right) \left( \hat S_{p_y} \ket{\uparrow}_p \right) = \left(- i \frac{\hbar}{2}\ket{\uparrow}_e \right) \left(i \frac{\hbar}{2} \ket{\downarrow}_p \right) = \frac{\hbar^2}{4}\ket{\uparrow \downarrow} = \frac{\hbar^2}{4}\ket{3} $$