# 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.

Homework Helper
Gold Member
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 $$\hat S_{1_y} \hat S_{2_y} \ket{2} = \hat S_{e_y} \hat S_{p_y}\ket{2} = \hat S_{e_y}\hat S_{p_y}\ket{\downarrow \uparrow} \equiv \left( \hat S_{e_y} \ket{\downarrow}_e \right) \left( \hat S_{p_y} \ket{\uparrow}_p \right) = \left(- i \frac{\hbar}{2}\ket{\uparrow}_e \right) \left(i \frac{\hbar}{2} \ket{\downarrow}_p \right) = \frac{\hbar^2}{4}\ket{\uparrow \downarrow} = \frac{\hbar^2}{4}\ket{3}$$

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