# Density matrix

## Main Question or Discussion Point

Hey,

I am studying Spin 1/2 system, the case of 2 electrons in a magnetic field, since we have 2 electrons, we expect that the matrix will be of the size 4x4, which is what I have got.

As I know that I could use the density matrix to calculate the expectation values of any physical quantity such us <Sz> , <Sx> and <Sy> by taking the trace of the product of the density matrix with the one we want to find out.

but the point that disappointed me is that Sz operator for example is 2x2 matrix , and the density matrix is 4x4 ! I cant take a product of two matrices !

What is the best way to proceed ?

I am really struggling with this, I hope someone would help ()

Thank you

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DrDu
If you want to calculate the expectation value of e.g. $\sigma_z$ for e.g. the first spin, you have to form the tensor product $\sigma_z \otimes \sigma_0$, where $\sigma_0$ is the 2x2 unit matrix and calculate the expectation value using the density matrix and this operator.

If you want to calculate the expectation value of e.g. $\sigma_z$ for e.g. the first spin, you have to form the tensor product $\sigma_z \otimes \sigma_0$, where $\sigma_0$ is the 2x2 unit matrix and calculate the expectation value using the density matrix and this operator.
Thank You Doctor.

If you want to calculate the expectation value of e.g. $\sigma_z$ for e.g. the first spin, you have to form the tensor product $\sigma_z \otimes \sigma_0$, where $\sigma_0$ is the 2x2 unit matrix and calculate the expectation value using the density matrix and this operator.
I am wondering if there is a way to re write puali matrices in 4x4 matrix size with choosing 4 basis instead of using product ?

DrDu
$$\sigma_z \otimes \sigma_0=\begin{pmatrix} \sigma_z & 0 \\ 0 & \sigma_z\end{pmatrix}$$. I am too lazy to write this explicitly as a 4x4 matrix.
The general way to write a tensor product of $d \times d$ matrices as a $d^2 \times d^2$ matrix uses the Kronecker product.
But if you are only interested in the observables for each spin, it might be easier to just calculate the $2 \times 2$ reduced density matrices $\rho_A = \text{Tr}_B ( \rho_{AB})$ and $\rho_B = \text{Tr}_A ( \rho_{AB})$ where the trace is the partial trace.