I Solving 2 Electron Spin 1/2 System: Need Help!

[Noora Alameri]

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 <S_z> , <S_x> and <S_y> 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 S_z operator for example is 2x2 matrix , and the density matrix is 4x4 ! I can't 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

 

[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.

 

[Noora Alameri]

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.
Appreciate your helpful reply ()

 

[Noora Alameri]

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]

Yes,
$$ \sigma_z \otimes \sigma_0=\begin{pmatrix} \sigma_z & 0 \\ 0 & \sigma_z\end{pmatrix}$$. I am too lazy to write this explicitly as a 4x4 matrix.

 

[Truecrimson]

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.