Can a 4x4 matrix act on a 2x2 matrix in a specific way?

[Replusz]

TL;DR

So what I don't quite understand, is how direct products in Quantum mechanics can be acted upon with operators. Specifically spin up and spin down states, which I believe are (1,0) and (0,1) vectors.
Now when we have an H atom, we have a proton and electron, so the wavefunction now is the direct product of the spin states of electron and proton. Assuming these are (0,1) and (0,1), so both are in down state, what does the direct product look like? (0,1,0,1)?

And when we act on such a direct product with the sigma (Pauli) matrices, and sigma+ and sigma-, we act on the individually, is that right?
Thank you!

PS. this is NOT homework help, term hasnt even started and this is a past question sheet. Also, I have answers, but they are brief and incorrect.

THANK YOU! :)

 

[jambaugh]

The direct (tensor) product of spaces is acted upon by the direct (tensor) product of operator algebras. An operator A acting only on, say, the first factor vector would then be appended with the identity of the other factor algebra.
A \mapsto A \otimes \boldsymbol{1}
Where A\psi = \phi then ( A\otimes \boldsymbol{1} )\psi\otimes \xi =\phi\otimes\xi.

 

[A. Neumaier]

how direct products in Quantum mechanics can be acted upon with operators. Specifically spin up and spin down states, which I believe are (1,0) and (0,1) vectors.

The simplest way to look at the tensor product of two Hilbert spaces of n-dimensional vectors representing two systems 1 and 2 is as a space of $n\times n$ matrices. n=2 for two spins. The general pure state is such a matrix $\psi$, and the inner product is $\langle\phi|\psi\rangle=Tr~\phi^*\psi$, where the star denotes conjugate transpose.

The tensor product $\psi:=\psi_1\otimes \psi_2$ of two single spin states $\psi_1$ and $\psi_2$ is the outer product matrix $\psi=\psi_1\psi_2^T$. For example, if $\psi_1=|up\rangle={1 \choose 0}$ and $\psi_2=|down\rangle={0 \choose 1}$ then $\psi:=\psi_1\otimes \psi_2=\pmatrix{0 & 1 \cr 0 & 0}$.

The operators of system 1 act by multiplication on the left, those of system 2 act by multiplication with the transpose on the right. The tensor product $A_1\otimes A_2$ thus maps $\psi$ to $A_1\psi A_2^T$. It is easily checked that it therefore maps $\psi_1\otimes \psi_2$ to $A_1\psi_1\otimes A_2\psi_2$, which is the usual definition.

 

[Replusz]

Ah okay! Thank you.
But how can I act with a 4x4 matrix on a 2x2 matrix? As in the problem

 

[A. Neumaier]

But how can I act with a 4x4 matrix on a 2x2 matrix? As in the problem

In this case you need to rearrange the 4 entries of the matrix into a vector of length 4 in a way conforming to the content, usually rowwise or columnwise.