# The dot or cross product of two operators acting on a state

1. May 21, 2014

### Robert_G

If a system is made up by two subsystems, for example, the atom and the photon. and let's assume the state of the atoms is described by $|\phi\rangle$, while the state of the photons can be described by $|n\rangle$, The Kronecker product of the $|\phi\rangle$ and $|n\rangle$ can be used to describe the whole system. and that would be:

$|\Psi\rangle=|\phi, n\rangle=|\phi\rangle\otimes|n\rangle$

I always treat the $|\phi\rangle$ and $|n\rangle$ as vectors, so the operation of $\otimes$ means the elements of the first vector (here $|\phi\rangle$) times the "whole" following vector which is $|n\rangle$ here; that will gives us a vector which is $|\phi, n\rangle$. so if the numbers of the elements of $|\phi\rangle$ and $|n\rangle$ is $m$ and $n$ respectively, the vector $|\phi, n\rangle$ has $m\times n$ elements.

now for example, we have two operators, $\hat{\mathbf{A}}$ and $\hat{\mathbf{N}}$, and they satisfy the following equations:
$\hat{\mathbf{A}}|\phi\rangle=\mathbf{a}|\phi'\rangle$
$\hat{\mathbf{N}}|n\rangle=\mathbf{n}|n'\rangle$.

Of course, $\hat{\mathbf{A}}$ can only act on the atomic states, and $\hat{\mathbf{N}}$ can only act on the photons states.

Now, my question, what is $\hat{\mathbf{A}}\cdot \hat{\mathbf{N}}|\phi, n\rangle$, and what is $\hat{\mathbf{A}}\times \hat{\mathbf{N}}|\phi, n\rangle$? The idea just not clear to me, if the operation $\otimes$ is involved.
if $\hat{U}=\hat{\mathbf{A}}\cdot \hat{\mathbf{N}}$, for example, how to write $\langle \phi, n|\;|U|^2\; |\phi', n'\rangle$ on the base of $|\phi\rangle$ and $|n\rangle$?

2. May 21, 2014

### strangerep

You need the tensor product of operators. (See the section "Tensor product of linear maps".)

3. May 21, 2014

### Robert_G

are you talking about a section of a book?

4. May 22, 2014

### micromass

Staff Emeritus