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

[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?

 

[strangerep]

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

 

[Robert_G]

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

are you talking about a section of a book?

 

[micromass]

are you talking about a section of a book?

He's talking about this http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_linear_maps